FloydWarshall_DP-1

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Floyd-Warshall Algorithm PROF. NAVRATI SAXENA ASSISTANT PROFESSOR, DEPT. OF COMPUTER SCIENCE SAN JOSE STATE UNIVERSITY, CA.
Floyd-Warshall Algorithm A weighted, directed graph is a collection vertices connected by weighted edges (where the weight is some real number). One of the most common examples of a graph in the real world is a road map. Each location is a vertex and each road connecting locations is an edge. We can think of the distance traveled on a road from one location to another as the weight of that edge. SFO San Jose LA SFO 0 1.7 3.5 San Jose 1.5 0 LA 4 2.5 0 SFO San Jose LA 3.5 4 1.5 1.7 2.5
Storing a Weighted, Directed Graph Adjacency Matrix: Let D be an edge-weighted graph in adjacency-matrix form D( i,j ) is the weight of edge (i, j), or if there is no such edge. Update matrix D , with the shortest path through immediate vertices. 0 1 2 3 0 0 6 5 1 0 4 3 2 ∞ ∞ 0 2 3 ∞ ∞ ∞ 0 2 1 3 0 6 5 4 3 2 D =
Floyd-Warshall Algorithm Given a weighted graph, we want to know the shortest path from one vertex in the graph to another. The Floyd-Warshall algorithm determines the shortest path between all pairs of vertices in a graph. What is the difference between Floyd-Warshall and Dijkstra’s?
Floyd-Warshall vs. Dijkstra Dijkstra’s runs in ( V 2 ), where V: number of vertices We can just call Dijkstra |V| times , passing a different source vertex each time. ( V V 2 ) = ( V 3 ) (runtime of Floyd-Warshall Algorithm is the same) BUT , Dijkstra’s doesn’t work with negative -weight edges . Floyd Warshall can work with negative edges.
Floyd Warshall Algorithm: Basics Working Principle of Floyd Warshall Vertices in a graph: Numbered from 1 … n . Consider the subset {1,2,…, k} of these n vertices. Imagine finding the shortest path from vertex i to vertex j using vertices in the set {1,2,…,k} only. There are two situations: 1) k is an intermediate vertex on the shortest path . 2) k is not an intermediate vertex on the shortest path . j k i
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