Bellman-Ford Shortest-Path Algorithm

External

• https://www.coursera.org/learn/algorithms-npcomplete/lecture/x0YZd/single-source-shortest-paths-revisted
• https://www.coursera.org/learn/algorithms-npcomplete/lecture/g8N36/optimal-substructure
• https://www.coursera.org/learn/algorithms-npcomplete/lecture/9YeyY/the-basic-algorithm-i
• https://www.coursera.org/learn/algorithms-npcomplete/lecture/WhILJ/the-basic-algorithm-ii
• https://www.coursera.org/learn/algorithms-npcomplete/lecture/AB5wH/detecting-negative-cycles
• https://www.coursera.org/learn/algorithms-npcomplete/lecture/TrNPq/a-space-optimization

Internal

• Shortest Path in a Graph
• Dynamic Programming
• Floyd-Warshall Algorithm
• Johnson's Algorithm

Overview

A dynamic programming algorithm that can compute shortest path in graphs with negative length edges.

There are n² subproblems, and we might spend more than linear time for each subproblem: we have to do brute force search through a list of candidates that might be super-constant: each arc that comes into the vertex v provides a candidate for what the correct solution to the subproblem may be, and the number of candidates is proportional to the degree of vertex v. The running time is O(mn). In a sparse graph, m=O(n), and in a dense graph is O(n²).