shortest path problem

带权图
1. 单源点最短路径问题： Dijkstra,(可用 priority_queue 优化)
2. 任意两点最短路径问题：Floyd

邻接矩阵的 Dijkstra
**算法：
第一步：在S-V中，选择一个d[v]最小的点v，将v加入S，d[v]为它的最终值
第二步：在S-V中，找到一个待更新最短路径的点u（因为S中添入了V），更新u的最短路径（暂时的）和前置点**

#include<iostream>
using namespace std;#define MAX 10000  //定义无穷大
#define N 100
int path[N];struct Graph{int vexnum;int arcnum;int arcs[N][N];char vexs[N];
};int Loc(Graph G,char a)
{for(int i=1;i<=G.vexnum;i++){if(G.vexs[i] == a){return i;}}
}void Input(Graph &G)
{cin>>G.vexnum>>G.arcnum;for(int i=1;i<=G.vexnum;i++){cin>>G.vexs[i];}for(int i=1;i<=G.vexnum;i++){for(int j=1;j<=G.vexnum;j++){G.arcs[i][j] = MAX;}}for(int i=1;i<=G.arcnum;i++){int x,y,w;char a,b;cin>>a>>b>>w;x = Loc(G,a);y = Loc(G,b);G.arcs[x][y] = w;}
}void OutPath(Graph G, int v)
{if(path[v] == 0){cout<<G.vexs[v]<<" ";return ;}else{OutPath(G,path[v]);cout<<G.vexs[v]<<" ";}
} // 求a->b的最短路径
void Dijkstra(Graph &G,char a,char b)
{int s[N];int d[N];int v0 = Loc(G,a);// 初始化for(int i=1;i<=G.vexnum;i++){s[i] = 0;d[i] = G.arcs[v0][i];if(G.arcs[v0][i] != MAX){path[i] = v0;       }else{path[i] = 0;}}s[v0] = 1;d[v0] = 0;for(int i=1;i<=G.vexnum-1;i++){// 在 S-V 中选择一点 u ,d[u]最小    ----> 将 u 加入s int min = MAX;int v,j;for(j=1;j<=G.vexnum;j++){if(s[j]==0){if(d[j] < min){min = d[j];v = j;}} }// 得到 v s[v] = 1;//u 为S-v 中的点，若d[u] > d[v]+arcs[v][u], 则d[u] = d[v] + arcs[v][u],path[u] = v; for(j=1;j<=G.vexnum;j++){if(s[j] == 0){if(d[j] > d[v]+G.arcs[v][j]){path [j] = v;d[j] = d[v] + G.arcs[v][j];}}}}int vt = Loc(G,b);cout<<d[vt]<<endl;OutPath(G,vt);cout<<endl;
}int main()
{Graph G;Input(G);char a,b;cin>>a>>b;Dijkstra(G,a,b);return 0;
}

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