Cs 341 algorithms. this highly inefficient representation.
Cs 341 algorithms. This homework is counted 5% of the course grade.
Cs 341 algorithms. almost. graph! Throw away We are interested in designing efficient algorithms for the following basic questions: 1. ca 1 •Input: points 𝑞1,…,𝑞 in 2D space that form a convex -gon •Assume points are sorted clockwise around the center of P •Find: a triangulation of such that the sum of the CS 341: Algorithms Module 5: Greedy Algorithms Eugene Zima Based on lecture notes by many previous CS 341 instructors David R. CS 341 –Algorithms Lecture 8 –Greedy Algorithms 11 June 2021. ca 1 2 ALL PAIRS SHORTEST PATHS (APSP) PROBLEM 3 5-4 4 2 3 12-1 , = 0 0 0 0 3 from: to: ∞ ∞∞ ∞ 1212 55 ∞ 12 5 4 0 −1 0 3 12 5 4 ∞ 0 3 4 Naïve algorithm attempt For simplicity, try edge-disjoint path problem first (unit capacities) Greedy idea: find a shortest 𝑠-𝑡 path (to use few edges), then repeat on the remaining edges CS 341: ALGORITHMS Lecture 9: dynamic programming III Readings: see website Trevor Brown https://student. Suppose lemma holds for − 1, show it holds for (where ≥ 2) Consider the edges with non-zero. Zima Assignment 1 Solutions 1 Asymptotics [10 marks] Prove or CS 341: Algorithms. dynamic programming (~4 lectures). If there are points in Q with the same x as R[1], and a lower y, then the algorithm would say they are dominated by R[1]. For written problems, please write pseudocode for the algorithms, prove their correctness and analyze their time complexity. 5. path. Meaning an algorithm for edge-disjoint paths can solve this. ca 1 DIJKSTRA’SALGORITHM Single-source shortest path in a graph with non-negative edge weights 2 PROBLEM: SINGLE SOURCE SHORTEST PATHS (SSSP) •Input: graph 𝐺 CS 341: Algorithms. 28. For the reduction, we need some graph structures for the variables and the truth assignments. Contribute to y87feng/CS341-spring-2018 development by creating an account on GitHub. ca 1 THIS TIME •Intractability (hardness of problems) •Decision problems •Complexity class P •Polynomial-time Turing reductions •Introductory reductions •Three flavours of the traveling 2023-11-27 1 CS 341: ALGORITHMS Lecture 22: intractability IV –poly transformations, NP completeness Readings: see website Trevor Brown https://student. CS 341: ALGORITHMS (S18) — LECTURE 4 FAST MULTIPLICATION ERIC BLAIS The MergeSort algorithm we studied in the last lecture is a classic example of the power of the divide and conquer technique. Determine if an input graph is a directed acyclic graph. Tuesdays and Thursdays 1:30-2:30, Zoom. cs 341: algorithms (s18) — lecture 2 big-o notation and reductions eric blais 1. path We prove this in two parts. Also add edges between all vertices on each canyon wall. Students will experiment with modern graphics programming CS 341: ALGORITHMS Lecture 13: graph algorithms IV –minimum spanning trees Readings: see website Trevor Brown https://student. graph algorithms (~4 lectures). Once we have finished visiting the list of neighbours of v, we change its status as explored. algorithms through the study of various classical algorithms. ca 1 THIS TIME •Intractability (hardness of problems) •Decision problems •Complexity class P •Polynomial-time Turing reductions •Introductory reductions •Three flavours of the traveling CS 341: ALGORITHMS Lecture 10: graph algorithms I Readings: see website Trevor Brown https://student. CS 341: ALGORITHMS Lecture 20: intractability II –complexity class NP Readings: see website Trevor Brown https://student. The course will give students some experience in program design and emphasize both pragmatic and mathematical aspects of program efficiency. such subset , verify(I,C) will return true. Enqueue all nodes with distance ∞ except for with distance 0. The main focus is to learn basic techniques to the design and analysis of efficient algorithms through the study of various classical algorithms. ≠ 0 and verify returns false. ca Module 5: dynamic programming 1/62 The key idea is: Consider polynomials ( ) and. Today’s Plan 1. Search the δxδ square points on the opposite side. pdf from CS 341 at University of Waterloo. in 𝑆𝑖𝑧𝑒(𝑅1) In this case, 𝑺𝒊𝒛𝒆𝑹𝟑∈𝚯𝟐𝑺𝒊𝒛𝒆𝑹𝟏 We should . This is at least 3/10ths of our 50-element input, or . CS 341: ALGORITHMS Lecture 5: finishing D&C, greedy algorithms I Readings: see website Trevor Brown https://student. But multiplying polynomials of degrees 1, 2 results in a polynomial of CS 401 Accelerated Fundamentals of Algorithms I credit: 3 Hours. CS 341: Algorithms Fall 2022 General Information This course studies the major algorithmic design paradigms and mathematical tools for analyzing the running times of algorithms and detecting computational problems for which no efficient deterministic algorithm. CS 341: Algorithms @ Waterloo Final Exam Winter 2016 Date: Apr 15, 4pm-6:30pm There are totally 115 marks. Correctness proof. pseudo-linear. ca/~cs341 Decrease the flow of each edge in by 1. cs. Homework. Decrease the flow of each edge in by 1. Note that in practice it can be di cult to distinguish in nite from large nite. e. So, subset-sum ∈. of size n T(I) runtime of a particular instance. So on and so forth R might contain multiple points with the same x value but with different y values. Wednesdays and Fridays, 1:30-2:30. PROOF. 𝑮′so that for any two paths 𝑃1 and 𝑃2 in 𝐺,𝑃1 and 𝑃2 are vertex-disjoint. the neighbours. Contents 1 IntroductionandOrderNotation 2 2 DivideandConquer 4 3 DynamicProgramming 10 CS 341: Algorithms. rule out. 13. Consider recurrence: 1 = ; = + Θ where ≥ 1, ≥ 2 and is a power of (i. Bear cannot get through the canyon if North and South walls are connected. CS 341 - Algorithms - Spring 2021. The main focus is on the design and analysis of algorithms. Topics include counting, sets, functions, decision trees, recursion, binary numbers, basic graph theory, depth first search and breadth first search, algorithms for computing shortest paths, data structures CS 341 - Algorithms - University of Waterloo. greedy algorithms (~3 lectures). I do not make any warranties about the completeness, reliability and accuracy of CS 341 - Algorithms - Winter 2023. 6. Prim’s algorithm. This homework is counted 5% of the course grade. for the purpose of proving polynomial runtime. Repeatedly selecting the lightest edge. in 𝑆𝑖𝑧𝑒𝑅3. T (n) = maxI of size n T (I) Tavg(n) = PI. ca 1 DIJKSTRA’SALGORITHM Single-source shortest path in a graph with non-negative edge weights 2 PROBLEM: SINGLE SOURCE SHORTEST PATHS (SSSP) •Input: graph 𝐺 CS 341: ALGORITHMS Lecture 12: graph algorithms III –DAG testing, topsort, SCC Readings: see website Trevor Brown https://student. This is a good example of a greedy algorithm. Towards the end, we will also study the notion of NP-completeness and prove that many natural parameterized by an integer n, called the size. ca 1 DFS APPLICATION: TESTING WHETHER A GRAPH IS A DAG 2 Back edge: points Recall: nodes become to an ancestor in the DFS forest 3 •Case ⇐: Suppose ∃directed cycle. GRAPHS 2. ca 1 •Input: points 𝑞1,…,𝑞 in 2D space that form a convex -gon •Assume points are sorted clockwise around the center of P •Find: a triangulation of such that the sum of the For CS 341 in Winter 2016 only, do not distribute! 1 CS 341: Algorithms @ Waterloo Last Updated: Mar 21, 2016 Homework 3 Solutions (Written Part) The full mark is 50. Determine if an input graph is a strongly connected graph. So the algorithm terminates, and each 𝒙𝒊 is matched with some 𝒚𝒋 View Test prep - winter2016final. Cheriton School of Computer Science, University of Waterloo Spring 2023 A. So the algorithm terminates, and each 𝒙𝒊 is matched with some 𝒚𝒋 CS 341 Algorithms Keven Qiu Instructor: Mark Petrick Fall 2022. Cheriton School of Computer Science, University of Waterloo Winter 2020 E. there are two corresponding edge-disjoint paths in 𝐺′ CS 341: ALGORITHMS Lecture 20: intractability II –complexity class NP Readings: see website Trevor Brown https://student. We start with a simplified version. Big- O notation We ended the first lecture with definitions for algorithms, for solving a problem, and for the model of computation (Word RAM) that we will use to measure the run time of an algorithm on a given input. Provides a formula for solving many recurrence relations. Tuesdays and Thursdays, 2:30-3:30, DC 3120. This is ( ) --- multiplication of polynomials. Algorithms. 4, 26. This means that there is a directed path P from w to v consisting of tree edges. Tuesdays and Thursdays, 11:30-12:50, MC 2017. Zima (SCS, UW) Module 5: Greedy Algorithms Winter 2020 1 / 14 CS 341: Algorithms (S ’18) Tutorial 3 Solution If vw is a back edge, then v is a descendant of w in the depth-first forest. Wednesdays and Fridays, 11:20-12:50. 11, 38, 6, 21, 20, 17, 14, 9, 7, 5, 8, 34 CS 341: Algorithms @ Waterloo Last Updated: Jan 27, 2023 Homework 2 Due Date: Feb 10, 2023, 11pm You are allowed to discuss with others but are not allowed to use any references other than the course notes and the three reference books. Tuesdays and Thursdays, 1:00-2:20, MC 2017. Idea: determine whether runtime is polynomial in CS341 is an advanced project based course, framed as the natural continuation of CS246 - Mining Massive Data Sets. CS 341: Algorithms (S '20) Tutorial 2 1 Solving recurrences Solve the following recurrences to obtain a closed-form big-Θ expression for T Announcements. 1/1 0/1 0/1 Removing path with flow 1 1/1 2/3 changes flow value from to − 1 1/1 1/1 Every vertex still satisfies 1/1 1/1 conservation of flow an - flow There must exist some Sorry, looks like something is wrong on our end – try again in a few minutes. Jamshidpey (CS, UW) Lec 17: NPC Part So, some algorithms could be. an . Repeat 1 & 2 for the next lowest y-valued point. No subset of sums to 0. ⇝. Case 1: Let 𝐼 be any yes-instance;Find 𝐶 such that 𝑣𝑒𝑟𝑖𝑓𝑦𝐼,𝐶=𝑡𝑟𝑢𝑒. Theorem: At the end of the algorithm, for all , [ ] is exactly the total weight of the shortest. Trivial algorithm yields 𝑂(𝑚) time for each 𝑎𝑖, totaling 𝑂(𝑛𝑚 Formally, Π is undecidable IFF there cannot exist an algorithm such that, for every instance , ( ) returns the correct answer in finite time. divide and conquer (~4 lectures). You must write your own solutions. Recall: median of each row. CS 341: Algorithms Fall 2023 General Information This course studies the major algorithmic design paradigms and mathematical tools for analyzing the running times of algorithms and detecting computational problems for which no efficient deterministic algorithm. Incrementally extend a tree T into an MST, by: Use priority queue to Initializing T to contain any arbitrary node in G store outgoing edges. Minimizing Total Completion Time CS 466/666 Algorithm Design and Analysis Graph Structure. ca 1 •A point dominates everything to the southwest PROBLEM: NON-DOMINATED POINTS + + No other point dominates me So, I am a non-dominated point 2 MORE FORMALLY •Given two points ( 1, 1) and ( 2, 2), Shamos’ algorithm (1975) This selection step preserves the y-sort order Observe PxL and PyL contain the same points (specifically the points with x <= xmid) Moreover PxL is sorted by x while PyL is sorted by y And similarly for PxR, PyR… No need to sort in Recurse! 27 x-coord uniqueness used Somehow we can implement an exhaustive search algorithm as efficient as a greedy algorithm! This is why dynamic programming is so useful and powerful, because it is very systematic and yet it provides very competitive algorithms. CS 341: ALGORITHMS Lecture 14: graph algorithms V –single source shortest path Readings: see website Trevor Brown https://student. It will turn out that all these problems can be solved in 𝑂( I+ J)time, OTHER NOTABLE MST ALGORITHMS. linear time algorithm, which leverages techniques from convex optimization and sophisticated data structures View Test prep - MidtermSolutions. The idea is to create a long “two-way path” for each variable, so that going the path from left to right corresponds to setting the variable to True, while from right to left corresponds to setting it to False. LATEXed byS ̊i ̃bfle ̈lˇi ̊u ̄s Pe›ng. Generate every subset certificate S. CS341: ALGORITHMS (F23) Lecture 1 Trevor Brown https://student. We will also study the theory of NP-completeness in the end. CS 341 – Algorithms. be a no-instance & be any certificate. A simpler implementation with union-by-rank only yields CS 341: ALGORITHMS Lecture 15: graph algorithms VI –all pairs shortest paths Readings: see website Trevor Brown https://student. T (I) = runtime on input I. where. Design a poly-time 𝑖 (𝐼, )algorithm 3. For an efficient union-find algorithm (with union by rank and path compression), we get a total running time for Kruskal’s algorithm of 𝑂𝛼𝑚+𝑛𝑚+𝑛,where 𝛼(𝑥) is the inverse Ackermann function. Worst possible runtime happens if every step in the reduction is a call to the oracle. Equivalently, Π is undecidable IFF, for every algorithm , there exists some input CS 341: ALGORITHMS Lecture 19: intractability I Readings: see website Trevor Brown https://student. Lecture 15: graph algorithms VI – all pairs shortest paths. The tentative schedule is: 1. Jan 27: Homework 2 is posted. Runtime is ( | |), which is in ( 2) = ( + 2) DUMB SUBSET-SUM ALGORITHM: PRETEND YOU’RE AN ORACLE AND MAKE CERTS. worst-case runtime. Zima Midterm Solutions 1 True or false (10 marks) State whether each of CS 341: ALGORITHMS Lecture 16: max flow Readings: CLRS 26. brown@uwaterloo. the disjoint intervals). ca 1 FLOWS AND PATHS 2 FLOWS AND PATHS •Edge-disjoint paths problem •Input: digraph 𝐺=(𝑉,𝐸)and two vertices , ∈𝑉 •Output: A maximal number of edge-disjoint paths in 𝐺 •Paths 𝑃1 and 𝑃2 CS 341: ALGORITHMS Lecture 16: max flow Readings: CLRS 26. Blais, N. ca 1 THE SELECTION PROBLEM 2 THE SELECTION PROBLEM •Input: An array A containing n distinct integer values, and an integer k between 1 and n •Output: The k-th smallest integer in A Naïve algorithm attempt For simplicity, try edge-disjoint path problem first (unit capacities) Greedy idea: find a shortest 𝑠-𝑡 path (to use few edges), then repeat on the remaining edges CS 341: ALGORITHMS Lecture 4: divide & conquer III Readings: see website Trevor Brown https://student. time complexity, computation model, 3-SUM. Imagine sorting each row: /. MASTER THEOREM FOR RECURRENCES. Problem A speci cation of an in nite set of inputs and corresponding outputs. cs CS 341: Algorithms @ Waterloo Last Updated: May 31, 2021 Homework 2 Due Date: June 14, 2021, 11:00pm You are allowed to discuss with others but are not allowed to use any references other than the course notes and the three reference books. CS 341: ALGORITHMS Lecture 12: graph algorithms III –DAG testing, topsort, SCC Readings: see website Trevor Brown https://student. ca 1 FLOWS AND PATHS 2 FLOWS AND PATHS •Edge-disjoint paths problem •Input: digraph 𝐺=(𝑉,𝐸) and two vertices , ∈𝑉 •Output: A maximal number of edge-disjoint paths in 𝐺 •Paths 𝑃1 and 𝑃2 For all pairs of vertices u, v in V: if dist(u,v) < d, add edge uv. But multiplying polynomials of degrees 1, 2 results in a polynomial of CS 341: Algorithms NP-completeness part 2 Armin Jamshidpey ´ Based on lecture notes by Eric Schost, and many other CS 341 instructors David R. Lecture notes. Design a poly-time 𝑣𝑒𝑟𝑖𝑓𝑦(𝐼,𝐶)algorithm. this highly inefficient representation. We will cover elementary rendering algorithms such as rasterization and raytracing, examine mathematical concepts and algorithms for geometric modeling, and then study concepts and algorithms for computer animation. Case 2: Let 𝐼 be any no-instance, and 𝐶 be any certificate;Prove 𝑣𝑒𝑟𝑖𝑓𝑦𝐼,𝐶=𝑓𝑎𝑙𝑠𝑒. ca 1 THE SELECTION PROBLEM 2 THE SELECTION PROBLEM •Input: An array A containing n distinct integer values, and an integer k between 1 and n •Output: The k-th smallest integer in A CS 341: ALGORITHMS (S18) — LECTURE 11 GRAPH EXPLORATION: BREADTH-FIRST SEARCH 3 its neighbours, marking all of the ones that are undiscovered as discovered and ignoring the other ones. Maintain nodes in priority order, ordered by smallest distance. Test connectivity using BFS from any point on the North wall, and checking if any point on the South wall is visited. ca 1 •Consider an undirected graph in which each edge has a weight (or cost) WEIGHTED UNDIRECTED GRAPH d b c a e f g i h j l k 7 2 3 8 11 15 6 18 14 4 1 9 17 10 20 12 13 CS 341: ALGORITHMS Lecture 10: graph algorithms I Readings: see website Trevor Brown https://student. g. representing the runtime of a reduction and its oracle, respectively, on an input of size . average runtime, not used much in. Harms, E. When we add vw to P, a directed cycle is formed. of case 2: . Key insight: after relaxing all, the smallest (that we didn’t already know was optimal) is now optimal. IFF . ca CS 341 Lecture 2 1 Analyzing Algorithms We begin with some de nitions. # elements ≤ 23 is at least 3(5). INDUCTIVE STEP. ca 1 GRAPHS 2 GRAPHS 7 5 8 1 2 3 6 4 •A graph is a pair 𝐺=(𝑉,𝐸) •𝑉contains vertices •𝐸contains edges •An edge connects two distinct vertices , •Also denoted ( , ) CS 341: ALGORITHMS Lecture 15: graph algorithms VI –all pairs shortest paths Readings: see website Trevor Brown https://student. new graph . ca 1 TABLE OF CONTENTS •Course mechanics •Models of computation •Worked example: Bentley’s problem •Multiple solutions, demonstrating different algorithm design techniques •Analyzed in different models of computation 2 COURSE CS 341: ALGORITHMS Lecture 14: graph algorithms V –single source shortest path Readings: see website Trevor Brown https://student. Remind yourself of how the algorithm works (use any source you like). Mar 10: Homework 4 is posted. ca 1 THIS TIME •Finishing TSP reductions •Complexity class NP •Oracles, certificates, polytime verification algorithms 2 RECALL •So far we know •TSP-Dec ≤ 𝑇 TSP 2023-11-21 1 CS 341: ALGORITHMS Lecture 21: intractability III –complexity class NP, poly transformations Readings: see website Trevor Brown https://student. For example, there are interesting questions we can ask about games like chess and Rubik’s cube. Goal: transform. CS 341: ALGORITHMS Lecture 4: divide & conquer III Readings: see website Trevor Brown https://student. Input to verify is , . ca 1 THE CLOSEST PAIR PROBLEM classroom You overworked student When someone near you 2 THE CLOSEST PAIR PROBLEM Input: Set P of n 2D points For all pairs of vertices u, v in V: if dist(u,v) < d, add edge uv. ca 1 THIS TIME •Finishing TSP reductions •Complexity class NP •Oracles, certificates, polytime verification algorithms 2 RECALL •So far we know •TSP-Dec ≤ 𝑇 TSP 2. Jan 2: Course homepage is up. Oct 26, 2011 · [10 marks]Huffman Coding CS 240 you learned about Huffman encoding to build an optimal prefix code for a set of characters with given frequencies. Correctness proof •Case 1: Let 𝐼be any yes-instance; Find such that 𝑖 𝐼, = •Case 2: Let 𝐼be any no-instance, and be any certificate; Prove 𝑖 𝐼, = •Contrapositive of case 2: The key idea is: Consider polynomials ( ) and. 3. Simplified Master Theorem. ca 1 DFS APPLICATION: TESTING WHETHER A GRAPH IS A DAG 2 Back edge: points Recall: nodes become to an ancestor in the DFS forest 3 •Case ⇐: Suppose ∃ directed cycle. , length of an array that we want to work with. CS 341} Lap Chi Lau. global . Verify certificate S. Lecture 22: intractability V – More NPC transformations. ≤ the total weight of the shortest. Please list your collaborators for each question. Find all strongly connected components of a directed graph. Exercise: Write a program to print out an optimal solution (i. (valid + sums to zero) If any certificate S sums to zero, it is a yes-certificate (a proof that the answer to the decision problem is “true CS 341: ALGORITHMS. ca 1. Disclaimer Much of the information on this set of notes is transcribed directly/indirectly from the lectures of CS 341 during Spring 2021 as well as other related resources. from T (and repeatedly extract the minimum that crosses cut (T, V\T) weight one) Visualization: https://www. 2 Trevor Brown https://student. 5) In 2022, researchers found . So, after restructuring, pivot 23 must have at least / elements before and after it This is a good pivot! We recurse on or , and both have size at most /. CS 341: Algorithms. Towards the end, we will also study the notion of NP-completeness and prove that many natural problems are "hard" using polynomial time reductions. Preface. Jan 13: Homework 1 is posted. but exponential . course outline [pdf] Lecture 1 (May 11): course introduction [pdf] [one] [pptx] [pdf] course information and overview. uwaterloo. ca/~cs341 Notable algorithms: mergesort, quicksort, binary search, … Divide-and-Conquer Design Strategy divide: Given a problem instance 𝑰,construct one or more smaller problem instances 𝑰𝟏, …, 𝑰𝒂 Case 1: Let be a yes-instance. Start from lowest y valued point in the strip. Eventually dequeue all nodes (no more enqueues) Each dequeued node. 4. = log . Please note that any term-specific content of this document is decided tentatively at the beginning of the term, and is subject to change. CS 341: ALGORITHMS Lecture 9: dynamic programming III Readings: see website Trevor Brown https://student. This technique is a general approach for solving problems with a three-step approach: Divide: the original problem into smaller subproblems, Conquer View Homework Help - Assignment1Solutions. , = for integer ) Example corresponding algorithm. polynomial . is a subset in that sums to 0. bipartite matching (~3 lectures). CS 341: Algorithms (S 18) E. See the course website for current, up-to-date information. 1/1 0/1 0/1 Removing path with flow 1 1/1 2/3 changes flow value from to − 1 1/1 1/1 Every vertex still satisfies 1/1 1/1 conservation of flow So this is an - flow There must View Tutorial2. Both interesting datasets as well as computational infrastructure (Google Cloud) will be provided to the students by the CS 341: ALGORITHMS Lecture 19: intractability I Readings: see website Trevor Brown https://student. Students will work on Data Mining and Machine Learning algorithms for analyzing very large amounts of data. Wrong! We can find all of the points with the same x as R[1] in linear time. ca/~cs341 trevor. has optimal. the input graph 𝐺 into a . There will be five assignments, contributing to 40% of the course grade. For all practical x, 𝛼𝑥≤5, so this is . Feb 24: Homework 3 is posted. Lecture 12: graph algorithms III – DAG testing, Don’t want to run path finding algorithm on the entire . The first class in a sequence of two classes that introduces students to the theoretical foundations of computer science. 0/1 0/1 1/1 1/1. If there are multiple such points, and some are in Q, then they are. If it is not possible to design an algorithm that solves decision problem Π, then we say Π is undecidable. Notes will usually be posted the day before lecture. homework 1 [pdf] supplementary exercises [pdf] homework 2 [pdf] supplementary exercises [pdf] homework 3 [pdf] supplementary exercises [pdf] CS 341: ALGORITHMS Lecture 3: divide & conquer II Readings: see website Trevor Brown https://student. Shamos’ algorithm (1975) This selection step preserves the y-sort order Observe PxL and PyL contain the same points (specifically the points with x <= xmid) Moreover PxL is sorted by x while PyL is sorted by y And similarly for PxR, PyR… No need to sort in Recurse! 27 x-coord uniqueness used This course provides an introduction to the field of Computer Graphics. ca 1 2 ALL PAIRS SHORTEST PATHS (APSP) PROBLEM 3 5-4 4 2 3 12-1 , = 0 0 0 0 3 from: to: ∞ ∞∞ ∞ 1212 55 ∞ 12 5 4 0 −1 0 3 ∞ 12 5 4 ∞ 0 3 CS 341: ALGORITHMS Lecture 10: graph algorithms I Readings: see website Trevor Brown https://student. Contrapositive. There are more sophisticated algorithms with 𝑂(𝑉2𝐸) and even 𝑂(𝑉3) runtimes (optional: CLRS 26. 2. 1. CS 341: Algorithms University of Waterloo Eric Schost´ eschost@uwaterloo. ca 1 GRAPHS 2 GRAPHS 7 5 8 1 2 3 6 4 •A graph is a pair 𝐺 = (𝑉,𝐸) •𝑉 contains vertices •𝐸 2contains edges •An edge connects two distinct vertices , •Also denoted ( , ) CS 341 - Algorithms - Winter 2023. e. tymculxvqfwqcznumppp