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 1.1, Problem 1E
Program Plan Intro
To describe a real world situation that uses sorting and also give a situation that requires computation of convex hull.
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You are organizing a programming competition, where contestants implement Dijkstra's algorithm. Given adirected graph G = (V, E) with integer-weight edges and a starting vertex s ∈ V , their programs are supposedto output triplets (v, v.d, v.π) for each vertex v ∈ V . Design an O(V +E) time algorithm that takes as inputthe original graph G in both adjacency matrix (G.M) and adjacency list (G.Adj) representations, startingvertex s, and the output of a contestant's program (given as an array A of triplets), and returns whetherA is the correct output for G. Write down the pseudocode for your algorithm, explain why it correctlyveri es the output, and analyze your algorithm's running time. You may assume that all edge weights of the input graph provided to the contestantsare nonnegative and A (the output of their programs) is in the valid format, i.e., you don't need to verifythat A is actually an array of triplets, with v and v.π being valid vertices and v.d being an integer.Can you…
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In this question you will explore Graph Colouring algorithms. Given a graph G, we say that G is k-colourable if every vertex of G can be assigned one of k colours so that for every pair u, v of adjacent vertices, u and v are assigned different colours. The chromatic number of a graph G, denoted by χ(G), is the smallest integer k for which graph G is k-colorable. To show that χ(G) = k, you must show that the graph is k-colourable and that the graph is not (k − 1)-colourable.
Question: It is NP-complete to determine whether an arbitrary graph has chromatic number k, where k ≥ 3. However, determining whether an arbitrary graph has chromatic number 2 is in P. Given a graph G on n vertices, create an algorithm that will return TRUE if χ(G) = 2 and FALSE if χ(G) 6= 2. Clearly explain how your algorithm works, why it guarantees the correct output, and determine the running time of your algorithm.
Chapter 1 Solutions
Introduction to Algorithms
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