最小生成树

最小生成树

Prim

朴素版(O(n^2))和堆优化版(O(mlogn))

#include <iostream>
#include <algorithm>
#include <cstring>
using namespace std;

const int N = 510, INF = 0x3f3f3f3f;
int n, m;
int g[N][N], dist[N];
bool st[N];

int prim() {
    int res = 0;
    memset(dist, 0x3f, sizeof dist);
    for (int i = 0; i < n; ++i) {
        int t = -1;
        for (int j = 1; j <= n; ++j) {
            if (!st[j] && (t == -1 || dist[t] > dist[j])) t = j;
        }
        if (i && dist[t] == INF) return INF;
        if (i) res += dist[t];
        st[t] = true;
        for (int j = 1; j <= n; ++j) dist[j] = min(dist[j], g[t][j]);
    } 
    return res;
}

int main() {
    cin >> n >> m;
    memset(g, 0x3f, sizeof g);
    while (m--) {
        int a, b, c;
        cin >> a >> b >> c;
        g[a][b] = g[b][a] = min(g[a][b], c);
    }
    int t = prim();
    if (t == INF) puts("impossible");
    else cout << t << endl;
    
    return 0;
}

Kruskal

(O(mlogn))

• 将所有边按权重从小到大排序；
• 枚举每条边a、b、w，如果a、b不连通，将这条边加入集合。

#include <iostream>
#include <algorithm>
#include <cstring>
using namespace std;

const int N = 200010;
int n, m;
int p[N];

int find(int x) {
    if (p[x] != x) p[x] = find(p[x]);
    return p[x];
}

struct edge {
    int a, b, w;
    
    bool operator< (const edge &W) const {
        return w < W.w;
    }
} edges[N];

int main() {
    cin >> n >> m;
    for (int i = 0; i < m; ++i) {
        int a, b, w;
        cin >> a >> b >> w;
        edges[i] = {a, b, w};
    }
    
    sort(edges, edges + m);
    
    for (int i = 1; i <= n; ++i) p[i] = i;
    
    int res = 0, cnt = 0;
    for (int i = 0; i < m; ++i) {
        int a = edges[i].a, b = edges[i].b, w = edges[i].w;
        a = find(a), b = find(b);
        
        if (a != b) {
            p[a] = b;
            res += w;
            cnt++;
        }
    }
    
    if (cnt < n - 1) puts("impossible");
    else cout << res << endl;
    return 0;
}