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Introduction to algorithms class. Please show your work. And please DO NOT copy any solutions to this problem that may already exist on Chegg Study.
Introduction to algorithms class. Please show your work. And please DO NOT copy any solutions to this problem that may already exist on Chegg Study.
A walk of length k in a graph is a sequence of vertices vo, V1, ... , Vk such that v is a neighbor of vi+1 for i = 0, 1, 2, ...,k - 1. Suppose the product of two n x n matrices can be computed in time O(n^) for a constant w > 2. Give an algorithm that counts the number of walks of length k in a graph with n vertices in time O(nW log k). HINT: If A is the adjacency matrix, prove that the (i, j)'th entry of Ak is exactly the number of walks of length k that start at i and end at j. Repeatedly square the adjacency matrix to compute Ak. A walk of length k in a graph is a sequence of vertices vo, V1, ... , Vk such that v is a neighbor of vi+1 for i = 0, 1, 2, ...,k - 1. Suppose the product of two n x n matrices can be computed in time O(n^) for a constant w > 2. Give an algorithm that counts the number of walks of length k in a graph with n vertices in time O(nW log k). HINT: If A is the adjacency matrix, prove that the (i, j)'th entry of Ak is exactly the number of walks of length k that start at i and end at j. Repeatedly square the adjacency matrix to compute AkStep by Step Solution
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