洛谷 P4234 最小差值生成树(LCT)

题面

luogu

题解

LCT

动态树Link-cut tree(LCT)总结

考虑先按边权排序,从小到大加边

如果构成一颗树了,就更新答案

当加入一条边,会形成环.

贪心地想,我们要最大边权-最小边权最小

最大边权固定就是新加入的这条边,我们要让最小边权尽量地大

那么我们可以去掉原先路径上最小的那一条边,这样一定不会差

以上,可以用LCT维护

ps:LCT只有点权,所以对于每条边,新建一个节点

Code

#include<bits/stdc++.h>
#define mp make_pair
#define LL long long
#define RG register
const int inf = 2147483647;
using namespace std;
template<class T> inline void read(T &x) {
	x = 0; RG char c = getchar(); bool f = 0;
	while (c != '-' && (c < '0' || c > '9')) c = getchar(); if (c == '-') c = getchar(), f = 1;
	while (c >= '0' && c <= '9') x = x*10+c-48, c = getchar();
	x = f ? -x : x;
	return ;
}
template<class T> inline void write(T x) {
	if (!x) {putchar(48);return ;}
	if (x < 0) x = -x, putchar('-');
	int len = -1, z[20]; while (x > 0) z[++len] = x%10, x /= 10;
	for (RG int i = len; i >= 0; i--) putchar(z[i]+48);return ;
}

const int N = 250000 + 10;

struct node {
	int v, m, fa, ch[2], w;//m为最小值,w为位置
	bool rev;
}t[N];
int S[N], top, val[N];
set<pair<int, int> > Min;
void putrev(int x) {
	swap(t[x].ch[0], t[x].ch[1]);
	t[x].rev ^= 1;
}
void pushup(int x) {
	t[x].m = val[x], t[x].w = x;
	if (t[x].m > t[t[x].ch[0]].m && t[x].ch[0]) t[x].m = t[t[x].ch[0]].m, t[x].w = t[t[x].ch[0]].w;
	if (t[x].m > t[t[x].ch[1]].m && t[x].ch[1]) t[x].m = t[t[x].ch[1]].m, t[x].w = t[t[x].ch[1]].w;
}
#define get(x) (t[t[x].fa].ch[1] == x)
bool isroot(int x) {
	return (t[t[x].fa].ch[0] != x) && (t[t[x].fa].ch[1] != x);
}
void pushdown(int x) {
	if (t[x].rev) {
		t[x].rev = 0;
		if (t[x].ch[0]) putrev(t[x].ch[0]);
		if (t[x].ch[1]) putrev(t[x].ch[1]);
	}
}
void rotate(int x) {
	int k = get(x), y = t[x].fa, z = t[y].fa;
	if (!isroot(y)) t[z].ch[get(y)] = x;
	t[x].fa = z;
	t[t[x].ch[k^1]].fa = y; t[y].ch[k] = t[x].ch[k^1];
	t[y].fa = x; t[x].ch[k^1] = y;
	pushup(y);
}
void splay(int x) {
	S[top = 1] = x;
	for (RG int i = x; !isroot(i); i = t[i].fa) S[++top] = t[i].fa;
	for (RG int i = top; i; i--) pushdown(S[i]);
	while (!isroot(x)) {
		int y = t[x].fa;
		if (!isroot(y))
			(get(x) ^ get(y)) ? rotate(x) : rotate(y);
		rotate(x);
	}
	pushup(x);
}

void access(int x) {for (int y = 0; x; y = x, x = t[x].fa)splay(x), t[x].ch[1] = y, pushup(x);}

void makeroot(int x) {access(x); splay(x); putrev(x);}
void link(int x, int y) {
	makeroot(x);
	t[x].fa = y;
}
void cut(int x, int y) {
	makeroot(x);
	access(y);
	splay(y);
	t[x].fa = t[y].ch[0] = 0; pushup(y);
}
void split(int x, int y) {makeroot(x); access(y); splay(y);}

struct Node {
	int u, v, w;
	bool operator <(Node z) const {
		return w < z.w;
	}
}p[N];
int fa[N];
int find(int x) {
	return fa[x] == x ? x : fa[x] = find(fa[x]);
}
int main() {
	int n, m, cnt = 0, ans = inf;

	read(n), read(m);
	for (int i = 1; i <= m; i++)
		read(p[i].u), read(p[i].v), read(p[i].w);
	sort(p+1, p+1+m);
	for (int i = 1; i <= n; i++) fa[i] = i, val[i] = inf;
	for (int i = 1; i <= m; i++) {
		int x = p[i].u, y = p[i].v;
		if (x == y) continue;
		val[i + n] = p[i].w;
		if (find(x) != find(y)) {
			cnt++; fa[find(y)] = find(x);
			link(i + n, x);	link(i + n, y);
			Min.insert(mp(p[i].w, i));
		} else {
			split(x, y);
			int wz = t[y].w;
			cut(wz, p[wz - n].u); cut(wz, p[wz - n].v);
			link(i + n, x); link(i + n, y);
			Min.erase(mp(p[wz - n].w, wz - n));
			Min.insert(mp(p[i].w, i));
		}
		if (cnt == n-1)
			ans = min(ans, p[i].w - (Min.begin()->first));
	}
	printf("%d\n", ans);
	return 0;
}