算法导论 ch25 每对顶点间的最短路径

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本文介绍了一种使用邻接矩阵表示的图结构，并通过Floyd-Warshall算法计算所有顶点之间的最短路径，还展示了如何通过调整算法来获取图的传递闭包。

1. add method for class AdjMatrixGraph

/* * All-Pairs Shortest Paths */ void AdjMatrixGraph::floydWarshall() { unsigned int *d = new unsigned int[v * v]; unsigned int *p = new unsigned int[v * v]; // d0 = A for (int i = 0; i < v; i++) { for (int j = 0; j < v; j++) { d[i * v + j] = A[i * v + j]; p[i * v + j] = v + 1; } } int n = v; for (int k = 0; k < n; k++) { for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { unsigned int distance = d[i * v + k] + d[k * v + j]; if (distance < d[i * v + j]) { d[i * v + j] = distance; p[i * v + j] = k; } } } } cout << "All-Pairs Shortest Paths distance matrix is "<< endl; for (int i = 0; i < v; i++) { for (int j = 0; j < v; j++) { cout << d[i * v + j]<< " "; } cout << endl; } cout << endl; cout << "All-Pairs Shortest Paths parent matrix is "<< endl; for (int i = 0; i < v; i++) { for (int j = 0; j < v; j++) { cout << p[i * v + j]<< " "; } cout << endl; } }

2. test suite

#include "../ch22/Edge.h" #include "../ch22/Graph.h" #include <iostream> #include <vector> #include <algorithm> using namespace std; int main() { Edge e[] = { { 1, 2, 10 }, { 1, 4, 5 }, { 2, 3, 1 }, { 2, 4, 2 }, { 3, 5, 4 }, { 4, 2, 3 }, { 4, 3, 9 }, { 4, 5, 2 }, { 5, 1, 7 }, { 5, 3, 6 } }; Edges* edges = Edges::getInstance(); for (unsigned int i = 0; i < sizeof(e)/sizeof(Edge); i++) { edges->insert(&(e[i])); } int numberOfVertexes = 5; AdjMatrixGraph *g1 = new AdjMatrixGraph(numberOfVertexes, edges, true, true); MatrixGraphBuilder* mgb = MatrixGraphBuilder::getInstance(); g1->build(mgb); g1->display(); g1->floydWarshall(); }

3. run result

0 10 2147483647 5 2147483647 2147483647 0 1 2 2147483647 2147483647 2147483647 0 2147483647 4 2147483647 3 9 0 2 7 2147483647 6 2147483647 0 All-Pairs Shortest Paths distance matrix is 0 8 9 5 7 11 0 1 2 4 11 19 0 16 4 9 3 4 0 2 7 15 6 12 0 All-Pairs Shortest Paths parent matrix is 6 3 3 6 3 4 6 6 6 3 4 4 6 4 6 4 6 1 6 6 6 3 6 0 6

4. 推广：如果将E中每条边的权值赋为1，然后运行Floyd-Warshall算法，可以得到有向图的传递闭包。

4.1 add method transitiveClosure

/* * Transitive Closure */ void AdjMatrixGraph::transitiveClosure() { int *d = new int[v * v]; // init d0 for (int i = 0; i < v; i++) { for (int j = 0; j < v; j++) { if (i == j || ((A[i * v + j] > 0) && (A[i * v + j] < std::numeric_limits<int>::max()))) { d[i * v + j] = 1; } else { d[i * v + j] = 0; } } } int n = v; for (int k = 0; k < n; k++) { for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { d[i * v + j] |= d[i * v + k] & d[k * v + j]; } } } cout << "Transitive Closure Matrix is "<< endl; for (int i = 0; i < v; i++) { for (int j = 0; j < v; j++) { cout << d[i * v + j]<< " "; } cout << endl; } }

4.2 test suite

4.3 test result

0 10 2147483647 5 2147483647 2147483647 0 1 2 2147483647 2147483647 2147483647 0 2147483647 4 2147483647 3 9 0 2 7 2147483647 6 2147483647 0 Transitive Closure Matrix is 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1