(15 pts.) We are given a graph G = /,E); G could be a directed graph or undirected graph. Let M be the adjacency matrix of G. Let n be the number of vertices so that the matrix M is n x n matrix. For any matrix A, let us denote the element of i-th row and j-th column of the matrix A by A[i, j]. 1. Consider the square of the adjacency matrix M. For all i and j, show that M²[i, j] is the number of different paths of length 2 from the i-th vertex to the j-th vertex. It should be explained or proved as clearly as possible. 2. For any positive integer k, show that Mk[i, j] is the number of different paths of length k from the i-th vertex to the j-th vertex. You may use induction on k to prove it. 3. Assume that we are given a positive integer k. Design an algorithm to find the number of different paths of length k from the i-th vertex to j-th vertex for all pairs of (i, j). The time complexity of your algorithm should be O(n³ log k). You can get partial credits if you design an algorithm of O(n³k).

(15 pts.) We are given a graph G = /,E); G could be a directed graph or undirected graph. Let M be the adjacency matrix of G. Let n be the number of vertices so that the matrix M is n x n matrix. For any matrix A, let us denote the element of i-th row and j-th column of the matrix A by A[i, j]. 1. Consider the square of the adjacency matrix M. For all i and j, show that M²[i, j] is the number of different paths of length 2 from the i-th vertex to the j-th vertex. It should be explained or proved as clearly as possible. 2. For any positive integer k, show that Mk[i, j] is the number of different paths of length k from the i-th vertex to the j-th vertex. You may use induction on k to prove it. 3. Assume that we are given a positive integer k. Design an algorithm to find the number of different paths of length k from the i-th vertex to j-th vertex for all pairs of (i, j). The time complexity of your algorithm should be O(n³ log k). You can get partial credits if you design an algorithm of O(n³k).

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

2. (15 pts.)
the adjacency matrix of G. Let n be the number of vertices so that the matrix M is n x n matrix. For any
matrix A, let us denote the element of i-th row and j-th column of the matrix A by A[i, j].
We are given a graph G = (V, E); G could be a directed graph or undirected graph. Let M be
1. Consider the square of the adjacency matrix M. For all i and j, show that M²[i, j] is the number of
different paths of length 2 from the i-th vertex to the j-th vertex. It should be explained or proved as
clearly as possible.
2. For any positive integer k, show that Mk[i, j] is the number of different paths
vertex to the j-th vertex. You may use induction on k to prove it.
length k from the i-th
3. Assume that we are given a positive integer k. Design an algorithm to find the number of different
paths of length k from the i-th vertex to j-th vertex for all pairs of (i, j). The time complexity of
your
algorithm should be O(n³ log k). You can get partial credits if you design an algorithm of O(n°k).

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// Java program for implementation

// of Lagrange's Interpolation

import java.util.*;

class GFG

{

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