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 22.3, Problem 9E
Program Plan Intro
To give a counter example to the conjecture that if a directed graph G contains a path from u to v then any depth-first search must result in
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Suppose are you given an undirected graph G = (V, E) along with three distinct designated vertices u, v, and w. Describe and analyze a polynomial time algorithm that determines whether or not there is a simple path from u to w that passes through v. [Hint: By definition, each vertex of G must appear in the path at most once.]
Given 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.
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).
Chapter 22 Solutions
Introduction to Algorithms
Ch. 22.1 - Prob. 1ECh. 22.1 - Prob. 2ECh. 22.1 - Prob. 3ECh. 22.1 - Prob. 4ECh. 22.1 - Prob. 5ECh. 22.1 - Prob. 6ECh. 22.1 - Prob. 7ECh. 22.1 - Prob. 8ECh. 22.2 - Prob. 1ECh. 22.2 - Prob. 2E
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- Give a counterexample to the following claim and justify it: Claim: If a directed graph G contains a path from u to v, then any depth-first search must result in pre(v) ≤ post(u). That is, provide a graph G and vertices u, v where this claim does not hold, and explain why not.arrow_forwardSuppose that the graph G has n nodes and m edges, and that the completion of a depth-first search takes n minutes.arrow_forwardProve that if G is a connected graph, then there always is a closed walk that passes through each edge at least once and at most twice.arrow_forward
- LONG-Route is the issue of finding whether or not there is a simple path in G from u to v with a length of at least k given the inputs (G, u, v, k), where G is a graph, u and v are vertices, and k is an integer.Demonstrate that LONG-PATH is an NP-complete problem.arrow_forwardNot handwritten Consider a graph G = (V, E) with nonnegative integer function c : V → N. Find an augmenting pathmethod to compute a subgraph H = (V, F) (F ⊆ E) with maximum number of edges such that for everyv ∈ V , deg(v) ≤ c(v).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_forward
- Assume 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_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_forwardGiven a directed graph G=(V, E), design an algorithm to find out whether there is a route between two nodes, say, u and v. Your design should be based on depth first search and should be given in pseudo code.arrow_forward
- Say that a graph G has a path of length three if there exist distinct vertices u, v, w, t with edges (u, v), (v, w), (w, t). Show that a graph G with 99 vertices and no path of length three has at most 99 edges.arrow_forwardWrite a function in a directed graph represented by adjacency lists that returns true (1) if an edge exists between two provided vertices u and v and false (0) otherwise.arrow_forwardConsider 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_forward
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