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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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.
Problem 2. Give a counterexample to the conjecture that if a directed graph G contains a
path from u to v, then any depth-first search must result in v.d ≤ u.f.
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.]
Chapter 22 Solutions
Introduction to Algorithms
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- Let G = (V, E) be an edge-weighted undirected multigraph, that is, an undirected graph with weighted edges that may have selfloops and parallel edges. We call an edge e of G futile if either e is a selfloop or if e is not the lightest edge between its two endpoints (equivalently, it is not the lightest edges among parallel edges). Since all futile edges can be considered as red edges in the MST Meta-Algorithm (every futile edge is the heaviest edge in a cycle of length 1 or 2), we can remove all futile edges from G without changing its minimum spanning tree. The operation of simplifying G removes all futile edges. Show that any edge-weighted multigraph with |V | vertices and |E| edges can be simplified in O(|V | + |E|) time. Describe an algorithm for doing so and present data structures used.arrow_forwardou are given a directed graph G = (V, E) and two vertices s and t. Moreover, each edge of this graph is colored either blue or red. Your goal is to find whether there is at least one path from s to t such that all red edges in this path appear after all blue edges (the path may not contain any blue edges or any red edges, but if it has both types of edges, all red edges should appear after all blue edges). Design and analyze an algorithm for solving this problem in O(n + m) time.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_forward
- Given a graph G = (V, E) find whether there exists an independent set of size 8. Is the problem N-P complete or not. Why?arrow_forwardConsider a connected, undirected graph G with n vertices and m edges. The graph G has a unique cycle of length k (3 <= k <= n). Prove that the graph G must contain at least k vertices of degree 2.arrow_forwardGiven any directed graph G, with weighted edges and marked vertices s and t, there is a known linear time algorithm to compute the longest path from s to t in G.arrow_forward
- Suppose we have a graph G = (V, E) with m edges. Prove that there exists a partition of V into three subsets A, B, C such that there are 2m edges between these subsets (i.e. between A and B, between B and C, or between A and C). 3arrow_forwardA Hamiltonian path on a directed graph G = (V, E) is a path that visits each vertex in V exactly once. Consider the following variants on Hamiltonian path: (a) Give a polynomial-time algorithm to determine whether a directed graph G contains either a cycle or a Hamiltonian path (or both). Given a directed graph G, your algorithm should return true when a cycle or a Hamiltonian path or both and returns false otherwise. (b) Show that it is NP-hard to decide whether a directed graph G’ contains both a cycle and a Hamiltonian Path, by giving a reduction from the HAMILTONIAN PATH problem: given a graph G, decide whether it has a Hamiltonian path. (Recall that the HAMILTONIAN PATH problem is NP-complete.)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
- Let G be a graph with n vertices. The k-coloring problem is to decide whether the vertices of G can be labeled from the set {1, 2, ..., k} such that for every edge (v,w) in the graph, the labels of v and w are different. Is the (n-4)-coloring problem in P or in NP? Give a formal proof for your answer. A 'Yes' or 'No' answer is not sufficient to get a non-zero mark on this question.arrow_forwardProve by induction that for any connected graph G with n vertices and m edges, we have n< m + 1.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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