洛谷P3366 【模板】最小生成树 题解 Prim+堆优化

题目链接：https://www.luogu.com.cn/problem/P3366

标准Prim（(O(n^2+m))）代码：

#include <bits/stdc++.h>
using namespace std;
const int maxn = 5050, maxm = 400040;
struct Edge {
    int v, w, nxt;
    Edge() {};
    Edge(int _v, int _w, int _nxt) { v = _v; w = _w; nxt = _nxt; }
} edge[maxm];
int n, m, head[maxn], ecnt;
void init() {
    ecnt = 0;
    memset(head, -1, sizeof(int)*(n+1));
}
void addedge(int u, int v, int w) {
    edge[ecnt] = Edge(v, w, head[u]); head[u] = ecnt ++;
    edge[ecnt] = Edge(u, w, head[v]); head[v] = ecnt ++;
}
int cost[maxn], ans;
bool vis[maxn];
void prim() {
    memset(cost, -1, sizeof(int)*(n+1));
    cost[1] = 0;
    for (int t = 0; t < n; t ++) {
        int u = -1;
        for (int i = 1; i <= n; i ++) {
            if (!vis[i] && cost[i] != -1 && (u == -1 || cost[u] > cost[i])) u = i;
        }
        ans += cost[u];
        vis[u] = true;
        for (int i = head[u]; i != -1; i = edge[i].nxt) {
            int v = edge[i].v, w = edge[i].w;
            if (cost[v] == -1 || cost[v] > w) cost[v] = w;
        }
    }
}
int main() {
    cin >> n >> m;
    init();
    while (m --) {
        int u, v, w;
        cin >> u >> v >> w;
        addedge(u, v, w);
    }
    prim();
    cout << ans << endl;
    return 0;
}

然后我寻思 Prim 和 Dijkstra 思想是一样的（不知道这么理解对不对囧~），所以可以采用 Dijkstra 一样的方法对 Prim 进行堆优化。

Prim+优先队列（(O(m cdot log m))）代码：

#include <bits/stdc++.h>
using namespace std;
const int maxn = 5050, maxm = 400040;
struct Edge {
    int v, w, nxt;
    Edge() {};
    Edge(int _v, int _w, int _nxt) { v = _v; w = _w; nxt = _nxt; }
} edge[maxm];
int n, m, head[maxn], ecnt;
void init() {
    ecnt = 0;
    memset(head, -1, sizeof(int)*(n+1));
}
void addedge(int u, int v, int w) {
    edge[ecnt] = Edge(v, w, head[u]); head[u] = ecnt ++;
    edge[ecnt] = Edge(u, w, head[v]); head[v] = ecnt ++;
}
int cost[maxn], ans;
bool vis[maxn];
struct Node {
    int u, cost;
    Node() {};
    Node(int _u, int _cost) { u = _u; cost = _cost; }
    bool operator < (const Node x) const {
        return cost > x.cost;
    }
};
priority_queue<Node> que;
void prim() {
    memset(cost, -1, sizeof(int)*(n+1));
    que.push(Node(1, 0));
    while (!que.empty()) {
        Node nd = que.top();
        que.pop();
        int u = nd.u;
        if (vis[u]) continue;
        vis[u] = true;
        ans += nd.cost;
        for (int i = head[u]; i != -1; i = edge[i].nxt) {
            int v = edge[i].v, w = edge[i].w;
            if (cost[v] == -1 || cost[v] > w) {
                cost[v] = w;
                que.push(Node(v, w));
            }
        }
    }
}
int main() {
    cin >> n >> m;
    init();
    while (m --) {
        int u, v, w;
        cin >> u >> v >> w;
        addedge(u, v, w);
    }
    prim();
    cout << ans << endl;
    return 0;
}