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最小生成树

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最小生成树

最小生成树（英语：Minimum spanning tree，简称MST）是最小权重生成树（英语：Minimum weight spanning tree）的简称，是一个连通加权无向图中一棵权值最小的生成树

Kruskal算法

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#include <bits/stdc++.h>
#define int long long
using namespace std;

// Kruskal算法

// 1, 将所有边按权重从小到大排序
// 2, 枚举每条边 a-b, 权重是c
// 如果a, b不连通, 将这条边加入在集合当中

const int N = 1000010;

int n, m;
int p[N];

struct Edge{
	int a, b, w;
} edge[N];

bool cmp(Edge x, Edge y) {
	return x.w < y.w;
}

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

void solve() {
	cin >> n >> m;
	for (int i = 0; i < m; i++) {
		int a, b, w;
		cin >> a >> b >> w;
		edge[i] = {a, b, w};
	}
	sort(edge, edge + m, cmp);
	for (int i = 1; i <= n; i++) p[i] = i;
	
	int ans = 0, cnt = 0;
	for (int i = 0; i < m; i++) {
		int a = edge[i].a, b = edge[i].b, w = edge[i].w;
		a = find(a); b = find(b);
		if (a != b) {
			p[a] = b;
			ans += w;
			cnt++;
		} 
	}
	if (cnt < n - 1) {
		cout << "impossible" << endl;
	} else {
		cout << ans << endl;
	}
}

signed main() {
	ios::sync_with_stdio(0);
	cin.tie(0);
	cout.tie(0);
	int t = 1;
//	cin >> t;
	while (t--) {
		solve();
	}
	
	return 0;
}

经典例题

口袋的天空

https://www.luogu.com.cn/problem/P1195

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#include <bits/stdc++.h>
#define int long long
using namespace std;

int n, m, k;
const int N = 1e5;

int p[N];

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

struct Edge{
	int x, y, l;
} edge[N];

bool cmp(Edge a, Edge b) {
	return a.l < b.l;
}

void solve() {
	cin >> n >> m >> k;
	for (int i = 0; i < m; i++) {
		int x, y, l;
		cin >> x >> y >> l;
		edge[i] = {x, y, l};
	}
	for (int i = 1; i <= n; i++) {
		p[i] = i;
	}
	sort(edge, edge + m, cmp);
	int cnt = 0, ans = 0;
	for (int i = 0; i < m; i++) {
		int x = edge[i].x, y = edge[i].y, l = edge[i].l;
		x = find(x); y = find(y);
		if (x != y) {
			p[x] = y;
			cnt++;
			ans += l;
		}
		if (cnt >= n - k) break;
	}
	// 棉花糖可以理解为连通块
	if (cnt >= n - k) {
		cout << ans << endl;
	} else {
		cout << "No Answer" << endl;
	}
}

signed main() {
	ios::sync_with_stdio(0);
	cin.tie(0);
	cout.tie(0);
	int t = 1;
//	cin >> t;
	while (t--) {
		solve();
	}
	
	return 0;
}