Consider a weighted, directed graph G with n vertices and m edges that have integer weights. A graph walk is a sequence of not-necessarily-distinct vertices vi, v2, --- , Vk such that each pair of consecutive vertices vi, Vi+1 are connected by an edge. This is similar to a path, except a walk can have repeated vertices and edges. The length of a walk in a weighted graph is the sum of the weights of the edges in the walk. Let s, t be given vertices in the graph, and L be a positive integer. We are interested counting the number of walks from s to t of length exactly L. Assume all the edge weights are positive. Describe an algorithm that computes the number of graph walks from sto t of length exactly L in O((n+ m)L) time. Prove the correctness and analyze the running time. (Hint: Dynamic Programming solution)

Consider a weighted, directed graph G with n vertices and m edges that have integer weights. A graph walk is a sequence of not-necessarily-distinct vertices vi, v2, --- , Vk such that each pair of consecutive vertices vi, Vi+1 are connected by an edge. This is similar to a path, except a walk can have repeated vertices and edges. The length of a walk in a weighted graph is the sum of the weights of the edges in the walk. Let s, t be given vertices in the graph, and L be a positive integer. We are interested counting the number of walks from s to t of length exactly L. Assume all the edge weights are positive. Describe an algorithm that computes the number of graph walks from sto t of length exactly L in O((n+ m)L) time. Prove the correctness and analyze the running time. (Hint: Dynamic Programming solution)

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Consider a weighted, directed graph G with n vertices and m edges that have
integer weights. A graph walk is a sequence of not-necessarily-distinct vertices v1,
v2, ... , Vk such that each pair of consecutive vertices Vi, Vi+1 are connected by an
edge. This is similar to a path, except a walk can have repeated vertices and edges.
The length of a walk in a weighted graph is the sum of the weights of the edges in
the walk. Let s, t be given vertices in the graph, and L be a positive integer. We are
interested counting the number of walks from s to t of length exactly L.
Assume all the edge weights are positive. Describe an algorithm that computes
the number of graph walks from s to t of length exactly L in O((n+ m)L) time. Prove
the correctness and analyze the running time. (Hint: Dynamic Programming
solution)

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