算法储备之Floyd Warshall算法

Floyd Warshall算法是动态规划的经典算法

该算法可以解决图中每个顶点到其他顶点的距离，图中可以有负权值边，但不能有负循环。

时间复杂度为O(V的三次方)

算法思想

dist[V][V]初始化为二维数组edge[V][V]的内容

for循环执行V次，每次以一个顶点为中间顶点，更新所有顶点通过中间顶点到其他顶点的距离

for(int k=0;i<V;k++)

  for(int i=0;i<V;i++)

    for(int j=0;j<V;j++)

      if(edge[i][k] ! = INT_MAX && edge[k][j] != INT_MAX && edge[i][j]>edge[i][k]+edge[k][j])

        edge[i][j]=edge[i][k]+edge[k][j];

算法的理解

设顶点v0到顶点v5之间的最短路径为 v0->v6->v3->v4->v5

当中间顶点为除v0和v5的其他顶点时就能把与中间顶点相邻的两个顶点连起来，必能得到v0到v5的最短路径

#include <iostream>#include <vector>using namespace std;struct Graph {int V;vector<vector<int>>adjMatrix;public:Graph(int v) :V(v), adjMatrix(V) { }void floydWarshall();};void printSolution(vector<vector<int>> & dist){cout << "Following matrix shows the shortest""distances between every pair of vertices" << endl;for (int i = 0;i < dist.size();i++){for (int j = 0;j < dist.size();j++){if (dist[i][j] == INT_MAX)printf("%7s", "INF");elseprintf("%7d", dist[i][j]);}cout << endl;}}void Graph::floydWarshall(){auto dist = adjMatrix;for (int k = 0;k < V;k++)for (int i = 0;i < V;i++)for (int j = 0;j < V;j++)if (dist[i][k] != INT_MAX &&dist[k][j] != INT_MAX &&dist[i][k] + dist[k][j] < dist[i][j])dist[i][j] = dist[i][k] + dist[k][j];printSolution(dist);}int main(){Graph graph(4);graph.adjMatrix = {{ 0,5,INT_MAX, 10 },{ INT_MAX, 0,3, INT_MAX },{ INT_MAX, INT_MAX, 0,1 },{ INT_MAX, INT_MAX, INT_MAX, 0 }};graph.floydWarshall();return 0;}