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 26.4, Problem 5E
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Kruskal's algorithm can return different spanning trees for the input Graph G.Show that for every minimal spanning tree T of G, there is an execution of the algorithm that gives T as a result.
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Suppose X is a subset of vertices in a G = (V,E). Give a polynomial-time algorithm to test whether there exists a matching in G covering X.
The reverse of a directed graph G = (V,E) is another directed graph G^R = (V,E^R) on the same
vertex set, but with all edges reversed; that is, E^R = {(v, u) : (u, v) ∈ E}.Give a linear-time algorithm for computing the reverse of a graph in adjacency list format.
Chapter 26 Solutions
Introduction to Algorithms
Ch. 26.1 - Prob. 1ECh. 26.1 - Prob. 2ECh. 26.1 - Prob. 3ECh. 26.1 - Prob. 4ECh. 26.1 - Prob. 5ECh. 26.1 - Prob. 6ECh. 26.1 - Prob. 7ECh. 26.2 - Prob. 1ECh. 26.2 - Prob. 2ECh. 26.2 - Prob. 3E
Ch. 26.2 - Prob. 4ECh. 26.2 - Prob. 5ECh. 26.2 - Prob. 6ECh. 26.2 - Prob. 7ECh. 26.2 - Prob. 8ECh. 26.2 - Prob. 9ECh. 26.2 - Prob. 10ECh. 26.2 - Prob. 11ECh. 26.2 - Prob. 12ECh. 26.2 - Prob. 13ECh. 26.3 - Prob. 1ECh. 26.3 - Prob. 2ECh. 26.3 - Prob. 3ECh. 26.3 - Prob. 4ECh. 26.3 - Prob. 5ECh. 26.4 - Prob. 1ECh. 26.4 - Prob. 2ECh. 26.4 - Prob. 3ECh. 26.4 - Prob. 4ECh. 26.4 - Prob. 5ECh. 26.4 - Prob. 6ECh. 26.4 - Prob. 7ECh. 26.4 - Prob. 8ECh. 26.4 - Prob. 9ECh. 26.4 - Prob. 10ECh. 26.5 - Prob. 1ECh. 26.5 - Prob. 2ECh. 26.5 - Prob. 3ECh. 26.5 - Prob. 4ECh. 26.5 - Prob. 5ECh. 26 - Prob. 1PCh. 26 - Prob. 2PCh. 26 - Prob. 3PCh. 26 - Prob. 4PCh. 26 - Prob. 5PCh. 26 - Prob. 6P
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- Prove that in a breadth-first search on a undirected graph G, every edge iseither a tree edge or a cross edge, where x is neither an ancestor nor descendant of y, in cross edge (x, y).arrow_forwardImplement the following algorithm for connectivity of undirected graphs and produce the attached output.(Both algorithm and output attached below)arrow_forwardSuppose you are given a connected weighted undirected graph, G, with n vertices and m edges, such that the weight of each edge in G is an integer in the interval [1, c], for a fixed constant c > 0. Show how to solve the single-source shortest-paths problem, for any given vertex v, in G, in time O(n + m). Please don't copy and paste the other answers to this question. I'm posting it because the other answers either don't answer the question or have a complicated way of solving the problem.arrow_forward
- Connecting a Graph, Let G be an undirected graph. Give a linear time algorithm to compute the smallest number of edges that one would need to add to G in order to make it a connected graph??arrow_forwardGive a high-level analysis of the running time of depth-first-search, assuming that the graph G has n nodes and m edges.arrow_forwardwe have a graph with optimal coloring. how would you prove that every color in said coloring colors a vertex which has neighbours in all colors except its own?arrow_forward
- Kruskal's method may generate multiple spanning trees for the same input graph G \sdepending on how it breaks ties when the edges are sorted into order. Show that for any least spanning tree T of G , there is a technique to arrange the edges of G in Kruskal's algorithm such that the algorithm produces T .arrow_forwardConsider a graph G that has k vertices and k −2 connected components,for k ≥ 4. What is the maximum possible number of edges in G? Proveyour answer.arrow_forwardConsider a directed graph with n vertices that is represented using an adjacency list. Describe an algorithm to compute the outdegree of every vertex. what is the worst case complexity of your algorithm by making bound as tight as possible. Explain.arrow_forward
- When we want to calculate the shortest paths from a vertex using the Bellman-Ford algorithm, it is possible to stop early and not do all |V| - 1 iterations on graphs without a negative cycle. How can we modify the Bellman-Ford Algorithm so that it stops early when all distances are correct?arrow_forwardGiven a directed graph G=(V,E) with positive weights in the vertex and two subsets S and T of V, propose an algorithm with worst case time complexity O(|E| * log |V|) to find the minimum path of some vertex of S to some vertex of Tarrow_forwardConsider a graph G that is comprised only of non-negative weight edges such that (u, v) € E, w(u, w) > 0. Is it possible for Bellman-Ford and Dijkstra's algorithm to produce different shortest path trees despite always producing the same shortest-path weights? Justify your answer.arrow_forward
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