最短路径算法实现

最短路径算法

1、Dijkstra算法

目的是求解固定起点分别到其余各点的最短路径

步骤如下：

1. 准备工作：构建二位矩阵edge,edge[i][j]存储i->j的权重，如果i==j则edge[i][j]=0,如果i和j不是直达的，则edge[i][j]=MAX_INT
2. 构建数组dis[],其中dis[i]表示其实点start->i之间的权重，不断更新，得到最小的权重
3. 选取离start最近的直达点，（注，非直达的点一定会经过中间的跳变点，间接到达，首先考虑的一定是经过离start最近的点进行跳变）
4. 判断dis[i]与dis[离start最近的点index]+edge[离start最近的点index][i]的大小，更新dis[i]
5. 重复3-4(注意标记，防止重复计算)

#include <iostream>

using namespace std;

const int max_int = ~(1<<31);
const int min_int = (1<<31);

int main(){
    int n,m,s;//n is the number of nodes, m is the number of edges and s is the start node.
    int t1,t2,t3;//t1 is the start node, t2 is the end node and t3 is the weight between t1 and t2
    cout<<"Please input the number of node(n), edges(m) and start node(s):"<<endl;
    cin>>n>>m>>s;
    int edge[n+1][n+1];//Store the edges for the n nodes
    int dis[n+1], is_visited[n+1];// dis[k] store the min distance between s and k,\
                                     is_visited store the status of the node(whether it is visited or not)

    //Init the edge[][] with the max_int
    for(int i=1; i<=n; i++){
        for(int j = 1; j <= n; j++)
            if(i==j) edge[i][j] = 0;
            else edge[i][j] = max_int;
    }

    //Input the Edge data
    cout<<"Please input the edge data: t1(start node), t2(end node), t3(weight)"<<endl;
    for(int i=1; i<=m; i++){
        cin>>t1>>t2>>t3;
        edge[t1][t2] = t3;
    }

    /*
     * Init the is_visited[] with 0
     * Init the dis[]
     */
    for(int i=1; i<=n; i++){
        is_visited[i] = 0;
        dis[i] = edge[s][i];
    }

    is_visited[s] = 1;

    //The Dijkstra algorithm
    for(int i=1; i<=n; i++){
        int u; // Store the min value index in dis[] which is not visited
        int min = max_int; // Store the min value in dis[] which is not visited
        for(int j=1; j<=n; j++){
            if(is_visited[j]==0 && dis[j]<min){
                min = dis[j];
                u = j;
            }
        }

        is_visited[u] = 1;
        for(int k=1; k<=n; k++){
            //第一层判断防止dis[u]+edge[u][k]越界
            if(edge[u][k] < max_int){
                if(is_visited[k]==0 && dis[k]>dis[u]+edge[u][k]){
                    dis[k] = dis[u] + edge[u][k];
                    cout<<u<<" "<<k<<" "<<dis[k]<<endl;
                }
            }
        }
    }

    // Print the result
    for(int i=1; i<=n; i++){
        cout<<"The min weight between "<<s<<" and "<<i<<" is: "<<dis[i]<<endl;
    }

}

2、Floyd算法

目的是求解任意两点的最短路径，核心思想是经过任意数量的节点进行中转，检查路径是否为最短

 for(k=1;k<=n;k++)//经过k进行中转
        for(i=1;i<=n;i++)
            for(j=1;j<=n;j++)
                if(e[i][j]>e[i][k]+e[k][j])
                     e[i][j]=e[i][k]+e[k][j];