Consider the following graph: 2 5 6 4 NI 2 E /10 H 5 F Find the shortest path from node C to all the other nodes of the graph. Use the alphabetical order when you have two possibilities with the same weight. (Show your work). For every node of the graph, give the list of edges of the shortest path from A to that node as well as its weight 3 2

Consider the following graph: 2 5 6 4 NI 2 E /10 H 5 F Find the shortest path from node C to all the other nodes of the graph. Use the alphabetical order when you have two possibilities with the same weight. (Show your work). For every node of the graph, give the list of edges of the shortest path from A to that node as well as its weight 3 2

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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We are given a graph G = (V, E); G could be a directed graph or undirected graph. Let M bethe adjacency matrix of G. Let n be the number of vertices so that the matrix M is n ×n matrix. For anymatrix 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 M 2[i, j] is the number ofdifferent paths of length 2 from the i-th vertex to the j-th vertex. It should be explained or proved asclearly as possible.2. For any positive integer k, show that M k[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).
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.]
You are given a simple connected undirected graph with N nodes(or vertices) for computing a Minimum Spanning Tree (MST) of the graph. Write down the upper and lower bound of the possible number of edges of that MST. Now consider the same for a “simple disconnected undirected graph”. What is the lower and upper bound now?
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Given a digraph, find a bitonic shortest path from s to every other vertex (if one exists). A path is bitonic if there is an intermediate vertex v suchthat the edges on the path from s to v are strictly increasing and the edges on the pathfrom v to t are strictly decreasing. The path should be simple
Consider an undirected graph G with 100 nodes. The maximum number of edges to be included in G so that the graph is not connected is
Given a digraph, find a bitonic shortest path from s to every other vertex (if one exists). A path is bitonic if there is an intermediate vertex v suchthat the edges on the path from s to v are strictly increasing and the edges on the pathfrom v to t are strictly decreasing. The path should be simple (no repeated vertices).
Let G = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ∈ V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-set
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