Q1:We define the capacity of a path in a weighted graph G(V, E) to be the minimum weight ofan edge along that path. For example, the capacity of the path [V0, V1, V2] in the graphbelow is 5 and the capacity of the path V1, V0, V3, V2 is 3. A maximum-capacity pathbetween u and v is a simple path between u and v whose capacity is larger than or equal tothose of all paths from u to v. In the graph below, the maximum capacity path from V0 toV3 is [V0, V1, V2, V3].(a) Show that if [a, b, ... , u, v] is a maximum-capacity path from a to v, then a maximum-capacity path from a to u followed by the edge (u, v) is a maximum-capacity path froma to v.(b) Is it true that if [a, b, c, u, v] is a maximum-capacity path from a to v, then [a, b, c, u]is a maximum-capacity path from a to u? Why?(c) Can you think of an algorithm you have seen before that can be modified to find themaximum-capacity paths from a particular vertex s to all other vertices in the graph?How? 10Vo3V225V3V4

Path Capacities Maximum Capacities Vo V3 V4 3, 8 3 Vo V1 V2 V3 10, 5, 25 25 Vo V1 V2 V3…

Q1: We define the capacity of a path in a weighted graph G(V, E) to be the minimum weight of an edge along that path. For example, the capacity of the path [V0, V1, V2] in the graph below is 5 and the capacity of the path V1, V0, V3, V2 is 3. A maximum-capacity path between u and v is a simple path between u and v whose capacity is larger than or equal to those of all paths from u to v. In the graph below, the maximum capacity path from V0 to V3 is [V0, V1, V2, V3].

Q1: We define the capacity of a path in a weighted graph G(V, E) to be the minimum weight of an edge along that path. For example, the capacity of the path [V0, V1, V2] in the graph below is 5 and the capacity of the path V1, V0, V3, V2 is 3. A maximum-capacity path between u and v is a simple path between u and v whose capacity is larger than or equal to those of all paths from u to v. In the graph below, the maximum capacity path from V0 to V3 is [V0, V1, V2, V3].

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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Concept explainers

Minimum Spanning Tree

A spanning tree is a sub-graph of a connected, undirected graph. It has all the vertices of a graph with the minimum possible number of edges. If a vertex is skipped, then it will not be a spanning tree. The edges of a spanning tree may or may not have weights.

Question

Q1:
We define the capacity of a path in a weighted graph G(V, E) to be the minimum weight of
an edge along that path. For example, the capacity of the path [V0, V1, V2] in the graph
below is 5 and the capacity of the path V1, V0, V3, V2 is 3. A maximum-capacity path
between u and v is a simple path between u and v whose capacity is larger than or equal to
those of all paths from u to v. In the graph below, the maximum capacity path from V0 to
V3 is [V0, V1, V2, V3].
(a) Show that if [a, b, ... , u, v] is a maximum-capacity path from a to v, then a maximum-
capacity path from a to u followed by the edge (u, v) is a maximum-capacity path from
a to v.
(b) Is it true that if [a, b, c, u, v] is a maximum-capacity path from a to v, then [a, b, c, u]
is a maximum-capacity path from a to u? Why?
(c) Can you think of an algorithm you have seen before that can be modified to find the
maximum-capacity paths from a particular vertex s to all other vertices in the graph?
How?

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