Presented by-Kapil Kumar Cse-iii rd year kapilkumar0303@gmail.com TOPIC-ALL PAIR SHORTEST PATH Floyd – Warshall Alogorithm Presented by-Kapil Kumar Cse-iii rd year kapilkumar0303@gmail.com

Graph - Basis A Non linear data structure Set of vertices and edges Classification According to Direction of edges Directed Graph Undirected graph Classification According to weight on edges Weighted graph Unweighted graph vertex edge

Graph – Basis (continue..) Representation Incident matrix Adjacency matrix Adjacency list Path matrix

What is Floyd warshall algorithm? An example of dynamic programming An algorithm for finding shortest path in a weighted graph with positive or negative edge weights No negative weight cycle Find the length of shortest path between all pair of vertices

History Naming Bernard Roy in 1959 Robert Floyd in 1962 Stephen Warshall in 1962 Naming The Floyd's algorithm the Roy–Warshall algorithm the Roy–Floyd algorithm, or the WFI algorithm

Path 3: A -> B -> C -> D = 6 What is shortest path? There are several paths between A and D: 7 B D 1 2 3 Path 1: A -> B -> D = 9 Path 2: A -> C -> D = 7 Path 3: A -> B -> C -> D = 6 A C 4

Floyd Warshall Algorithm function Floyd(L[l.. n, 1..n]): array [1.. n,1. n] array D[L..n,1 ..n] Time D  L 0(1) for k  1 to n do 0(n) for i  1 to n do 0(n) for j  1 to n do 0(n) D[i,j] min(D[i, j],D[i, k]+D[k,j]) return D

Floyd Warshall Algorithm The running time is O(n3). The space requirements are O(n2)

Example 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 12345 12345 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 12345

Application Fast computation of path finder networks. Optimal routing. Detecting negative-weight cycles in graph. Shortest paths in directed graphs Transitive closure of directed graphs. Inversion of real matrices Maximum bandwidth paths

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