Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 34.1, Problem 1E
Program Plan Intro
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Given an undirected graph G = (V, E), a vertex cover is a subset of V
so that every edge in E has at least one endpoint in the vertex cover.
The problem of finding a minimum vertex cover is to find a vertex cover
of the smallest possible size. Formulate this problem as an integer linear
programming problem.
3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E)
with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v)
such that 1
Given a directed graph with positive edge lengths (weights) and two distinct vertices u and v in the graph, the “all-pairs u-constrained v-avoiding shortest path problem” is the problem of computing for each pair of vertices i and j the length of the shortest path from i to j that goes through the vertex u and avoids vertex v. If no such path exists, the answer is ∞. Describe an algorithm that takes a graph G = (V, E) and vertices u and v as input parameters and computes values L(i,j) that represent the length of u-constrained v-avoiding shortest path from i to j for all 1 ≤ i, j ≤ |V |, i ̸= u, j ̸= u, i ̸= v, j ̸= v. Prove your algorithm correct. Your algorithm must have running time in O(|V |2). Detailed pseudocode is required.
Chapter 34 Solutions
Introduction to Algorithms
Ch. 34.1 - Prob. 1ECh. 34.1 - Prob. 2ECh. 34.1 - Prob. 3ECh. 34.1 - Prob. 4ECh. 34.1 - Prob. 5ECh. 34.1 - Prob. 6ECh. 34.2 - Prob. 1ECh. 34.2 - Prob. 2ECh. 34.2 - Prob. 3ECh. 34.2 - Prob. 4E
Ch. 34.2 - Prob. 5ECh. 34.2 - Prob. 6ECh. 34.2 - Prob. 7ECh. 34.2 - Prob. 8ECh. 34.2 - Prob. 9ECh. 34.2 - Prob. 10ECh. 34.2 - Prob. 11ECh. 34.3 - Prob. 1ECh. 34.3 - Prob. 2ECh. 34.3 - Prob. 3ECh. 34.3 - Prob. 4ECh. 34.3 - Prob. 5ECh. 34.3 - Prob. 6ECh. 34.3 - Prob. 7ECh. 34.3 - Prob. 8ECh. 34.4 - Prob. 1ECh. 34.4 - Prob. 2ECh. 34.4 - Prob. 3ECh. 34.4 - Prob. 4ECh. 34.4 - Prob. 5ECh. 34.4 - Prob. 6ECh. 34.4 - Prob. 7ECh. 34.5 - Prob. 1ECh. 34.5 - Prob. 2ECh. 34.5 - Prob. 3ECh. 34.5 - Prob. 4ECh. 34.5 - Prob. 5ECh. 34.5 - Prob. 6ECh. 34.5 - Prob. 7ECh. 34.5 - Prob. 8ECh. 34 - Prob. 1PCh. 34 - Prob. 2PCh. 34 - Prob. 3PCh. 34 - Prob. 4P
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- Given an undirected graph G = <V,E>, a vertex cover is a subset of vertices S V such that for each edge (u,v) belongs to E, either u S or v S or both. The Vertex Cover Problem is to find minimum size of the set S. Consider the following algorithm to Vertex Cover Problem: (1) Initialize the result as {} (2) Consider a set of all edges in given graph. Let the set be E’. (3) Do following while E’ is not empty ...a) Pick an arbitrary edge (u,v) from set E’ and add u and v to result ...b) Remove all edges from E which are either incident on u or v. (4) Return result. It claim that this algorithm is exact for undirected connected graphs. Is this claim True or False? Justify the answer.arrow_forwardThe minimum vertex cover problem is stated as follows: Given an undirected graph G = (V, E) with N vertices and M edges. Find a minimal size subset of vertices X from V such that every edge (u, v) in E is incident on at least one vertex in X. In other words you want to find a minimal subset of vertices that together touch all the edges. For example, the set of vertices X = {a,c} constitutes a minimum vertex cover for the following graph: a---b---c---g d e Formulate the minimum vertex cover problem as a Genetic Algorithm or another form of evolutionary optimization. You may use binary representation, OR any repre- sentation that you think is more appropriate. you should specify: • A fitness function. Give 3 examples of individuals and their fitness values if you are solving the above example. • A set of mutation and/or crossover and/or repair operators. Intelligent operators that are suitable for this particular domain will earn more credit. • A termination criterion for the…arrow_forward5. Fleury's algorithm is an optimisation solution for finding a Euler Circuit of Euler Path in a graph, if they exist. Describe how this algorithm will always find a path or circuit if it exists. Describe how you calculate if the graph is connected at each edge removal. Fleury's Algorithm: The algorithm starts at a vertex of v odd degree, or, if the graph has none, it starts with an arbitrarily chosen vertex. At each step it chooses the next edge in the path to be one whose deletion would not disconnect the graph, unless there is no such edge, in which case it picks the remaining edge (a bridge) left at the current vertex. It then moves to the other endpoint of that edge and adds the edge to the path or circuit. At the end of the algorithm there are no edges left ( or all your bridges are burnt). (NOTE: Please elaborate on the answer and explain. Please do not copy-paste the answer from the internet or from Chegg.)arrow_forward
- A Vertex Cover of an undirected graph G is a subset of the nodes of G,such that every edge of G touches one of the selected nodes.The VERTEX-COVER problem is to decide if a graph G has a vertex cover of size k.VERTEX-COVER = { <G,k> | G is an undirected graph with a k-node vertex cover }The VC3 problem is a special case of the VERTEX-COVER problem where the value of k is fixed at 3.VERTEX-COVER 3 = { <G> | G is an undirected graph with a 3-node vertex cover }Use parts a-b below to show that Vertex-Cover 3 is in the class P.a. Give a high-level description of a decider for VC3.A high-level description describes an algorithmwithout giving details about how the machine manages its tape or head.b. Show that the decider in part a runs in deterministic polynomial time.arrow_forwardGiven a directed graph with positive edge lengths and two distinct vertices uand v in the graph, the “all-pairs uv-constrained shortest path problem” is the problemof computing for each pair of vertices i and j the length of the shortest path from i toj that goes through the vertex u or through the vertex v. If no such path exists, theanswer is ∞. Describe an algorithm that takes a graph G = (V, E) and vertices u and v asinput parameters and computes values L(i, j) that represent the length of uv-constrainedshortest path from i to j for all 1 ≤ i, j ≤ |V|, i ! = u, j ! = u, i != v, j ! = v. Provide clearpseudocode solution. Prove your algorithm correct. Your algorithm must have runningtime in O(|V| ^2).arrow_forward(1) T F Given a directed graph G and a vertex v in the graph, breath first search (BFS) can be used to detect if v is part of a cycle in the graph. (2) T F Let P be a shortest path from some vertex s to some other vertex t in a directed graph. If the weight of each edge in the graph is decreased by one, then P will still be a shortest path from s to t. (3) T F edge Kruskal's algorithm is always correct even in graphs with negative weights. (4) T F For any flow network, there is only one unique way to assign flow value to the edges so as to achieve the maximum flow for the network. NP problems are those problems that cannot be solved in polynomial (5) T F time.arrow_forward
- Part 2: Random GraphsA tournament T is a complete graph whose edges are all oriented. Given a completegraph on n vertices Kn, we can generate a random tournament by orienting each edgewith probability 12 in each direction.Recall that a Hamiltonian path is a path that visits every vertex exactly once. AHamiltonian path in a directed graph is a path that follows the orientations of thedirected edges (arcs) and visits every vertex exactly once. Some directed graphs havemany Hamiltonian paths.In this part, we give a probabilistic proof of the following theorem:Theorem 1. There is a tournament on n vertices with at least n!2n−1 Hamiltonian paths.For the set up, we will consider a complete graph Kn on n vertices and randomlyorient the edges as described above. A permutation i1i2 ...in of 1,2,...,n representsthe path i1 −i2 −···−in in Kn. We can make the path oriented by flipping a coin andorienting each edge left or right: i1 ←i2 →i3 ←···→in.(a) How many permutations of the vertices…arrow_forwardGiven a graph that is a tree (connected and acyclic). (I) Pick any vertex v.(II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance.(III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are truea. p is the longest path in the graphb. p is the shortest path in the graphc. p can be calculated in time linear in the number of edges/verticesarrow_forwardQuestion 9. Present an algorithm for the following problem. The input is a weighted graph G, two vertices s and t, and a positive number k. The goal is to find a path from s to t such that all edges along the path have weight ≤ k (if there is such a path), or to print "no good path", if there is no such path.arrow_forward
- Let G = (X ∪ Y, E) be a bipartite graph such that the vertices are partitioned into two groups Xand Y , and each edge has one end point in X and one end point in Y .A 2-1 generalized matching is a set of edges S ⊂ E satisfying the following two conditions:1. Every vertex in X belongs to at most two edges in S.2. Every vertex in Y belongs to at most one edge in S.Give an algorithm to find the size (number of edges) of maximum 2-1 generalized matchingarrow_forwardThe Floyd-Warshall algorithm is a dynamic algorithm for searching the shortest path in a graph. Each vertex pair has its assigned weight. You are asked to draw the initial directed graph and show the tables for each vertex from Mo to Ms by finding all the shortest paths. Below is the algorithm as a guide. Algorithm 1: Pseudocode of Floyd-Warshall Algorithm Data: A directed weighted graph G(V, E) Result: Shortest path between each pair of vertices in G for each de V do | distance|d][d] «= 0; end for each edge (s, p) € E do | distance[s][p] + weight(s, p); end n = cardinality(V); for k = 1 to n do for i = 1 to n do for j = 1 to n do if distancefi][j] > distance/i][k] + distance/k][j] then | distance i]lj] + distancefi|[k] + distance/k|[j]; end end end end Consider the relation R = {(1,4) =4, (2,1)=3, (2,5)=-3, (3,4)=2, (4,2)=1, (4,3)=1, (5,4)=2 } on A = (1,2,3,4,5) solve the Floyd-Warshall Algorithm.arrow_forwardGiven a graph that is a tree (connected and acyclic). (1) Pick any vertex v. (II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance. (III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are true a. p is the longest path in the graph b. p is the shortest path in the graph c. p can be calculated in time linear in the number of edges/vertices a,c a,b a,b,c b.carrow_forward
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