Given tle following Floyd Warshall algorithm, prove that there exists a vertex x with dist[x][x] < 0 iff there exists a negative weight cycle C in G such that x ∈ V (C).
Given tle following Floyd Warshall algorithm, prove that there exists a vertex x with dist[x][x] < 0 iff there exists a negative weight cycle C in G such that x ∈ V (C).
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
Given tle following Floyd Warshall
![let dist be a |V| × |V| array of minimum distances initialized to ∞ (infinity)
let next be a |V × |V] array of vertex indices initialized to null
procedure FloydWarshallWithPathReconstruction() is
for each edge (u, v) do
dist[u][v]
w (u, v)
// The weight of the edge (u, v)
next[u][v] + v
for each vertex v do
dist[v][v]
+ 0
next[v][v]
V
for k from 1 to v do // standard Floyd-Warshall implementation
for i from 1 to v
for j from 1 to v
if dist[i][j] > dist[i][k] + dist[k][j] then
dist[i][j]
dist[i][k] + dist[k][j]
next[i][k]
next[i][j]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F093c6a52-dd65-4b33-818c-1afddd1b00ec%2F1c73e8ec-1f36-4459-81df-ed37fe01d381%2Fl3z99x5_processed.png&w=3840&q=75)
Transcribed Image Text:let dist be a |V| × |V| array of minimum distances initialized to ∞ (infinity)
let next be a |V × |V] array of vertex indices initialized to null
procedure FloydWarshallWithPathReconstruction() is
for each edge (u, v) do
dist[u][v]
w (u, v)
// The weight of the edge (u, v)
next[u][v] + v
for each vertex v do
dist[v][v]
+ 0
next[v][v]
V
for k from 1 to v do // standard Floyd-Warshall implementation
for i from 1 to v
for j from 1 to v
if dist[i][j] > dist[i][k] + dist[k][j] then
dist[i][j]
dist[i][k] + dist[k][j]
next[i][k]
next[i][j]
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