最短路-Dijkstra算法整理

维基说的很全面：https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm

理解：

　　先设置访问数组vis[]和距离数组dist[]，开始时把源点(source)先加入已访问数组(vis[source]=1),源点到源点的距离数组设为0(dist[source]=0);然后其它点到源点的dist[]为源点到该点的距离。然后找与源点相连的最近的点，并将其加入访问数组，再用这个点利用松弛操作更新其它未访问的点。循环执行顶点数-1次结束。

下面代码假设源点为1，终点为N。

#include<iostream>

#include<cstring>

#include<cstdio>

#include<vector>

using namespace std;

const int INF = 0x3f3f3f3f;

const int maxn = ;

int matrix[maxn][maxn];

int vis[maxn], dist[maxn];

int N, M;

void Dijkstra(int source)

{

    memset(vis, , sizeof(vis));

    vis[source] = ;

    for (int i = ; i <= N; i++) dist[i] = matrix[source][i];

    int cost, k;

    for (int i = ; i < N; i++)

    {

        cost = INF;

        for (int j = ; j <= N; j++)

        {

            if (!vis[j] && dist[j]<cost)

            {

                cost = dist[j];

                k = j;

            }

        }

        vis[k] = ;

        for (int j = ; j <= N; j++)

        {

            if (!vis[j] && matrix[k][j] != INF&&dist[j]>matrix[k][j] + cost)

            {

                dist[j] = matrix[k][j] + cost;

            }

        }

    }

}

int main()

{

    while (scanf("%d%d", &N, &M))

    {

        for (int i = ; i <= N; i++)

            for (int j = ; j <= N; j++)

                if (i == j) matrix[i][j] = ;

                else matrix[i][j] = INF;

        for (int i = ; i<M; i++) {

            int u, v, w;

            scanf("%d%d%d", &u, &v, &w);

            matrix[u][v] = matrix[v][u] = w;

        }

        Dijkstra();

        printf("%d\n", dist[N]);

    }

    return ;

}

用优先队列和vector：

#include<iostream>

#include<cstdio>

#include<cstring>

#include<queue>

#include<vector>

using namespace std;

const int INF=0x3f3f3f3f;

const int maxn=;

struct Node

{

    int num,dis;//num存储当前节点的编号，dis存储路径长度

    Node(){}

    Node(int a,int b){num=a;dis=b;}

    bool operator < (const Node& rhs)const{

        dis>rhs.dis;

    }

};

priority_queue<Node>que;

vector<vector<Node> >v;

int vis[maxn],dist[maxn];

int N,M;

void Dijkstra(int s)

{

    for (int i=;i<=N;i++) dist[i]=INF;

    dist[s]=;

    que.push(Node(s,dist[s]));

    while(!que.empty())

    {

        Node p=que.top();que.pop();

        for (int i=;i<v[p.num].size();i++)

        {

            Node q;

            q.num=v[p.num][i].num;

            if(dist[q.num]>dist[p.num]+v[p.num][i].dis)

            {

                dist[q.num]=dist[p.num]+v[p.num][i].dis;

                que.push(Node(q.num,dist[q.num]));

            }

        }

    }

}

int main()

{

    scanf("%d%d",&N,&M);

    v.clear();

    v.resize(N+);

    for (int i=;i<M;i++)

    {

        int fr,to,w;

        scanf("%d%d%d",&fr,&to,&w);

        v[fr].push_back(Node(to,w));

        v[to].push_back(Node(fr,w));

    }

    Dijkstra();

    printf("%d\n",dist[N]);

}

以hdu1874畅通工程续为例，一个pair较为偷懒的写法：

 #include<iostream>

 #include<cstdio>

 #include<cstring>

 #include<queue>

 using namespace std;

 const int maxn = ;

 vector<pair<int, int> > E[maxn];

 int d[maxn];

 int n, m;

 void Dijkstra(int s)

 {

     priority_queue<pair<int, int> >Q;

     d[s] = ;

     Q.push(make_pair(-d[s],s));

     while (!Q.empty())

     {

         int now = Q.top().second; Q.pop();

         for (int i = ; i < E[now].size(); i++)

         {

             int v = E[now][i].first;

             if (d[v] > d[now] + E[now][i].second)

             {

                 d[v] = d[now] + E[now][i].second;

                 Q.push(make_pair(-d[v], v));

             }

         }

     }

 }

 int main()

 {

     while (scanf("%d%d",&n,&m)==)

     {

         for (int i = ; i < n; i++) E[i].clear();

         for (int i = ; i < n; i++) d[i] = 1e9;

         for (int i = ; i < m; i++)

         {

             int x, y, z;

             scanf("%d%d%d", &x, &y, &z);

             E[x].push_back(make_pair(y,z));

             E[y].push_back(make_pair(x, z));

         }

         int s, t;

         scanf("%d%d", &s, &t);

         Dijkstra(s);

         if (d[t] == 1e9)

             printf("-1\n");

         else

             printf("%d\n", d[t]);

     }

     return ;

 }

巴特西

最短路-Dijkstra算法整理

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