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2. For a graph G = (V, E), suppose for any pair of vertices, the shortest paths is unique. Let P be the list of edges on the shortest path from some vertex s to t. (a) (b) If we double each edge weight (i.e., set the new weight to be 2x original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample. If we increase all edge weights by 1 (i.e., set the new weight to be 2x original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample.

2. For a graph G = (V, E), suppose for any pair of vertices, the shortest paths is unique. Let P be the list of edges on the shortest path from some vertex s to t. (a) (b) If we double each edge weight (i.e., set the new weight to be 2x original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample. If we increase all edge weights by 1 (i.e., set the new weight to be 2x original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample.

Database System Concepts
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
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I need question two

In this question, we'll see how some changes to the edge weight can affect the MST or SSSP of the graph. (V, E), suppose all edges have distinct weights. Let T be the set of edges in 1. For a graph G the MST of G. (a) If we double each edge weight (i.e., set the new weight to be 1+ original edge weight), will T still be the MST of the new graph? If so, prove it. If not, provide a counterexample. 2. = (b) If we increase all edge weights by 1 (i.e., set the new weight to be 1+ original edge weight), will T still be the MST of the new graph? If so, prove it. If not, provide a counterexample. For a graph G (V, E), suppose for any pair of vertices, the shortest paths is unique. Let P be the list of edges on the shortest path from some vertex s to t. = (a) (b) If we double each edge weight (i.e., set the new weight to be 2× original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample. If we increase all edge weights by 1 (i.e., set the new weight to be 2× original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample.
Transcribed Image Text:In this question, we'll see how some changes to the edge weight can affect the MST or SSSP of the graph. (V, E), suppose all edges have distinct weights. Let T be the set of edges in 1. For a graph G the MST of G. (a) If we double each edge weight (i.e., set the new weight to be 1+ original edge weight), will T still be the MST of the new graph? If so, prove it. If not, provide a counterexample. 2. = (b) If we increase all edge weights by 1 (i.e., set the new weight to be 1+ original edge weight), will T still be the MST of the new graph? If so, prove it. If not, provide a counterexample. For a graph G (V, E), suppose for any pair of vertices, the shortest paths is unique. Let P be the list of edges on the shortest path from some vertex s to t. = (a) (b) If we double each edge weight (i.e., set the new weight to be 2× original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample. If we increase all edge weights by 1 (i.e., set the new weight to be 2× original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample.
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Could you also do question one please

In this question, we'll see how some changes to the edge weight can affect the MST or SSSP of the graph. (V, E), suppose all edges have distinct weights. Let T be the set of edges in 1. For a graph G the MST of G. (a) If we double each edge weight (i.e., set the new weight to be 1+ original edge weight), will T still be the MST of the new graph? If so, prove it. If not, provide a counterexample. 2. = (b) If we increase all edge weights by 1 (i.e., set the new weight to be 1+ original edge weight), will T still be the MST of the new graph? If so, prove it. If not, provide a counterexample. For a graph G (V, E), suppose for any pair of vertices, the shortest paths is unique. Let P be the list of edges on the shortest path from some vertex s to t. = (a) (b) If we double each edge weight (i.e., set the new weight to be 2× original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample. If we increase all edge weights by 1 (i.e., set the new weight to be 2× original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample.
Transcribed Image Text:In this question, we'll see how some changes to the edge weight can affect the MST or SSSP of the graph. (V, E), suppose all edges have distinct weights. Let T be the set of edges in 1. For a graph G the MST of G. (a) If we double each edge weight (i.e., set the new weight to be 1+ original edge weight), will T still be the MST of the new graph? If so, prove it. If not, provide a counterexample. 2. = (b) If we increase all edge weights by 1 (i.e., set the new weight to be 1+ original edge weight), will T still be the MST of the new graph? If so, prove it. If not, provide a counterexample. For a graph G (V, E), suppose for any pair of vertices, the shortest paths is unique. Let P be the list of edges on the shortest path from some vertex s to t. = (a) (b) If we double each edge weight (i.e., set the new weight to be 2× original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample. If we increase all edge weights by 1 (i.e., set the new weight to be 2× original edge weight), will P still be the SSSP from s to t on the new graph? If so, prove it. If not, provide a counterexample.
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