2 Consider the following directed graphs on a set V of all integers from 0 to and including 99. G₁ = (V, E₁) where E₁ is the set of all pairs (a, b) V x V where a mod 5=b mod 5 G₂ (V, E₂) where E2 is the set of all pairs (a, b) VXV where (a - b) mod 5 = 1 G3 = (V.E3) where E3 is the set of all pairs (a, b) € V x V where a - b ≤0 G is a DAG on V. Fill in the number of connected strongly and weakly connected components for each of the graphs. There are no negative points for incorrect answers. If there is no single correct answer (since G₁ can be several different graphs) then fill in undefined. Otherwise fill in a number. G₁ has G₂ has G3 has G₁ has weakly connected components and strongly connected components. weakly connected components and strongly connected components. weakly connected components and strongly connected components. weakly connected components and strongly connected components

2 Consider the following directed graphs on a set V of all integers from 0 to and including 99. G₁ = (V, E₁) where E₁ is the set of all pairs (a, b) V x V where a mod 5=b mod 5 G₂ (V, E₂) where E2 is the set of all pairs (a, b) VXV where (a - b) mod 5 = 1 G3 = (V.E3) where E3 is the set of all pairs (a, b) € V x V where a - b ≤0 G is a DAG on V. Fill in the number of connected strongly and weakly connected components for each of the graphs. There are no negative points for incorrect answers. If there is no single correct answer (since G₁ can be several different graphs) then fill in undefined. Otherwise fill in a number. G₁ has G₂ has G3 has G₁ has weakly connected components and strongly connected components. weakly connected components and strongly connected components. weakly connected components and strongly connected components. weakly connected components and strongly connected components

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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Given N cities represented as vertices V₁, V2, un on an undirected graph (i.e., each edge can be traversed in both directions). The graph is fully-connected where the edge eij connecting any two vertices vį and vj is the straight-line distance between these two cities. We want to search for the shortest path from v₁ (the source) to VN (the destination). ... Assume that all edges have different values, and €₁,7 has the largest value among the edges. That is, the source and destination have the largest straight-line distance. Compare the lists of explored vertices when we run the uniform-cost search and the A* search for this problem. Hint: The straight-line distance is the shortest path between any two cities. If you do not know how to start, try to run the algorithms by hand on some small cases first; but remember to make sure your graphs satisfy the conditions in the question.
TRY YOUR BEST PLEASE Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B. 1. Starting with no edges between A and B, if |E| many edges are added between A and B uniformly at random, what is the expected number of perfect matchings in the resulting graph? Hint: if S is a set of edges in a potential perfect matching, let X sub S = 1 if all the edges in S are added to the graph, and X sub S= 0 if any of them are missing. What isE[X sub S]?
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Consider the following weighted undirected graph with 7 vertices and 11 edges. A C 7 8 В 9 7 15 E D 8. 9. F 11 G Let's convert the above graph to a directed graph, where each edge is pointed in one direction – i.e., xy means the edge goes from x to y. We will also make some of our edge weights negative. Suppose our edges (and weights) are listed in this order: GE=+9, FG=+11, EF=+8, ED=-15, EC=+5, DF=+6, DB=+9, CB=-8, BE=+7, BA=-7, AD=+5. Using the above order, apply the Bellman-Ford Algorithm to determine the shortest distance from the source vertex A to each of the other six vertices in the graph. Clearly show your steps. LO
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