/*@jsxRuntime automatic @jsxImportSource react*/ const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = Object.assign({ p: "p", strong: "strong", h3: "h3", ul: "ul", li: "li", img: "img", pre: "pre", code: "code", h4: "h4", span: "span", math: "math", semantics: "semantics", mrow: "mrow", mi: "mi", mo: "mo", mn: "mn", annotation: "annotation", ol: "ol", em: "em", a: "a", msub: "msub", mfrac: "mfrac", msup: "msup" }, _provideComponents(), props.components), {Section, Box} = _components; if (!Box) _missingMdxReference("Box", true); if (!Section) _missingMdxReference("Section", true); return _jsxs(_Fragment, { children: [_jsx(_components.p, { children: "This post serves as a study guide. All the links for the information can be found at the end of each section." }), "\n", _jsx(_components.p, { children: _jsx(_components.strong, { children: "I do not claim it as my own." }) }), "\n", _jsx(_components.p, { children: "This page is an organization of topics." }), "\n", _jsx("br", {}), "\n", _jsxs(Section, { section: "Process Synchronization", children: [_jsx(_components.h3, { children: "Process" }), _jsx(Box, { question: "What is a process?", children: _jsx(_components.p, { children: "We can define a process as a program that is currently being executed." }) }), _jsxs(Box, { question: "What are the states of a process?", children: [_jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Ready: ready for execution" }), "\n", _jsx(_components.li, { children: "Running: executing process" }), "\n", _jsx(_components.li, { children: "Blocked: waiting or blocked until another process gives the all clear" }), "\n"] }), _jsx(_components.p, { children: _jsx(_components.img, { src: "https://media.geeksforgeeks.org/wp-content/uploads/20220114144811/state-660x365.PNG", alt: "States of a Process" }) })] }), _jsxs(Box, { question: "What are the types of concurrent unit control?", children: [_jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Physical Concurrency" }), "\n", _jsx(_components.li, { children: "Logical Concurrency" }), "\n"] }), _jsx(Box, { question: "when do they occurr?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Phyical occurs when two or more processors are running simultaneously" }), "\n", _jsx(_components.li, { children: "Logical happens when execution happens in intervals in a single processor" }), "\n"] }) })] }), _jsx(Box, { question: "How do concurrent processes interact with each other?", children: _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["shared variables: processes read and write to shared memory", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: _jsx(_components.img, { src: "https://web.mit.edu/6.005/www/fa14/classes/17-concurrency/figures/shared-memory.png", alt: "Shared Svariables" }) }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["message passing: messages are sent through communication channels\n-", _jsx(_components.img, { src: "https://web.mit.edu/6.005/www/fa14/classes/17-concurrency/figures/message-passing.png", alt: "Message passing" })] }), "\n"] }) }), _jsx(_components.h3, { children: "Critical Sections" }), _jsx(Box, { question: "What is a critical section?", children: _jsx(_components.p, { children: "Region of code that needs to be executed atomically since resources are shared" }) }), _jsx(Box, { question: "Solution to the critical section problem?", children: _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["\n", _jsx(Box, { question: "Busy Waiting", children: _jsxs(_components.p, { children: ["Technique in which a process/task waits and constantly checks for a condition to be satisfied before proceeding with its execution\n", _jsx(_components.img, { src: "https://www.baeldung.com/wp-content/uploads/sites/4/2021/08/Busywaiting.svg", alt: "Busy Waiting" })] }) }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsx(Box, { question: "Disabling Interrupts", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "The CPU will be unable to switch processes" }), "\n", _jsx(_components.li, { children: "Processes can use shared variables without another process accessing it" }), "\n"] }) }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsxs(Box, { question: "Test-and-Set Intruction", children: [_jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Shared variable is lock which is initialized to false" }), "\n", _jsx(_components.li, { children: "Algorithm returns value that was sent and sets lock to true" }), "\n"] }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-c", children: " boolean lock = false;\n\n boolean TestAndSet (boolean &target){\n boolean rv = target;\n target = true;\n return rv;\n }\n\n while(1){\n while (TestAndSet(lock));\n critical section\n lock = false;\n remainder section\n }\n" }) })] }), "\n"] }), "\n"] }) }), _jsx(_components.h3, { children: "Semaphores" }), _jsxs(Box, { question: "What is a semaphore?", children: [_jsx(_components.p, { children: "It is similar to an integer but differs in:" }), _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "value can only be increased or decreased. No reading occurs" }), "\n", _jsx(_components.li, { children: "after decreasing, if the value is negative then the thread is blocked until the semaphore is positive" }), "\n", _jsx(_components.li, { children: "once a thread is released and increases the value of the semaphore, one waiting thread gets accesss" }), "\n"] })] }), _jsxs(Box, { question: "Does the semaphore have atomic operations?", children: [_jsx(_components.p, { children: "Yes" }), _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "P(S): decrements the value" }), "\n", _jsx(_components.li, { children: "V(S): increments the value" }), "\n"] }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-c", children: "sem = Semaphore(1)\n\n// Process\nsem.P()\n // critical section\nsem.V()\n" }) })] }), _jsx(Box, { question: "Why use semaphores?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Solutions are clean" }), "\n", _jsx(_components.li, { children: "Solutions are portable and efficient" }), "\n"] }) }), _jsx(Box, { question: "Difference between locks, mutex semaphore?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "lock allows only one thread to enter the part that's locked and the lock is not shared with any other processes" }), "\n", _jsx(_components.li, { children: "mutex is the same as a lock but it can be system wide" }), "\n", _jsx(_components.li, { children: "semaphore does the same as a mutex but allows x number of threads to enter" }), "\n"] }) }), _jsx(_components.h4, { children: "Producer Consumer Problem" }), _jsx(Box, { question: "What is this Producer Consumer Problem?", children: _jsx(_components.p, { children: "Producer threads create items and place them in a buffer, meanwhile consumer threads remove them from the buffer.\nProducer threads should not be able to create items if the buffer is full, and consumers should not be able to remove items if buffer is empty." }) }), _jsxs(Box, { question: "Infinite Buffer Example", children: [_jsx(_components.p, { children: "Code can be found on page 57 of Litte Book of semaphores" }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: " buffer: number[] = [];\n mutex: semaphore = 1;\n items: semaphore = 0;\n \n Producer {\n mutex.P() // lock the mutex\n buffer.add(item)\n items.V() // increase number of items for consumer threads\n mutex.V() // release mutex\n }\n\n Consumer {\n items.P() // decrease number of items\n mutex.P() \n buffer.pop() // remove item from buffer\n mutex.V()\n }\n" }) }), _jsxs(Box, { question: "What is wrong with this implementation? How to fix it?", children: [_jsxs(Box, { question: "Producer", children: [_jsxs(_components.p, { children: ["Assume ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "u" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "mutex = 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6151em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "u" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }) }), " at the start of the producer."] }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["mutex.P() inside Producer makes ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "u" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "mutex = 0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6151em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "u" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }) })] }), "\n", _jsx(_components.li, { children: "items.V() inside Producer can trigger a consumer thread" }), "\n", _jsxs(_components.li, { children: ["items.P() inside Consumer makes ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "u" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "mutex = -1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6151em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "u" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7278em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }) })] }), "\n", _jsxs(_components.li, { children: ["mutex.V() inside Producer makes ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "u" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "mutex = 0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6151em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "u" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }) })] }), "\n"] }), _jsx(_components.p, { children: "This means we cannot enter the critical section inside the consumer. In order to proceed the semaphore value has to be greater than 0" }), _jsx(_components.p, { children: "The fix is to signal the consumer after we are out of the critical section." }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: " buffer: number[] = [];\n mutex: semaphore = 1;\n items: semaphore = 0;\n \n Producer {\n mutex.P() // lock the mutex\n buffer.add(item)\n mutex.V() // release mutex\n items.V() // increase number of items and signal consumer threads\n }\n" }) })] }), _jsxs(Box, { question: "Consumer", children: [_jsx(_components.p, { children: "Various consumers can decrement value of items before popping. This leads to incorrect count. Consider this:" }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: " Consumer {\n mutex.P() \n items.P() // decrease number of items\n buffer.pop() // remove item from buffer\n mutex.V()\n }\n\n" }) }), _jsxs(_components.p, { children: ["Assume ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "items = 0" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }) })] }), _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "The consumer thread grabs the mutex" }), "\n", _jsx(_components.li, { children: "items.P() in Consumer gets stuck since the value is not greater than 0" }), "\n", _jsx(_components.li, { children: "Producer thread cannot increment the value of items, since it's waiting on the consumer to release the mutex" }), "\n"] }), _jsx(_components.p, { children: "We have reached a deadlock." })] })] })] }), _jsxs(Box, { question: "Finite Buffer Example", children: [_jsx(_components.p, { children: "Recall that a producer cannot create new items if buffer is full. Also, the consumer cannot retrieve items if buffer is empty." }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: "const bufferSize: number = N;\nitems: semaphore = 0;\n...\nif (items >= bufferSize)\n block()\n...\n" }) }), _jsx(Box, { question: "Would this work?", children: _jsx(_components.p, { children: "No, we cannot read the value of a semaphore. Only increase and decrease." }) }), _jsxs(Box, { question: "Solution", children: [_jsx(_components.p, { children: "We can add one more semaphore and set it equal to the size of the buffer." }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: " buffer: number[bufferSize];\n mutex: semaphore = 1;\n items: semaphore = 0;\n spaces: semaphore = bufferSize;\n\n Producer {\n spaces.P(); // decrement number of available spaces\n mutex.P();\n buffer.add()\n mutex.V();\n items.V() // signal the consumer\n }\n\n Consumer {\n items.P(); // decrement number of items in buffer\n mutex.P();\n buffer.pop();\n mutex.V();\n spaces.V(); // increment spaces\n }\n" }) })] })] }), _jsx(_components.h4, { children: "Readers-Writers Problem" }), _jsx(_components.p, { children: "Little Book of Semaphores 4.2" }), _jsx(Box, { question: "What is the readers-writers problem?", children: _jsx(_components.p, { children: "Occurs when a resource is read and written by concurrent threads. Ideally, you do not want to read the resource while it is being written." }) }), _jsx(Box, { question: "What are the constraints?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Multiple readers can be in the critical section simultaneously" }), "\n", _jsx(_components.li, { children: "Writers have exclusive access to the critical section" }), "\n"] }) }), _jsx(Box, { question: "Implementation", children: _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: " isRoomEmpty: semaphore = 1;\n mutex: semaphore = 1;\n readers: number = 0;\n\n Reader {\n mutex.P();\n readers++;\n if(readers == 1) isRoomEmpty.P(); // first reader locks the room\n mutex.V();\n\n \n\n mutex.P();\n readers--;\n if(readers == 0) isRoomEmpty.V(); // last readers frees the room\n mutex.V();\n }\n\n Writer {\n isRoomEmpty.P(); // writer waits for room to be empty\n \n isRoomEmpty.V();\n }\n" }) }) }), _jsxs(Box, { question: "Starvation", children: [_jsxs(_components.p, { children: ["In the previous implementation, writers can starve. This means that readers can keep coming in and writers ", _jsx(_components.em, { children: "might" }), " wait forever."] }), _jsx(Box, { question: "Make it so that when a writer arrives, exisiting readers finish.", children: _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: " isRoomEmpty: semaphore = 1;\n mutex: semaphore = 1;\n turnstile: semaphore = 1;\n readers: number = 0;\n\n Reader {\n turnstile.P(); // decrease number\n turnstile.V(); // the final increase to reach 1 is given by the writer\n\n mutex.P();\n readers++;\n if(readers == 1) isRoomEmpty.P(); // first reader locks the room\n mutex.V();\n\n \n\n mutex.P();\n readers--;\n if(readers == 0) isRoomEmpty.V(); // last readers frees the room\n mutex.V();\n }\n\n Writer {\n turnstile.P(); // let readers know a writer came in\n isRoomEmpty.P(); // writer waits for room to be empty\n \n turnstile.V(); // notify readers\n isRoomEmpty.V(); // room is ready\n }\n" }) }) })] }), _jsxs(Box, { question: "Writer priority", children: [_jsx(_components.p, { children: "Little Book of Semaphores page 79" }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: " readerMutex: semaphore = 1;\n readers: number = 0;\n\n writerMutex: semaphore = 1;\n writers: number = 0;\n\n allowReaders: semaphore = 1;\n allowWriters: semaphore = 1;\n\n Reader {\n allowReaders.P();\n readerMutex.P();\n readers++;\n if(readers == 1) allowWriters.P(); // notify writers that readers are there\n readerMutex.V();\n allowReaders.V();\n\n \n\n readerMutex.P();\n readers--;\n if(readers == 0) allowWriters.V(); //signal writers the are no more readers\n readerMutex.V();\n }\n\n Writer {\n writerMutex.P();\n writers++;\n if (writers == 1) allowReaders.P(); // notify readers to wait\n writerMutex.V();\n\n allowWriters.P();\n \n allowWriters.V();\n\n writerMutex.P();\n writers--;\n if(writers == 0) allowReaders.V();\n writerMutex.V();\n }\n" }) })] }), _jsx(_components.h4, { children: "Dining Philosophers Problem" }), _jsx(_components.p, { children: "Little Book of Semaphores 4.4" }), _jsxs(Box, { question: "What is the Dining Philosophers Problem?", children: [_jsx(_components.p, { children: _jsx(_components.img, { src: "https://w3.cs.jmu.edu/kirkpams/OpenCSF/Books/csf/html/_images/CSF-Images.8.3.png", alt: "Dining Philosophers Problem" }) }), _jsxs(_components.p, { children: ["A philosopher needs ", _jsx(_components.em, { children: "two" }), " forks to eat. Looking at the diagram, we can tell some of them will have to wait for an adjacent philosopher to finish eating."] }), _jsx(_components.p, { children: "Each philosopher, representing a thread, runs the following loop:" }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: " while (true) {\n think();\n getForks();\n eat();\n putForks();\n }\n" }) }), _jsxs(_components.p, { children: ["Each philosopher knows their number ranging from 0 to ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }) }), ". Each fork is also numbered from 0 to ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }) }), ". A philosopher ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }) }), " grabs the left fork ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }) }), " and the right fork ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i+1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7429em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }) })] }), _jsxs(_components.p, { children: ["The problem lies in the function ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "g" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "F" }), _jsx(_components.mi, { children: "o" }), _jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "getForks()" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "g" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "tF" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "or" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }) }), " and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mi, { children: "u" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "F" }), _jsx(_components.mi, { children: "o" }), _jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "putForks()" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "mord mathnormal", children: "u" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "tF" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "or" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }) }), ". We need to implement them such that:"] }), _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Only one philosopher can hold a fork at a time" }), "\n", _jsx(_components.li, { children: "No deadlocks" }), "\n", _jsx(_components.li, { children: "No starvation" }), "\n", _jsx(_components.li, { children: "Multiple philosophers eating" }), "\n"] })] }), _jsxs(Box, { question: "Solutions", children: [_jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: " const getForks = (i: number) => {\n forks[getLeft(i)].P();\n forks[getRight(i)].P();\n }\n\n const putForks = (i: number) => {\n forks[getLeft(i)].V();\n forks[getRight(i)].V();\n }\n" }) }), _jsx(Box, { question: "Does this solution work?", children: _jsx(_components.p, { children: "No. Imagine each philosopher picks up the fork to the right. There are no more forks available, and each philosphre only has one fork. No one can eat, thus leading to a deadlock." }) }), _jsxs(Box, { question: "Can it be improved?", children: [_jsxs(_components.p, { children: ["We can ensure no deadlock as long as we leave one fork available. Say there are 5 forks, 5 philosophers, but we only allow 4 to eat. In the scenario where everyone reaches for a fork on the right, there will be one spare fork since one philosopher is not eating.\nWe can implement this with a semaphore initialized to ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n-1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }) }), ", ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }) }), " being the number of philosophers."] }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: " maxPhilosophers: semaphore = n - 1;\n\n const getForks = (i: number) => {\n maxPhilosophers.P(); // only the first n - 1 philosophers can proceed\n\n forks[getLeft(i)].P();\n forks[getRight(i)].P();\n }\n\n const putForks = (i: number) => {\n forks[getLeft(i)].V();\n forks[getRight(i)].V();\n\n maxPhilosophers.V(); // notify the waiting philopher they can eat\n }\n" }) })] })] }), _jsx(_components.h3, { children: "Monitors" }), _jsxs(Box, { question: "What is a monitor?", children: [_jsx(_components.p, { children: "Collection of variables and procedures. Kinda like a class." }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: " interface Condition {\n wait, // Process performing will be suspended and placed in the local queue\n signal, // One of the queued processes is popped\n queue // local queue of pending processes \n }\n\n Monitor Test{\n variables: any;\n myConditions: Condition;\n\n procedure pn {...}\n }\n" }) })] }), _jsx(Box, { question: "Properties of monitors", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Outside processes cannot access the monitor's variables, but can call the procedures" }), "\n", _jsx(_components.li, { children: "Only one process at a time can execute code inside the monitor" }), "\n"] }) }), _jsx(_components.h4, { children: "Producer Consumer Problem" }), _jsx(Box, { question: "Implementation", children: _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: "Monitor ProducerConsumer {\n items: number = 0; // # of items inside buffer\n full: condition;\n empty: condition;\n\n Procedure Produce {\n if (items == N) full.wait(); // can't produce since buffer is full\n // should have released the wait\n items++;\n empty.signal(); // notify buffer is NOT empty & consumer can proceed\n }\n\n Procedure Consume {\n if (items == 0) empty.wait(); // can't consume since buffer is empty\n \n items--;\n full.signal(); // notify producer buffer is no longer full\n }\n}\n" }) }) }), _jsx(_components.h4, { children: "Readers-Writers Problem" }), _jsx(Box, { question: "Readers Priority", children: _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-ts", children: "Monitor ReaderWriter {\n readers: number = 0;\n busy: boolean = false;\n reader: condition;\n writer: condition;\n\n Procedure beginRead {\n if (busy) reader.wait(); // wait until writer is done\n readers++;\n reader.signal(); // signal readers\n }\n\n Procedure endRead {\n reader--;\n if(readers == 0) writers.signal(); // notify writers there are no more readers\n }\n\n Procedure beginWrite {\n if(readers > 0 || busy) writer.wait(); // either reading or writing in process\n busy = true; // preparation\n writer.signal(); // notify writer it can proceed\n }\n\n Procedure endWrite {\n busy = false; // writer has finished\n if(reader.queue) reader.signal(); // notify reader it can proceed\n else writer.signal(); // if no readers, go to writer\n }\n}\n" }) }) }), _jsx(_components.p, { children: "Geeks for Geeks:" }), _jsx(Box, { question: "Semaphores vs Monitors", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Mutual exclusion in monitors is automatic while in semaphores, mutual exclusion needs to be implemented explicitly." }), "\n", _jsx(_components.li, { children: "Shared variables are global to all processes in the monitor while shared variables are hidden in semaphores" }), "\n", _jsx(_components.li, { children: "Semaphores permit more than one thread to access the critical section, unlike monitors." }), "\n", _jsx(_components.li, { children: "In semaphores there is no spinning, hence no waste of resources due to no busy waiting." }), "\n"] }) }), _jsx(_components.h3, { children: "References" }), _jsx(Box, { question: "Sources", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/three-state-process-model-in-operating-system/", children: "Three State Process Model in Operating System" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.baeldung.com/cs/os-busy-waiting", children: "What Does “Busy Waiting” Mean in Operating Systems?" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.tutorialspoint.com/what-is-mutual-exclusion-by-using-interrupt-disabling#:~:text=Whenever%20the%20interrupts%20are%20disabled,without%20another%20process%20accessing%20it.", children: "What is mutual exclusion by using interrupt disabling?" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/hardware-synchronization-algorithms-unlock-and-lock-test-and-set-swap/", children: "Hardware Synchronization Algorithms : Unlock and Lock, Test and Set, Swap" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://greenteapress.com/semaphores/LittleBookOfSemaphores.pdf", children: "Little Book of Semaphores" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://stackoverflow.com/questions/2332765/what-is-the-difference-between-lock-mutex-and-semaphore", children: "What is the difference between lock, mutex and semaphore?" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/monitors-in-process-synchronization/", children: "Monitors in Process Synchronization" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/monitor-vs-semaphore/", children: "Monitor vs Semaphore" }) }), "\n"] }) })] }), "\n", _jsxs(Section, { section: "Process Communication", children: [_jsx(_components.p, { children: "Distributed Systems 3rd edition (2017) Chapter 4" }), _jsx(_components.h3, { children: "Message Passing Model" }), _jsx(Box, { question: "Why do Distributed Systems use message passing?", children: _jsx(_components.p, { children: "Distributed Systems send and receive low level messages due to the absence of shared memory." }) }), _jsxs(Box, { question: "In a very high level, how does it work?", children: [_jsxs(_components.p, { children: ["Say process ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "P" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" })] }) })] }) }), " wants to communicate with with process ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "Q" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Q" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" })] }) })] }) }), "."] }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "P" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" })] }) })] }) }), " builds message in local address space"] }), "\n", _jsxs(_components.li, { children: ["Sytem call sends message to ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "Q" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Q" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" })] }) })] }) })] }), "\n", _jsxs(_components.li, { children: ["Both ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "P" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" })] }) })] }) }), " and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "Q" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Q" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" })] }) })] }) }), " need to agree on what was sent"] }), "\n"] })] }), _jsx(_components.h3, { children: "OSI Model" }), _jsx(Box, { question: "What is the OSI Model?", children: _jsxs(_components.p, { children: ["The Open Systems Interconnection Model identifies the various levels of communication, their purpose, as well as naming them.\nIt is designed to allow open systems to communicate by using ", _jsx(_components.em, { children: "communication protocols" }), "."] }) }), _jsxs(Box, { question: "Why use layers in the OSI Model? How many are there?", children: [_jsx(_components.p, { children: "Each layer offers one or more specific communication services to the layer above it. In this way, the problem of getting a message\nfrom A to B can be divided into manageable pieces, each of which can be solved independently of the others." }), _jsx(_components.p, { children: "7 Layers:" }), _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Physical Layer" }), "\n", _jsx(_components.li, { children: "Data Link Layer" }), "\n", _jsx(_components.li, { children: "Network Layer" }), "\n", _jsx(_components.li, { children: "Transport Layer" }), "\n", _jsx(_components.li, { children: "Session Layer" }), "\n", _jsx(_components.li, { children: "Presentation Layer" }), "\n", _jsx(_components.li, { children: "Application Layer" }), "\n"] })] }), _jsx(Box, { question: "Common Protocols", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "File Transport Protocol (FTP)" }), "\n", _jsx(_components.li, { children: "HyperText Transport Protocol (HTTP)" }), "\n", _jsx(_components.li, { children: "Secure Shell (SSH)" }), "\n"] }) }), _jsx(_components.h3, { children: "Middleware Protocols" }), _jsx(Box, { question: "What is middleware", children: _jsx(_components.p, { children: "Application that lives in the OSI application layer, but contains many general-purpose protocols that warrant their own layers." }) }), _jsx(Box, { question: "What effect did it have on the OSI model?", children: _jsx(_components.p, { children: "It replaces the session and presentation layer" }) }), _jsx(_components.h3, { children: "Berkeley Sockets" }), _jsx(Box, { question: "What is a socket?", children: _jsx(_components.p, { children: "One endpoint of a two way communication link between two programs running on the network." }) }), _jsx(Box, { question: "What are the socket operations for TCP?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "socket: creates new communication endpoint" }), "\n", _jsx(_components.li, { children: "bind: attaches local address to socket" }), "\n", _jsx(_components.li, { children: "listen: readiness to accept client connection requests" }), "\n", _jsx(_components.li, { children: "accept: block caller until connection request arrives" }), "\n", _jsx(_components.li, { children: "connect: actively attempts to establish connection" }), "\n", _jsx(_components.li, { children: "send: send data over connection" }), "\n", _jsx(_components.li, { children: "receive: receive data over connection" }), "\n", _jsx(_components.li, { children: "close: release connection" }), "\n"] }) }), _jsxs(Box, { question: "Describe the communication pattern", children: [_jsx(_components.p, { children: "Server side executes the first 4 operations in order:" }), _jsx(_components.p, { children: _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "s" }), _jsx(_components.mi, { children: "o" }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "k" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "b" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "d" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "l" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "p" }), _jsx(_components.mi, { children: "t" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "socket \\rightarrow bind \\rightarrow listen \\rightarrow accept" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "soc" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "bin" }), _jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.01968em" }, children: "l" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8095em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mord mathnormal", children: "cce" }), _jsx(_components.span, { className: "mord mathnormal", children: "pt" })] })] })] }) }) }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["When ", _jsx(_components.em, { children: "socket" }), " is called, caller creates new communication endpoint. The OS reserves resources for the specified resource."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "bind" }), " associates local address with newly created socket. This tells the OS to receive messages on specified address and ports."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "listen" }), " us called only in the case of connection-oriented communication. Allows OS to reserve buffers for specified amount of pending connection requests that are to be accepted."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "accept" }), " blocks callers until receiving connection request. When request arrives, local OS creates copy of the socket and returns it to the caller."] }), "\n"] }), _jsx(_components.p, { children: "Now we look at the client side." }), _jsxs(_components.p, { children: ["5 . ", _jsx(_components.em, { children: "connect" }), " requires caller to specify where a connection request is to be sent. Client is blocked until connection is established."] }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Both client and server perform ", _jsx(_components.em, { children: "send" }), " and ", _jsx(_components.em, { children: "receive" }), " operations."] }), "\n"] }), _jsxs(_components.p, { children: ["8 . ", _jsx(_components.em, { children: "close" }), " is a symmetric operation."] }), _jsx(_components.p, { children: _jsx(_components.img, { src: "https://media.geeksforgeeks.org/wp-content/uploads/20200509144631/223-1.png", alt: "Socket Pattern" }) })] }), _jsx(_components.h3, { children: "Remote Procedure Call" }), _jsx(_components.p, { children: _jsx(_components.strong, { children: "Distributed Systems 3rd edition (2017) 4.2" }) }), _jsxs(Box, { question: "What is the basic idea of Remote Procedure Call?", children: [_jsx(_components.p, { children: "Allow programs to call procedures located on other machines." }), _jsx(_components.p, { children: "When a process on machine A calls a procedure on machine B:" }), _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "calling process on A is suspended, and" }), "\n", _jsx(_components.li, { children: "execution of the called procedure takes place on B" }), "\n"] }), _jsx(_components.p, { children: "Information can be transported from the caller to the callee in the parameters and can come back in\nthe procedure result" })] }), _jsx(Box, { question: "What can cause issues?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Procedures being executed in different address spaces" }), "\n", _jsx(_components.li, { children: "Passing of parameters and results" }), "\n", _jsx(_components.li, { children: "Either or both machines crashing" }), "\n"] }) }), _jsx(Box, { question: "Is RPC transparent?", children: _jsx(_components.p, { children: "Yes, the goal if that the calling procedure should not be aware that the called procedure is executing on a different machine or vice versa.\nIn simple words, we want to make it look like local procedure." }) }), _jsxs(Box, { question: "Single machine procedure call", children: [_jsx(_components.pre, { children: _jsx(_components.code, { className: "language-c", children: " newlist = append(data, dbList);\n" }) }), _jsx(_components.p, { children: "Assumptions:" }), _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "dbList is globally defined" }), "\n", _jsx(_components.li, { children: "data is the element to be appended" }), "\n"] }), _jsx(_components.p, { children: "Observations:" }), _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "dbList is a reference to a list" }), "\n", _jsx(_components.li, { children: "data is a value" }), "\n"] }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "p" }), _jsx(_components.mi, { children: "p" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "d" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "b" }), _jsx(_components.mi, { children: "L" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "append(data, dbList)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mord mathnormal", children: "pp" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }) }), " pushes the representations of data & dbList into the stack"] }), "\n", _jsx(_components.li, { children: "Representations are now accessible by append's implementation" }), "\n"] }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: ["stack before the call to append\n", _jsx(_components.img, { src: "https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQSZyRul-Q3hJJ681SBY9FBE2yfhrNg3Py9r8IpBf0xorHUk9RvpwkY1aq7mjk0qi_z7_c&usqp=CAU", alt: "Stack before" })] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: ["stack while the called procedure is active\n", _jsx(_components.img, { src: 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", alt: "Stack After" })] }), "\n"] }), "\n"] })] }), _jsx(Box, { question: "What is a stub?", children: _jsx(_components.p, { children: "A stub is a piece of code that translates parameters sent between the client and server during a remote procedure call." }) }), _jsx(Box, { question: "Why do stubs convert messages?", children: _jsx(_components.p, { children: "The client and server use different address spaces, so parameters used in function calls have to be converted, otherwise\nthe values of those parameters could not be used, because pointers to parameters in one computer's memory would point to\ndifferent data on the other computer. The client and server may also use different data representations, even for simple\nparameters (e.g., big-endian versus little-endian for integers)." }) }), _jsx(Box, { question: "Client stub vs server stub", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Client stubs convert parameters used in function calls and reconvert the result obtained from the server after function execution" }), "\n", _jsxs(_components.li, { children: ["Server stubs reconvert parameters passed by clients and convert results back after function execution\n", _jsx(_components.img, { src: "http://csis.pace.edu/~marchese/SE765/L7/L7_files/image016.jpg", alt: "RPC between Client and Server" })] }), "\n"] }) }), _jsxs(Box, { question: "Steps of a RPC", children: [_jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "Client calls client stub" }), "\n", _jsx(_components.li, { children: "Client stubs builds message" }), "\n", _jsx(_components.li, { children: "Client's OS sends message to remote OS" }), "\n", _jsx(_components.li, { children: "Remote OS gives message to server stub" }), "\n", _jsx(_components.li, { children: "Server stub unpacks message" }), "\n", _jsx(_components.li, { children: "Server performs operation and returns result to the stub" }), "\n", _jsx(_components.li, { children: "Server stub builds message" }), "\n", _jsx(_components.li, { children: "Server OS sends message to client OS" }), "\n", _jsx(_components.li, { children: "Client OS gives message to client stub" }), "\n", _jsx(_components.li, { children: "Client stub unpacks message" }), "\n"] }), _jsx(_components.p, { children: _jsx(_components.img, { src: "https://wachemo-elearning.net/wp-content/uploads/2022/08/image-824.png", alt: "Steps involved" }) })] }), _jsxs(Box, { question: "Stub Generation", children: [_jsx(_components.p, { children: "Conside the following function:" }), _jsx(_components.pre, { children: _jsx(_components.code, { className: "language-c", children: " void foobar(char x, float y, int z[5]){...}\n" }) }), _jsx(_components.p, { children: "Assuming a word is four bytes:" }), _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "char is one by" }), "\n", _jsx(_components.li, { children: "float is a whole word" }), "\n", _jsx(_components.li, { children: "array is group of words equal to array length" }), "\n"] }), _jsx(_components.p, { children: "Using these rules, the following message if formed:" }), _jsx(_components.p, { children: _jsx(_components.img, { src: "http://csis.pace.edu/~marchese/CS865/Lectures/Chap4/Chapter4_files/image024.jpg", alt: "Stub Generation" }) }), _jsx(_components.p, { children: "Server stub know the format that is being used." }), _jsx(Box, { question: "Is agreeing on the format enough?", children: _jsx(_components.p, { children: "No. Client and server need to agree on representation of data structures." }) })] }), _jsxs(Box, { question: "When to use Asynchronous RPC?", children: [_jsx(_components.p, { children: "To support situations in which there no result to return to the client" }), _jsxs(Box, { question: "How does it work?", children: [_jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Server immediately sends reply to client upon request arrival" }), "\n", _jsx(_components.li, { children: "Reply acts as acknowledgment to client" }), "\n", _jsx(_components.li, { children: "Client will unblock as soon as acknowledgment arrives" }), "\n"] }), _jsxs(_components.p, { children: [_jsx(_components.img, { src: "https://d3i71xaburhd42.cloudfront.net/6fa6b5bc123c47d91d3a7afa41b39235dbaa25f1/11-Figure2-7-1.png", alt: "Async RPC" }), "\na. shows traditional RPC (notice how the procedure finishes on the server and then sends results to client)"] }), _jsx(_components.p, { children: "b. Async RPC (acknowledgment is sent, then starts working on procedure)" })] })] }), _jsxs(Box, { question: "Deferrred synchronous RPC?", children: [_jsx(_components.p, { children: "Occurs when reply will be returned but client is not prepared to wait for it and do nothing in the meantime." }), _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "Client calls servers, waits for acceptance and continues" }), "\n", _jsx(_components.li, { children: "Server gets results" }), "\n", _jsx(_components.li, { children: "Server sends response to client" }), "\n", _jsxs(_components.li, { children: [_jsx(_components.em, { children: "callback" }), " in client's side is initiated"] }), "\n", _jsx(_components.li, { children: "Client sends acknowledgment to server" }), "\n"] }), _jsx(_components.p, { children: _jsx(_components.img, { src: "http://csis.pace.edu/~marchese/CS865/Lectures/Chap4/Chapter4_files/image030.jpg", alt: "Async RPC 2" }) })] }), _jsx(_components.h3, { children: "References" }), _jsx(Box, { question: "Sources", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.distributed-systems.net/index.php/books/ds3/", children: "Distributed Systems 3rd edition (2017)" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/socket-in-computer-network/", children: "Socket in Computer Network" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/stub-generation-in-distributed-system/", children: "Stub Generation in Distributed System" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.techopedia.com/definition/27680/stub-distributed-computing#:~:text=Client%20stubs%20convert%20parameters%20used,generated%20either%20manually%20or%20automatically.", children: "What Does Stub Mean?" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://en.wikipedia.org/wiki/Stub_(distributed_computing)", children: "Stub (distributed computing)" }) }), "\n"] }) })] }), "\n", _jsxs(Section, { section: "Clocks", children: [_jsx(_components.h3, { children: "Cristian's Clock" }), _jsx(_components.p, { children: "Distributed Systems 3rd edition (2017) 6.1" }), _jsx(Box, { question: "What is the idea?", children: _jsx(_components.p, { children: "Client's contact a server to retrieve the time." }) }), _jsx(Box, { question: "Issues with this?", children: _jsx(_components.p, { children: "Message delays will have outdated the reported time." }) }), _jsxs(Box, { question: "Example", children: [_jsx(_components.p, { children: _jsx(_components.img, { src: "https://www.researchgate.net/publication/301451426/figure/fig2/AS:748572651814912@1555485036134/Christian-algorithm.jpg", alt: "Cristian's Algorithm" }) }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math 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"katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mn, { children: "1" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_1 " })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: 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}) }), " with a timestamp, as well as the timestamp from ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mn, { children: "2" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_2" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: 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"katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mn, { children: "4" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_4" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: 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children: ")" })] }), _jsx(_components.mn, { children: "2" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\theta = \\frac{(T_2 - T_1) - (T_4 - T_3)}{2}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "θ" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.355em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: 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[_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mopen mtight", children: "(" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3173em" }, children: _jsxs(_components.span, { style: { top: "-2.357em", marginLeft: "-0.1389em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 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[_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.143em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose mtight", children: ")" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mopen mtight", children: "(" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3173em" }, children: _jsxs(_components.span, { style: { top: "-2.357em", marginLeft: "-0.1389em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "4" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.143em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3173em" }, children: _jsxs(_components.span, { style: { top: "-2.357em", marginLeft: "-0.1389em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.143em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose mtight", children: ")" })] }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) })] }), "\n"] })] }), _jsx(Box, { question: "If clocks are to be modified, would it be gradually or abruptly?", children: _jsx(_components.p, { children: "Graudally. Changing the time in one go can lead to serious problems such as:\nan object file compiled just after the clock change having a tim" }) }), _jsx(_components.h3, { children: "The Berkeley algorithm" }), _jsx(Box, { question: "How does it work?", children: _jsxs(_components.p, { children: ["Server constantly asks nodes what time it is. It then calculates the average time and notifies node whether they should speed up or slow down their local clocks.\n", _jsx(_components.img, { src: "https://www.abcassignmenthelp.com/webroot/uploadefiles/Samples/90Berkeley-Algorithm.png", alt: "Berkeley Algorithm" })] }) }), _jsx(Box, { question: "When to use this algorithm?", children: _jsx(_components.p, { children: "Typically to be used in an internal synchronization algorithm." }) }), _jsx(_components.h3, { children: "Logical clocks" }), _jsx(_components.p, { children: "Distributed Systems 3rd edition (2017) 6.2" }), _jsx(Box, { question: "Is it necessary to have an acurate account of the real time?", children: _jsxs(_components.p, { children: ["No, we just need nodes to agree on ", _jsx(_components.em, { children: "a" }), " time."] }) }), _jsx(Box, { question: "What two things did Lamport point out?", children: _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "If two processes do not interact, it is not necessary that their clocks are synchronized." }), "\n", _jsx(_components.li, { children: "What matters is that processes agree on the order in which events occur, rather than what time it is." }), "\n"] }) }), _jsx(_components.h3, { children: "Lamport's Logical Clocks" }), _jsx(_components.h4, { children: "Happens-before" }), _jsxs(Box, { question: "What does a || b mean?", children: [_jsxs(_components.p, { children: ["Also displayed as ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "b" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a \\rightarrow b" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] })] })] }) }), "\nIt is read as event ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "a" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] }) })] }) }), " happens before ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "b" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) })] }) }), "."] }), _jsx(_components.p, { children: "It means all events agree that a happens firtst, then afterwards b." }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["If ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "a" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] }) })] }) }), " and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "b" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) })] }) }), " are in the same process, then ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "b" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a \\rightarrow b" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] })] })] }) }), " is true"] }), "\n", _jsxs(_components.li, { children: ["If ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "a" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] }) })] }) }), " is an event that sends a message,and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "b" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) })] }) }), " is the event of receiving a message then ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "b" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a \\rightarrow b" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] })] })] }) }), " is true. (You can't receive a message without sending it!)"] }), "\n"] })] }), _jsxs(Box, { question: "Is Happens-before a transitive property?", children: [_jsxs(_components.p, { children: ["Yes.\nif ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "b" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a \\rightarrow b" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] })] })] }) }), " and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "b" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "c" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b \\rightarrow c" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "c" })] })] })] }) })] }), _jsx(_components.p, { children: _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "⇛" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "c" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: " \\Rrightarrow a \\rightarrow c" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7733em", verticalAlign: "-0.1367em" } }), _jsx(_components.span, { className: "mrel amsrm", children: "⇛" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "c" })] })] })] }) }) })] }), _jsxs(Box, { question: "When are events said to be concurrent?", children: [_jsx(_components.p, { children: "When they happen in separate processes that do not exchange messages." }), _jsxs(_components.p, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "b" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a \\rightarrow b" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] })] })] }) }), " is not true, nor is ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "b" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "a" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b \\rightarrow a" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] })] })] }) })] }), _jsx(_components.p, { children: "Pretty much nothing can be said about when the events happened first." })] }), _jsxs(_components.p, { children: ["For every event ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "a" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] }) })] }) }), ", we assign a value ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C(a)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }) }), " on which all processes agree."] }), _jsx(Box, { question: "What property must these values have?", children: _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["If ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "a" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] }) })] }) }), " and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "b" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) })] }) }), " are in the same process, and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "a" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] }) })] }) }), " occurs before ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "b" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) })] }) }), " then ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "b" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a \\rightarrow b" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] })] })] }) }), " then ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "b" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C(a) < C(b)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) })] }), "\n", _jsxs(_components.li, { children: ["If ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "a" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] }) })] }) }), " is a sending event to another process and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "b" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) })] }) }), " is the receiving event, then ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C(a)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }) }), " and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "b" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C(b)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }) }), " have to be assigned such that\neveryone agrees that ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "b" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C(a) < C(b)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) })] }), "\n"] }) }), _jsx(Box, { question: "How to make corrections to the clock time?", children: _jsxs(_components.p, { children: ["The clock time, ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "C" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" })] }) })] }) }), ", must always increase. Corrections to time can be made by adding a positive value, not substracting one."] }) }), _jsxs(Box, { question: "Lamport's Logical Clocks Implementation", children: [_jsxs(_components.p, { children: ["Each process, ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), ", mantains a local counter, ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) })] }), _jsx(Box, { question: "How are the clocks updated?", children: _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["Before executing an event, ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " increments ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), ": ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "←" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "d" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C_i \\leftarrow + d" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "←" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7778em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "+" }), _jsx(_components.span, { className: "mord mathnormal", children: "d" })] })] })] }) })] }), "\n", _jsxs(_components.li, { children: ["When process ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " sends message ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "m" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "m" })] }) })] }) }), " to ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), ", it sets m's timestamp ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mi, { children: "i" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "ts(m) = C_i" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] }) })] }), "\n", _jsxs(_components.li, { children: ["When ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " receives m, it adjusts its own local counter as ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mi, { children: "j" })] }), _jsx(_components.mo, { children: "←" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { children: "x" }), _jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mi, { children: "j" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "m" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C_j \\leftarrow max{C_j, ts(m)}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "←" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0361em", verticalAlign: "-0.2861em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ma" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsxs(_components.span, { className: "mord", children: [_jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] })] }) })] }), "\n"] }) }), _jsxs(Box, { question: "If a||b then C(a) < C(b)?", children: [_jsxs(_components.p, { children: ["True.\nBut the converse is false. ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "C" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "b" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "⇛" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "b" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C(a) < C(b) \\Rrightarrow a \\rightarrow b" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "b" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel amsrm", children: "⇛" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] })] })] }) })] }), _jsx(Box, { question: "Why?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Processes could be concurrent" }), "\n", _jsx(_components.li, { children: "Could be causally preceeding" }), "\n"] }) })] })] }), _jsx(_components.h4, { children: "Total-ordered multicasting" }), _jsx(_components.p, { children: "Consider a DB has been replicated across several sites in order to improve query performance. A query is forwarded to the nearest copy.\nEach update operation must be carried out at each replica in the same order." }), _jsx(Box, { question: "What is total-ordered multicasting?", children: _jsx(_components.p, { children: "Multicast operation by which all messages are delivered in the same order to each receiver" }) }), _jsx(_components.h4, { children: "Vector Clocks" }), _jsxs(Box, { question: "What can Lamport clocks say about the relationship of events a and be if C(a) < C(b)?", children: [_jsxs(_components.p, { children: ["Nothing. It does not imply ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "a" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] }) })] }) }), " happened before ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "b" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) })] }) }), "."] }), _jsxs(Box, { question: "Why? Hint: use three processes", children: [_jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "s" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: ":" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_s(m_i):" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "s" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: ":" })] }) })] }) }), " time at which ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " was sent"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "r" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_r(m_i)" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.02778em" }, children: "r" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }) }), " time of receipt"] }), "\n"] }), _jsxs(_components.p, { children: ["We know that for each ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), ": ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "s" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "<" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "r" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_s(m_i) < T_r(m_i)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "s" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.02778em" }, children: "r" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) })] }), _jsxs(_components.p, { children: ["Consider messages ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), ", what can we say from ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "r" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "<" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "s" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "j" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_r(m_i) < T_s(m_j)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.02778em" }, children: "r" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0361em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "s" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "?"] }), _jsxs(_components.p, { children: ["Let ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mn, { children: "1" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m_i = m_1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] }) }), " and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mi, { children: "j" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mn, { children: "3" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m_j = m_3" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] }) })] }), _jsxs(_components.p, { children: ["Looking at the figure below, we notice that ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mn, { children: "3" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m_3" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " was sent after receiving ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mn, { children: "1" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m_1" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), ". This means that ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mn, { children: "3" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m_3" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "3" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " ", _jsx(_components.em, { children: "may" }), "\nhave depended on both ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mn, { children: "1" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m_1" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "m" }), _jsx(_components.mn, { children: "2" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "m_2" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), ". We cannot be sure since Lamport clocks do ", _jsx(_components.strong, { children: "not" }), " capture causality."] }), _jsx(_components.p, { children: _jsx(_components.img, { src: "http://csis.pace.edu/~marchese/CS865/Lectures/Chap6/Chapter6_files/image046.jpg", alt: "Concurrent message transmission" }) })] })] }), _jsx(Box, { question: "How to capture causality?", children: _jsx(_components.p, { children: "Vector clocks" }) }), _jsxs(Box, { question: "How to track causality?", children: [_jsxs(_components.p, { children: ["Assign each event a unique name and an incrementing counter: ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mi, { children: "k" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "p_k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " is the ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "k" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "h" })] })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k^{th}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8491em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8491em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "t" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "h" })] }) })] }) }) }) }) })] })] }) })] }) }), " event that happened at process ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "P" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" })] }) })] }) }), "\nThen keep track of the ", _jsx(_components.em, { children: "causal histories" })] }), _jsxs(_components.p, { children: ["Example:\nTwo local events happen at process P, then causal history ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "H" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mo, { stretchy: "false", children: "{" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: "}" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "H(p_2) = \\{ p_1, p_2 \\} " })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.08125em" }, children: "H" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "{" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: "}" })] })] })] }) })] })] }), _jsxs(_components.p, { children: ["Assume process ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: 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_jsx(_components.mi, { children: "Q" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Q" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" })] }) })] }) }), ". At the time of arrival, the causal history of ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "Q" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Q" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" })] }) })] }) }), " was ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "q" }), _jsx(_components.mn, { children: "1" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "{q_1}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord", children: _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0359em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }) })] }) })] }) }), ". To track causality, ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "P" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" })] }) })] }) }), " sends\nits own causal history ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { separator: "true", children: "," }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mn, { children: "2" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "{p_1, p_2}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] }) })] }) }), ". Upon arrival, ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "Q" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Q" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8778em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "Q" })] }) })] }) }), " records the event as ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "q" }), _jsx(_components.mn, { children: "2" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "q_2" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0359em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " and updates its causal history to ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: 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_jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: "}" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\{p_1, p_2, p_3, q_1, q_2\\}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "{" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: 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_jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0359em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0359em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: "}" })] }) })] }) })] }), _jsx(Box, { question: "Why is the P's causality history before Q's?", children: _jsxs(_components.p, { children: ["To determine if an event ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "p" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "p" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "p" })] }) })] }) }), " ocurred before ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "q" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "q" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" })] }) })] }) }), " , we check that ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mo, { children: "∈" }), _jsx(_components.mi, { children: "H" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "q" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "p \\in H(q)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7335em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "∈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.08125em" }, children: "H" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "\nAssumming q is the last local event."] }) }), _jsx(Box, { question: "What is the problem with causal histories?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Representation is inneficient\nBut, there is no need to keep track of all successive events in a process, just the last one!" }), "\n"] }) }), _jsxs(Box, { question: "How to construct vector clocks?", children: [_jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Assign an index to each process" }), "\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "j" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "h" })] })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "j^{th}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0435em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05724em" }, children: "j" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8491em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "t" }), _jsx(_components.span, { className: "mord mathnormal mtight", children: "h" })] }) })] }) }) }) }) })] })] }) })] }) }), " entry represents number of events that occurred before ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) })] }), "\n"] }), _jsxs(_components.p, { children: ["We now construct the vector clock by: letting each process ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " mantain a vector ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "V" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mi, { children: "i" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "VC_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "V" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " with the following properties:"] }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "V" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "VC_i[i]" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "V" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) })] }) }), " number of events that have occurred so far at ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) })] }), "\n", _jsxs(_components.li, { children: ["If ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "V" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "j" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "k" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "VC_i[j] = k" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "V" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05724em" }, children: "j" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" })] })] })] }) }), " then ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " knows that ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" })] }) })] }) }), " events occured at ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) })] }), "\n"] })] }), _jsx(Box, { question: "Vector Clocks implementation rules", children: _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["Before executing an event, ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " executes: ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: 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className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "V" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "C" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0715em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing 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children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "d" })] })] })] }) }), ". Recording an event"] }), "\n", _jsxs(_components.li, { children: ["When ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " sends message ", _jsx(_components.span, { className: 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className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "m" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" }), _jsx(_components.span, { className: "mclose", children: "]}" })] })] })] }) })] }), "\n"] }) }), _jsx(Box, { question: "Example", children: _jsx(_components.p, { children: _jsx(_components.img, { src: "https://media.geeksforgeeks.org/wp-content/uploads/20200527210328/vector1.png", alt: "Vector Clock's example" }) }) }), _jsx(_components.h3, { children: "References" }), _jsx(Box, { question: "Sources", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/cristians-algorithm/", children: "Cristian’s Algorithm" }) }), "\n", _jsxs(_components.li, { children: ["Chapter 6 of ", _jsx(_components.a, { href: "https://www.distributed-systems.net/index.php/books/ds3/", children: "Distributed Systems 3rd edition (2017)" })] }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/vector-clocks-in-distributed-systems/", children: "Vector Clocks in Distributed Systems" }) }), "\n"] }) })] }), "\n", _jsxs(Section, { section: "Mutual Exclusion", children: [_jsx(Box, { question: "What categories are mutual exclusion algorithms classified in?", children: _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.p, { children: "Token Based Solutions: passing a special message between processes. Whoever holds the token gets to access the shared resource" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Pros: avoids starvation & deadlocks" }), "\n", _jsx(_components.li, { children: "Cons: if the token is lost, a procedure needs to create a new one" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.p, { children: "Permission Based Approach: process needs permission to access a resource" }), "\n"] }), "\n"] }) }), _jsx(_components.h3, { children: "A centralized algorithm" }), _jsxs(Box, { question: "Straightforward way to achieve mutual exclusin in DS?", children: [_jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Select one process as coordinator" }), "\n", _jsx(_components.li, { children: "When a certain process wants access to resource, ask the coordinator" }), "\n", _jsx(_components.li, { children: "If resource is not being accessed, reply with permission" }), "\n", _jsx(_components.li, { children: "If resource is being accessed, either refrain from replying or deny permission. Either way, queue the request" }), "\n", _jsx(_components.li, { children: "Once a process is done, notify coordinator" }), "\n", _jsx(_components.li, { children: "Coordinator then calls the next item in the queue." }), "\n"] }), _jsx(_components.p, { children: _jsx(_components.img, { src: 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", alt: "Centralizes Algorithm" }) })] }), _jsxs(Box, { question: "Pros and Cons?", children: [_jsx(_components.p, { children: "Pros:" }), _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "fair" }), "\n", _jsx(_components.li, { children: "no starvation" }), "\n", _jsx(_components.li, { children: "easy to implement" }), "\n", _jsx(_components.li, { children: "3 messages per resource (ask for permission, grant permission, release resource)" }), "\n"] }), _jsx(_components.p, { children: "Cons:" }), _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Coordinator is a single point of failure" }), "\n", _jsx(_components.li, { children: "In a big system, the coordinator is a performance bottleneck" }), "\n"] })] }), _jsx(_components.h3, { children: "Distributed Algorithms" }), _jsx(_components.h4, { children: "Lamport's Algorithm" }), _jsx(_components.p, { children: "Geeks for Geeks" }), _jsx(Box, { question: "What type of algorithm is Lamport's?", children: _jsx(_components.p, { children: "Permission based algorithm" }) }), _jsx(Box, { question: "How are critical sections executed?", children: _jsx(_components.p, { children: "They are executed in the increasing order of timestamps." }) }), _jsx(Box, { question: "What are the types of messages in the algorithm?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "REQUEST: message sent to all other sites to get their permission to enter CS" }), "\n", _jsx(_components.li, { children: "REPLY: message to requesting site that gives its permission to enter CS" }), "\n", _jsx(_components.li, { children: "RELEASE: message to all other sites when exiting CS" }), "\n"] }) }), _jsx(Box, { question: "How does each site keep track of CS requests?", children: _jsxs(_components.p, { children: ["Each site ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " has a local queue denoted by ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "q" }), _jsx(_components.mi, { children: "u" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "s" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "q" })] }), _jsx(_components.mi, { children: "u" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "u" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "i" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "request_queue_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9012em", verticalAlign: "-0.2861em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "re" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" }), _jsx(_components.span, { className: "mord mathnormal", children: "u" }), _jsx(_components.span, { className: "mord mathnormal", children: "es" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03588em" }, children: "q" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mord mathnormal", children: "u" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "u" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "\nThis queue orders requests requests by their timestamps."] }) }), _jsxs(Box, { question: "Summarize Lamport's Algorithm", children: [_jsxs(_components.p, { children: ["Let ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " be the site t"] }), _jsx(_components.p, { children: _jsx(_components.strong, { children: "Critical section request:" }) }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " sends request with timestamp to other sites while also putting it in its local queue"] }), "\n", _jsxs(_components.li, { children: ["Site ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " puts ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "'s request in the local request queue"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " replies to ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) })] }), "\n"] }), _jsx(_components.p, { children: _jsx(_components.strong, { children: "Critical section access:" }) }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " enters critical section when:", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " is on the top of the local queue"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: 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[_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) })] }), "\n"] }), "\n"] }), "\n"] }), _jsx(_components.p, { children: _jsx(_components.strong, { children: "Critical section release:" }) }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", 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children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " sends a timestamped RELEASE message to other sites"] }), "\n", _jsxs(_components.li, { children: ["All other sites remove ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { 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from ", _jsx(_components.a, { href: "https://therenegadecoder.com/code/distributed-mutual-exclusion-using-logical-clocks/", children: "Distributed Mutual Exclusion Using Logical Clocks" }), "\n", _jsx(_components.img, { src: "https://i2.wp.com/therenegadecoder.com/wp-content/uploads/2019/07/lamport-algorithm.png?fit=800%2C534&ssl=1", alt: "Lamport's Algorithm" })] }) }), _jsxs(Box, { question: "Proof by contradiction", children: [_jsx(_components.p, { children: "Assumptions:" }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), 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children: "S_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: 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children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: 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"katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: 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className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " and ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " are on top of their respective queues\nUsing FIFO, assumption 2, and the fact that ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " is executing, then the request from ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " is in ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "'s queue"] }), _jsxs(_components.p, { children: ["This is a contradiction since, ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " should be at the top of the queue since it has a smaller timestamp"] }), _jsx(_components.p, { children: "So there cannot be two processes executing at the same time." })] }), _jsx(_components.h4, { children: "Ricart Agrawala Algorithm" }), _jsx(_components.p, { children: "Geeks for Geeks" }), _jsx(Box, { question: "What type of algorithm is Ricart Agrawala's?", children: _jsx(_components.p, { children: "Permission based algorithm" }) }), _jsx(Box, { question: "How are critical sections executed?", children: _jsx(_components.p, { children: "They are executed in the increasing order of timestamps." }) }), _jsx(Box, { question: "What are the types of messages in the algorithm?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "REQUEST: message sent to all other sites to get their permission to enter CS" }), "\n", _jsx(_components.li, { children: "REPLY: message to requesting site that gives its permission to enter CS" }), "\n"] }) }), _jsx(Box, { question: "How does each site keep track of CS requests?", children: _jsxs(_components.p, { children: ["Each site ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " has a local queue denoted by ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "q" }), _jsx(_components.mi, { children: "u" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "s" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { children: "q" })] }), _jsx(_components.mi, { children: "u" }), _jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "u" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "e" }), _jsx(_components.mi, { children: "i" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "request_queue_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9012em", verticalAlign: "-0.2861em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "re" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" }), _jsx(_components.span, { className: "mord mathnormal", children: "u" }), _jsx(_components.span, { className: "mord mathnormal", children: "es" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "t" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03588em" }, children: "q" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mord mathnormal", children: "u" }), _jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "mord mathnormal", children: "u" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), "\nThis queue orders requests requests by their timestamps."] }) }), _jsxs(Box, { question: "Summarize Ricart Agrawala's Algorithm", children: [_jsxs(_components.p, { children: ["Let ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " be the site t"] }), _jsx(_components.p, { children: _jsx(_components.strong, { children: "Critical section request:" }) }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " sends message with timestamp to other sites"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: ["Sites ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " reply immediately to ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " with approval if ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), ":"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "is not interested in the critical section" }), "\n", _jsx(_components.li, { children: "not executing the CS" }), "\n", _jsx(_components.li, { children: "has a request for critical section with a larger timestamp than incoming request" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: ["The site ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " sends a delayed approval to ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " if it had a request for the critical section with a smaller timestamp than the incoming request; in this case the response is delayed until the critical section is done"] }), "\n"] }), "\n"] }), _jsx(_components.p, { children: _jsx(_components.strong, { children: "Critical section access:" }) }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " enters critical section after receiving approval from every other site"] }), "\n"] }), _jsx(_components.p, { children: _jsx(_components.strong, { children: "Critical section release:" }) }), _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " replies to all the defferred requests"] }), "\n"] })] }), _jsx(Box, { question: "Example", children: _jsxs(_components.p, { children: ["Image taken from ", _jsx(_components.a, { href: "https://therenegadecoder.com/code/distributed-mutual-exclusion-using-logical-clocks/", children: "Distributed Mutual Exclusion Using Logical Clocks" }), "\n", _jsx(_components.img, { src: "https://therenegadecoder.com/wp-content/uploads/2019/07/ricart-agrawala-algorithm-1024x683.png", alt: "Ricart Agrawala" })] }) }), _jsx(Box, { question: "Does this algorithm require FIFO?", children: _jsxs(_components.p, { children: ["No.\n\"That’s because each process is personally responsible for granting permission. Even with out of order messages due to transmission delays, the REPLY message will never come thanks to the logical clocks.\"\n", _jsx(_components.a, { href: "https://therenegadecoder.com/code/distributed-mutual-exclusion-using-logical-clocks/", children: "Distributed Mutual Exclusion Using Logical Clocks" })] }) }), _jsx(_components.h4, { children: "Maekawa's Algorithm" }), _jsx(Box, { question: "How does it differ from the previous algorithms?", children: _jsx(_components.p, { children: "Requests are not made to every other site, only a subset of them." }) }), _jsx(Box, { question: "What are the types of messages in the algorithm?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "REQUEST: message sent to sites in subset to get their permission to enter CS" }), "\n", _jsx(_components.li, { children: "REPLY: message to requesting site that gives its permission to enter CS" }), "\n", _jsx(_components.li, { children: "RELEASE: message to all other sites in the subset when exiting CS" }), "\n"] }) }), _jsxs(Box, { question: "Summarize the algorithm?", children: [_jsx(_components.p, { children: _jsx(_components.strong, { children: "Summary of critical section request:" }) }), _jsxs(_components.p, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " sends request to subset of sites\nSites ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " sends a reply to ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " if it has not already sent one since the last release\nIf it has sent a reply, it puts the request inside the queue."] }), _jsx(_components.p, { children: "**Summary of critical section access: **" }), _jsxs(_components.p, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: 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className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " enters critical section when after receiving replies from all sites in the subset"] })] }), _jsx(Box, { question: "Can deadlocks occur?", children: _jsx(_components.p, { children: "Yes." }) }), _jsx(_components.h3, { children: "Election" }), _jsx(_components.h4, { children: "Garcia-Molina" }), _jsx(Box, { question: "What is the bully algorithm?", children: _jsx(_components.p, { children: "Solution for the electing a coordinator." }) }), _jsxs(Box, { question: "Explained", children: [_jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Consider ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { 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className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " sends ELECTION messages to all higher processes (", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: 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children: "P_{k+1}, ..., P_{n-1}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8917em", verticalAlign: "-0.2083em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: 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children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mtight", children: "1" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2083em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), ")"] }), "\n", _jsxs(_components.li, { children: ["If no one responds, ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "k" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "P_k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " is the coordinator"] }), "\n", _jsx(_components.li, { children: "If a higher up responds, they take over." }), "\n", _jsx(_components.li, { children: "Repeat until coordinator gets elected" }), "\n"] }), _jsx(_components.p, { children: "\"Biggest guy in town wins\"" })] }), _jsx(_components.h4, { children: "Ring Algorithm" }), _jsxs(Box, { question: "Explained", children: [_jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Each process knows its successor\nWhen coordinator stops responding" }), "\n"] }), _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "Each process sends its ID to the successor" }), "\n", _jsx(_components.li, { children: "If successor node is suspendes, skip it until you find a running one" }), "\n", _jsx(_components.li, { children: "At each node, they add their IDS to nominate themselves" }), "\n", _jsx(_components.li, { children: "After returning the initial node, the node with the highest ID is elected" }), "\n", _jsx(_components.li, { children: "Second loop to inform everyone" }), "\n"] })] }), _jsx(_components.h3, { children: "References" }), _jsx(Box, { question: "Sources", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/lamports-algorithm-for-mutual-exclusion-in-distributed-system/?ref=rp", children: "Lamport’s Algorithm for Mutual Exclusion in Distributed System" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://therenegadecoder.com/code/distributed-mutual-exclusion-using-logical-clocks/", children: "Distributed Mutual Exclusion Using Logical Clocks" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/ricart-agrawala-algorithm-in-mutual-exclusion-in-distributed-system/", children: "Ricart–Agrawala Algorithm in Mutual Exclusion in Distributed System" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/maekawas-algorithm-for-mutual-exclusion-in-distributed-system/", children: "Maekawa’s Algorithm for Mutual Exclusion in Distributed System" }) }), "\n"] }) })] }), "\n", _jsxs(Section, { section: "Databases", children: [_jsx(Box, { question: "What is a database?", children: _jsx(_components.p, { children: "Collection of shared data that can be accessed by users." }) }), _jsx(_components.h3, { children: "Transactions" }), _jsx(Box, { question: "What are transactions?", children: _jsx(_components.p, { children: "Sequence of multiple operations performed on a database, and all served as a single logical unit of work" }) }), _jsx(Box, { question: "Actions", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Query: read-only" }), "\n", _jsx(_components.li, { children: "Update: modify object/s" }), "\n"] }) }), _jsx(Box, { question: "Assumptions?", children: _jsx(_components.p, { children: "They preserve consistency and take finite amount of time" }) }), _jsx(Box, { question: "ACID?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Atomic" }), "\n", _jsx(_components.li, { children: "Consistent: transactions only make changes in predefined ways" }), "\n", _jsx(_components.li, { children: "Isolate: transactions don’t interfere with or affect one another" }), "\n", _jsx(_components.li, { children: "Durability: changes made by successful transactions will be saved" }), "\n"] }) }), _jsx(Box, { question: "What is the transaction model?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "BEGIN_TRANSACTION" }), "\n", _jsx(_components.li, { children: "END_TRANSACTION" }), "\n", _jsx(_components.li, { children: "ABORT_TRANSACTION" }), "\n", _jsx(_components.li, { children: "READ" }), "\n", _jsx(_components.li, { children: "WRITE" }), "\n"] }) }), _jsx(Box, { question: "What are nested transactions?", children: _jsx(_components.p, { children: "When transactions are constructed as a number of subtransactions" }) }), _jsx(Box, { question: "What are distributed transactions?", children: _jsx(_components.p, { children: "When clients wrap a number of requests, possibly for different servers.\nThe idea is that either all the requests get executed, or none do." }) }), _jsx(_components.h3, { children: "Concurrency Control" }), _jsx(Box, { question: "What is concurrency control?", children: _jsx(_components.p, { children: "Protocols to mantain consistency" }) }), _jsxs(Box, { question: "What three modules(managers) does a DB system consists?", children: [_jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Transaction Manager: intercepts and executes all the submitted transactions." }), "\n", _jsx(_components.li, { children: "Scheduler: grands or releases locks on data objects as requested by a transaction" }), "\n", _jsx(_components.li, { children: "Data Manager: carries out the read-write requests issued by the transaction manager" }), "\n"] }), _jsx(Box, { question: "Which one has to enforce concurrency control?", children: _jsx(_components.p, { children: "Scheduler" }) })] }), _jsx(Box, { question: "Can the previous models be expanded to a Distributed System?", children: _jsxs(_components.p, { children: ["Yes\n", _jsx(_components.img, { src: "https://courses.cs.vt.edu/~cs5204/fall00/distributedDBMS/phanouriou/figure3.gif", alt: "Model of DS" })] }) }), _jsx(Box, { question: "When do transactions conflict with each other?", children: _jsx(_components.p, { children: "If they access the same resource, and at least one of them is a writing operation." }) }), _jsx(Box, { question: "What is serial execution?", children: _jsxs(_components.p, { children: ["Transaction ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "b" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) })] }) }), " executes after transaction ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "a" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] }) })] }) }), " has completed"] }) }), _jsx(Box, { question: "What is concurrent execution?", children: _jsxs(_components.p, { children: ["Transaction ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "b" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "b" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "b" })] }) })] }) }), " begins execution before ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "a" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "a" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" })] }) })] }) }), " has completed"] }) }), _jsx(_components.h3, { children: "Logs" }), _jsx(Box, { question: "What is a log?", children: _jsx(_components.p, { children: "Order of actions performed by transactions" }) }), _jsx(Box, { question: "When are logs equivalent?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "All transactions in both logs see the same state in DB" }), "\n", _jsx(_components.li, { children: "DB final state is the same" }), "\n"] }) }), _jsx(Box, { question: "Serial log vs Serializable Log?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Serial: when transactions wait for the previous transaction to terminate" }), "\n", _jsx(_components.li, { children: "Serializable: When actions are interleaved and the result is the same as the serial execution" }), "\n"] }) }), _jsxs(Box, { question: "What is a Serialization Graph?", children: [_jsx(_components.p, { children: "Directed graph to test if a log is serializable." }), _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Transactions serve as nodes" }), "\n", _jsxs(_components.li, { children: ["An edge ei is constructed between node ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " to ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "k" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " if one of the operations in ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "j" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_j" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " appears in the schedule before some conflicting operation in ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { children: "k" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T_k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03148em" }, children: "k" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " ."] }), "\n"] })] }), _jsx(Box, { question: "How to draw graph?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Number of nodes = Number of transactions" }), "\n", _jsxs(_components.li, { children: ["Directed edge ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { children: "j" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i \\rightarrow j" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.854em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05724em" }, children: "j" })] })] })] }) }), " for each conflicting operation ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "o" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { children: "→" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "o" }), _jsx(_components.mi, { children: "j" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "o_i \\rightarrow o_j" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "o" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "→" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7167em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "o" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.05724em" }, children: "j" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] })] })] })] }) })] }), "\n"] }) }), _jsx(_components.h3, { children: "Concurrency Control Algorithms" }), _jsx(Box, { question: "What do they do?", children: _jsx(_components.p, { children: "Control the interleaving of conflicting transactions in order to mantain DB consistency" }) }), _jsx(_components.h4, { children: "Lock Based" }), _jsx(Box, { question: "How do lock-based algorithms work?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Transactions lock data before accessing it" }), "\n"] }) }), _jsxs(Box, { question: "Types of Lock", children: [_jsx(Box, { question: "Static Lock", children: _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "Transaction locks all resources neede" }), "\n", _jsx(_components.li, { children: "Execution begins" }), "\n", _jsx(_components.li, { children: "Release all locks at the end of execution" }), "\n"] }) }), _jsx(Box, { question: "Dyncamic Lock", children: _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "Grab lock when needed" }), "\n", _jsx(_components.li, { children: "Releases lock after accessing resource" }), "\n"] }) })] }), _jsx(Box, { question: "What does it mean for a transaction to be well formed?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "locks data before accessing" }), "\n", _jsx(_components.li, { children: "does not lock data more than once" }), "\n", _jsx(_components.li, { children: "unlocks data before transaction completion" }), "\n"] }) }), _jsx(Box, { question: "Two-phase locking", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "once lock is released, it cannot be requested again" }), "\n", _jsx(_components.li, { children: "algorithm has a growing phase, then a lock point phase, finally a release phase\nThis means there are no local minimums in the graph" }), "\n"] }) }), _jsx(Box, { question: "How to know if all legal logs in a set of transactions are serializable?", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "if they are well formed" }), "\n", _jsx(_components.li, { children: "fllow a two-phase scheme" }), "\n"] }) }), _jsx(Box, { question: "Issues with two-phase locking", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Deadlocks" }), "\n", _jsx(_components.li, { children: "Cascaded rollbacks" }), "\n"] }) }), _jsx(_components.h4, { children: "Time Based Algorithms" }), _jsx(Box, { question: "What is the idea?", children: _jsx(_components.p, { children: "Use a transaction's timestamp to decide the order of execution" }) }), _jsxs(Box, { question: "How to timestamp transactions?", children: [_jsx(_components.p, { children: "Lamport's logical clock" }), _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Transaction Manager timestamps each tranaction" }), "\n"] })] }), _jsxs(Box, { question: "Basic Timestamp Ordering", children: [_jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Every transaction has an issued timestamp based on arrival" }), "\n", _jsx(_components.li, { children: "Conflicting operations follow transaction order" }), "\n"] }), _jsx(_components.p, { children: "Notes:\nW_TS(x): largest timestamp of any transaction that executed write(x)\nR_TS(x): largest timestamp of any transaction that executed read(x)" })] }), _jsx(_components.h3, { children: "References" }), _jsx(Box, { question: "Sources", children: _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://fauna.com/blog/database-transaction", children: "What is a database transaction?" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.databricks.com/glossary/acid-transactions", children: "ACID Transactions" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/concurrency-control-in-dbms/", children: "Concurrency Control in DBMS" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://courses.cs.vt.edu/~cs5204/fall00/distributedDBMS/phanouriou/dbms.html", children: "Distributed DBMS model Transactions, Architecture" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.tutorialspoint.com/explain-serial-execution-or-transaction-with-an-example-dbms", children: "Explain serial execution or transaction with an example(DBMS)" }) }), "\n", _jsx(_components.li, { children: _jsx(_components.a, { href: "https://www.geeksforgeeks.org/precedence-graph-for-testing-conflict-serializability-in-dbms/", children: "Precedence Graph For Testing Conflict Serializability in DBMS" }) }), "\n"] }) })] })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = Object.assign({}, _provideComponents(), props.components); return MDXLayout ? _jsx(MDXLayout, Object.assign({}, props, { children: _jsx(_createMdxContent, props) })) : _createMdxContent(props); } return { default: MDXContent }; function _missingMdxReference(id, component) { throw new Error("Expected " + (component ? "component" : "object") + " `" + id + "` to be defined: you likely forgot to import, pass, or provide it."); }

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