最短路算法模版集合

例题https://www.luogu.com.cn/problem/P1339

朴素dijkstra (邻接表)

dijkstra 正确性来自于贪心 也就是st数组内的数(dist) 必须逐渐变大这样才能保证后面的数更新的时候，当前的第三边dist[t]都是最小值 [详见](https://www.acwing.com/solution/content/94237/) 

dist[x]表示 x到start 的最短距离

#include <iostream>
#include <cstring>
#include <algorithm>
 
 
using namespace std;
 
const int N = 2510, M = 6220 * 2;
 
int h[N], w[M], ne[M], e[M], idx;
int n, m, start, ened;
bool st[N];
int dist[N];
 
void add(int a, int b, int c)
{
    e[idx] = b, ne[idx] = h[a], w[idx] = c, h[a] = idx ++ ;
}
 
int dijkstra()
{
    dist[start] = 0;
    int k = n;
    while (k -- ) // 运行n - 1 次就行 n次一样不错就是多算一遍 可改成 --k
    {
        int t = -1;
        for (int i = 1; i <= n; i ++ )
            if (!st[i] && (t == -1 || dist[t] > dist[i]))
                t = i;
                
        st[t] = true;
        
        for (int i = h[t]; i != -1; i = ne[i])
        {
            int j = e[i];
            dist[j] = min(dist[j], dist[t] + w[i]);
        }
    }
    
    return dist[ened];
}
 
int main()
{
    cin >> n >> m >> start >> ened;
    memset(h, -1, sizeof h);
    memset(dist, 0x3f, sizeof dist);
    while (m -- )
    {
        int a, b, c;
        scanf("%d%d%d", &a, &b, &c);
        add(a, b, c);
        add(b, a, c);
    }
    
    cout << dijkstra();
    
    return 0;
}

朴素dijkstra 邻接矩阵

#include <iostream>
#include <cstring>
#include <algorithm>
 
 
using namespace std;
 
const int N = 3510;
 
int g[N][N];
int n, m, start, ened;
bool st[N];
int dist[N];
 
 
int dijkstra()
{
    dist[start] = 0;
    int k = n;
    
    while (k -- )
    {
        int t = -1;
        for (int i = 1; i <= n; i ++ )
            if (!st[i] && (t == -1 || dist[t] > dist[i]))
                t = i;
                
        st[t] = true;
        
        for (int i = 1; i <= n; i ++ )
            dist[i] = min(dist[i], dist[t] + g[t][i]);
    }
    
    return dist[ened];
}
 
int main()
{
    cin >> n >> m >> start >> ened;
    memset(dist, 0x3f, sizeof dist);
    memset(g, 0x3f, sizeof g);
    while (m -- )
    {
        int a, b, c;
        scanf("%d%d%d", &a, &b, &c);
        g[a][b] = min(g[a][b], c);
        g[b][a] = min(g[b][a], c);
    }
    
    cout << dijkstra();
    
    return 0;
}

堆优化dijkstra

#include <iostream>
#include <algorithm>
#include <cstring>
#include <queue>
 
using namespace std;
 
typedef pair<int, int> PII;
 
const int N = 2510, M = 6550 * 2;
 
int h[N], ne[M], e[M], w[M], idx;
int dist[N];
bool st[N];
int n, m, S, T;
 
void add(int a, int b, int c)
{
    e[idx] = b, ne[idx] = h[a], w[idx] = c, h[a] = idx ++ ;
}
 
int dijkstra()
{
    priority_queue<PII, vector<PII>, greater<PII>> q;
    q.push({0, S});
    dist[S] = 0;
    while (q.size())
    {
        auto t = q.top();
        q.pop();
        int ver = t.second, distance = t.first;
        if (st[ver]) continue;
        st[ver] = true;      // 标记已经用这个点更新过了 （此点目前最小）
        
        for (int i = h[ver]; i != -1; i = ne[i])
        {
            int j = e[i];
            if (dist[j] > distance + w[i])
            {
                dist[j] = distance + w[i];
                q.push({dist[j], j});
            }
        }
    }
    return dist[T];
}
 
int main()
{
    cin >> n >> m >> S >> T;
    memset(h, -1, sizeof h);
    memset(dist, 0x3f, sizeof dist);
    while (m -- )
    {
        int a, b, c;
        cin >> a >> b >> c;
        add(a, b, c);
        add(b, a, c);
    }
    
    cout << dijkstra();
    return 0;
}

spfa

我更新过谁，我再拿被更新的这个更新别人。

#include <iostream>
#include <algorithm>
#include <cstring>
 
using namespace std;
 
const int N = 2510, M = 6210 * 2;
 
int h[N], ne[M], e[M], idx, w[M];
int n, m, S, T;
int dist[N];
int q[N];
bool st[N];
 
void add(int a, int b, int c)
{
    e[idx] = b, ne[idx] = h[a], w[idx] = c, h[a] = idx ++ ;
}
 
 
int spfa()
{
    int tt = 1, hh = 0;
    q[hh] = S;
    dist[S] = 0;
    
    while (hh != tt)
    {
        int t = q[hh ++ ];
        if (hh == N) hh = 0;
        st[t] = false; // 退出队列后还有可能进来
        
        for (int i = h[t]; i != -1; i = ne[i])
        {
            int j = e[i];
            if (dist[j] > dist[t] + w[i])
            {
                dist[j] = dist[t] + w[i];
                if (!st[j]) // 保证队列中这个点只有一个
                {
                    q[tt ++ ] = j;
                    if (tt == N) tt = 0;
                    st[j] = true;
                }
            }
        }
    }
    
    return dist[T];
}
int main()
{
    cin >> n >> m >> S >> T;
    memset(h, -1, sizeof h);
    memset(dist, 0x3f, sizeof dist);
    while (m -- )
    {
        int a, b, c;
        cin >> a >> b >> c;
        add(a, b, c);
        add(b, a, c);
    }
    
    cout << spfa();
    
    return 0;
}

bellman_ford

#include <iostream>
#include <cstring>
#include <algorithm>
 
using namespace std;
 
const int N = 2510, M = 6500 * 2;
 
 
int h[N], ne[M], e[M], w[M], idx;
int dist[N];
bool st[N];
int n, m, S, T;
 
 
void add(int a, int b, int c)
{
    e[idx] = b, ne[idx] = h[a], w[idx] = c, h[a] = idx ++ ;
}
int bellman_ford()
{
    int k = n;
    dist[S] = 0;
    while (k -- )
    {
        int backup[N];
        memcpy(backup, dist, sizeof dist); // 备份防止连续更新边 , 正常情况下没影响,限制边数时必须有
        for (int i = 1; i <= n; i ++ )
            for (int j = h[i]; j != -1; j = ne[j])
                dist[e[j]] = min(dist[e[j]], backup[i] + w[j]);
    }
    
    return dist[T];
}
 
int main()
{
    cin >> n >> m >> S >> T;
    memset(h, -1, sizeof h);
    memset(dist, 0x3f, sizeof dist);
    while (m -- )
    {
        int a, b, c;
        cin >> a >> b >> c;
        add(a, b, c);
        add(b, a, c);
    }
    
    cout << bellman_ford();
    
    return 0;
}

Floyd

原理是动态规划 时间复杂度O(n ^ 3)

#include <iostream>
#include <cstring>
#include <algorithm>
 
using namespace std;
 
const int N = 2510;
 
int g[N][N];
int n, m, S, T;
 
 
int main()
{
    memset(g, 0x3f, sizeof g);
    scanf("%d%d%d%d", &n, &m, &S, &T);
    
    while (m -- )
    {
        int a, b, c;
        scanf("%d%d%d", &a, &b, &c);
        g[a][b] = g[b][a] = min(c, g[a][b]);
    }
    for (int k = 1; k <= n; k ++ )
        for (int i = 1; i <= n; i ++ )
            for (int j = 1; j <= n; j ++ )
                if (i != j) g[i][j] = min(g[i][j], g[i][k] + g[k][j]);
    
    printf("%d", g[S][T]);
    
    return 0;
}