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Floyd's Algorithm: All-Pairs Shortest Path in Directed Weighted Graphs

This lecture discusses Floyd's Algorithm, a key method for solving the All-Pairs Shortest Path (APSP) problem in directed weighted graphs. The goal is to compute the minimal path from every node to every other node, utilizing a matrix representation of the graph where each entry denotes the weight of the edges between nodes. Additionally, the lecture touches upon related concepts like the Traveling Salesman Problem (TSP) and Dijkstra's Algorithm for single-source shortest paths, providing a comprehensive overview of graph optimization techniques.

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Floyd's Algorithm: All-Pairs Shortest Path in Directed Weighted Graphs

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  1. Lecture 12: Shortest-Path Problems

  2. Floyd's Algorithm Another popular graph optimization problem is All-Pairs Shortest Path.  In this problem, you are to compute the minimal path from every node to every other node in a directed weighted graph.  The array shown below is a representation of the directed weighted graph.  Each row and column represents a particular node in the graph, while each entry in the body of the array is the weight of an arc connected the corresponding nodes.

  3. Weighted Graphs

  4. A B C D E F G H A B C D E F G H - 5 7 6 4 10 8 9 8 - 14 9 3 4 6 2 7 9 - 11 10 9 5 7 16 6 8 - 5 7 7 9 1 3 2 5 - 8 6 7 12 8 5 3 2 - 10 13 9 5 7 9 6 3 - 4 3 9 6 8 5 7 9 - Traveling Salesperson Problem (TSP) A B H C G D E F

  5. Finding the Closest Available Next City publicstaticint minAvail(int row) { int minval = int.MaxValue; int imin = -1; for (int j = 0; j < n; j++) { if (row != j && !used[j] && M[row, j] < minval) { minval = M[row, j]; imin = j; } } return imin; } jth column M = ith row

  6. L A B C D E F G H itour 3 1 0 6 used 1 1 1 1 0 0 1 0 tour C B A G Greedy TSP publicstaticvoid doGreedyTSP(int p) { int k = 0; do { itour[k] = p; tour[k] = L[p]; used[itour[k]] = true; p = minAvail(p); k += 1; } while (k < n); }

  7. 2 v3 v2 v1 v0 v5 v4 5 1 3 1 1 4 1 6 3 The single source shortest path problem is as follows. We are given a directed graph with nonnegative edge weights G = (V,E) and a distinguished source vertex, . The problem is to determine the distance from the source vertex to every vertex in the graph. Single Source Shortest Path Given a weighted graph G find the minimum weight path from a specified vertex v0 to every other vertex.

  8. 2 v3 v4 v5 v0 v1 v2 v1 v2 v3 v4 v5 node minimum list path v1 v2 v3 v4 v5 5 1 4 - 6 {2} 3 42 6 {24} 3 3 5 {241} 35 {2413} 4 5 1 3 1 1 4 1 6 3 Dijkstra's Algorithm for SSSP v1 v2 v3 v4 v5 5 1 4 - 6 {2} 3 4 2 6 {24} 3 3 5 {241} 3 5 {2413} 4 v1 v2 v3 v4 v5 5 1 4 - 6 {2} 3 4 2 6 {24} 3 3 5 {241} 3 5 v1 v2 v3 v4 v5 5 1 4 - 6 {2} 3 4 2 6 {24} 3 3 5 v1 v2 v3 v4 v5 5 1 4 - 6 {2} 3 4 2 6 v1 v2 v3 v4 v5 5 1 4 - 6

  9. Eulers Formula Let G be a connected planar simple graph with e edges and v vertices. Let r be the number of regions in a planar representation of G. Then r = e - v + 2.

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