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 24.5, Problem 8E
Program Plan Intro
To perform infinite no. of relaxation of an edge of the weighted directed graph G that contains negative weight cycle and every relaxation can cause shortest path to change.
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Let G = (V, E) denote an weighted undirected graph, in which every edge has unit weight, and let T = (V, E') denote the minimum spanning tree of G. Prove formally that for all u, v ∈ V , the path between u and v in tree T is unique
Prove Proposition : For any vertex v reachable from s, BFS computes a shortest path from s to v (no path from s to v has fewer edges).
Given 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 T
Chapter 24 Solutions
Introduction to Algorithms
Ch. 24.1 - Prob. 1ECh. 24.1 - Prob. 2ECh. 24.1 - Prob. 3ECh. 24.1 - Prob. 4ECh. 24.1 - Prob. 5ECh. 24.1 - Prob. 6ECh. 24.2 - Prob. 1ECh. 24.2 - Prob. 2ECh. 24.2 - Prob. 3ECh. 24.2 - Prob. 4E
Ch. 24.3 - Prob. 1ECh. 24.3 - Prob. 2ECh. 24.3 - Prob. 3ECh. 24.3 - Prob. 4ECh. 24.3 - Prob. 5ECh. 24.3 - Prob. 6ECh. 24.3 - Prob. 7ECh. 24.3 - Prob. 8ECh. 24.3 - Prob. 9ECh. 24.3 - Prob. 10ECh. 24.4 - Prob. 1ECh. 24.4 - Prob. 2ECh. 24.4 - Prob. 3ECh. 24.4 - Prob. 4ECh. 24.4 - Prob. 5ECh. 24.4 - Prob. 6ECh. 24.4 - Prob. 7ECh. 24.4 - Prob. 8ECh. 24.4 - Prob. 9ECh. 24.4 - Prob. 10ECh. 24.4 - Prob. 11ECh. 24.4 - Prob. 12ECh. 24.5 - Prob. 1ECh. 24.5 - Prob. 2ECh. 24.5 - Prob. 3ECh. 24.5 - Prob. 4ECh. 24.5 - Prob. 5ECh. 24.5 - Prob. 6ECh. 24.5 - Prob. 7ECh. 24.5 - Prob. 8ECh. 24 - Prob. 1PCh. 24 - Prob. 2PCh. 24 - Prob. 3PCh. 24 - Prob. 4PCh. 24 - Prob. 5PCh. 24 - Prob. 6P
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- Suppose G = (V, E) is an undirected connected weighted graph such that all its edge weightsare distinct. Prove that that the minimum spanning tree of G is unique.arrow_forwardGiven a directed graph G(V, E), find an O(|V | + |E|) algorithm that deletes at most half of its edges so that the resulting graph doesn’t contain any directed cycles. Also follow up with proof of correctness.arrow_forwardDraw a (simple) directed weighted graph G with 8 vertices and 18 edges, such that G contains a minimum-weight cycle with at least 4 edges. Show that the Bellman-Ford algorithm will find this cycle.arrow_forward
- Consider a directed graph G with a starting vertex s, a destination t, and nonnegative edge lengths. Under what conditions is the shortest s-t path guaranteed to be unique? a) When all edge lengths are distinct positive integers. b) When all edge lengths are distinct powers of 2. c) When all edge lengths are distinct positive integers and the graph G contains no directed cycles. d) None of the other options are correct.arrow_forwardAssume that we are given an undirected graph G=(V,E). Consider that Dijkstra's algorithm found a shortest path in G, called SP, between two nodes A and X of V. Is it true or false that if we reverse the nodes on SP, we get a shortest path from X to A? Prove or disprove.arrow_forwardG = (V,E,W) is a weighted connected (undirected) graph where all edges have distinct weights except two edges e and e′ which have the same weight. Suppose there is a Minimum Spanning Tree of G containing both e and e′. Prove that G has a unique Minimum Spanning Tree.arrow_forward
- Algorithm : (1. Single-destination Shortest Path, 2. Bellman-Ford, 3. Negative-Weight Cycles), Dynamic Programming Define and prove a recurrence for the following problem: Given a directed graph G = (V, E) with edge weight function w : E → R and a source vertex s ∈ V , find a shortest path from s to v for every vertex v ∈ V .arrow_forwardDesign a linear-time algorithm for solving the single source shortest path problem on a given graph G, such that weights associated with edges are drawn from [1/2, 1/3, 1/5]arrow_forwardPropose an approximation algorithm for solving the independent-setproblem where an independent set of a graph G = (V, E) is a subset V’ is a subset of V of verticessuch that each edge in E is incident on at most one vertex in V’. Theindependent-set problem is to find a maximum-size independent set in G.arrow_forward
- Suppose you are given a connected graph G, with edge costs that are alldistinct. Prove that G has a unique minimum spanning tree.arrow_forwardGiven a weighted line graph (undirected connected graph, all vertices of degree 2, except two endpoints which have degree 1), devisean algorithm that preprocesses the graph in linear time and can return the distance ofthe shortest path between any two vertices in constant time.arrow_forwardGive a 4-approximation algorithm for finding a maximum cut in a directed graph D=(V;A).arrow_forward
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