Floyd算法
弗洛伊德算法，常用于多源最短路径问题，即求每对顶点间的最短路径。
/*
有权图的多源最短路径问题
*/
#include <iostream>
#include <queue>
using namespace std;
#define MaxVertexNum 10
#define infinity 1e9
struct Graph
{
int edgenum;
int vertexnum;
int edgeList[MaxVertexNum][MaxVertexNum];
int path[MaxVertexNum][MaxVertexNum]; // 前驱结点
int dist[MaxVertexNum][MaxVertexNum]; // 从源点到其他各顶点当前最短路径长度
};
void BuildGraph(Graph *G)
{
int start, end;
cout << "Please enter the number of vertices and edges" << endl;
cin >> G->vertexnum >> G->edgenum;
// 图的权重初始化
for (int i = 1; i <= G->vertexnum; i++)
{
for (int j = 1; j <= G->vertexnum; j++)
{
if (i == j)
{
G->edgeList[i][j] = 0;
}
else
{
G->edgeList[i][j] = infinity;
}
G->dist[i][j] = G->edgeList[i][j];
G->path[i][j] = -1;
}
}
// 输入权重信息
for (int i = 1; i <= G->edgenum; i++)
{
cout << "Please enter the Start number, end number, weight" << endl;
cin >> start >> end;
cin >> G->edgeList[start][end];
G->dist[start][end] = G->edgeList[start][end];
}
}
void Floyd(Graph *G)
{
for (int k = 1; k <= G->vertexnum; k++)
{
for (int i = 1; i <= G->vertexnum; i++)
{
for (int j = 1; j <= G->vertexnum; j++)
{
if (G->dist[i][k] + G->dist[k][j] < G->dist[i][j])
{
G->dist[i][j] = G->dist[i][k] + G->dist[k][j];
G->path[i][j] = k;
}
}
}
}
}
int main()
{
Graph G;
BuildGraph(&G);
Floyd(&G);
cout << "edge" << endl;
for (int i = 1; i <= G.vertexnum; i++)
{
for (int j = 1; j <= G.vertexnum; j++)
{
if (G.edgeList[i][j] != infinity)
{
printf("%10d", G.edgeList[i][j]);
}
else
{
printf("%10s", "infinity");
}
}
cout << endl;
}
cout << "distance" << endl;
for (int i = 1; i <= G.vertexnum; i++)
{
for (int j = 1; j <= G.vertexnum; j++)
{
cout << G.dist[i][j] << " ";
}
cout << endl;
}
cout << "path" << endl;
for (int i = 1; i <= G.vertexnum; i++)
{
for (int j = 1; j <= G.vertexnum; j++)
{
cout << G.path[i][j] << " ";
}
cout << endl;
}
system("pause");
return 0;
}
/*
4 8
1 4 4
4 1 5
4 3 12
3 4 1
1 3 6
3 1 7
1 2 2
2 3 3
*/
可以这么理解，将第一层循环的 k 看成是中转点，嵌套的2层循环里得到的 i，j 一对顶点，求的就是 i 到 j 的距离。判断从 i 出发经由 k 再到 j 的路径是否比当前储存的路径短，然后更新。