CF1628E Groceries in Meteor Town 题解

这为啥有 *3100。

考虑询问本质上在干什么。

树上两点之间边权最大值，相当于两点在 Kruskal 重构树（边权从小到大排序）上 LCA 的点权。

设 $l_{i,j}$ 表示重构树上 $i$ 与 $j$ 的 LCA，$val_i$ 表示重构树上 $i$ 的点权。

于是我们要求 $\max \limits_{u \in \text{white} \land u \neq x} val_{l_{u,x}}$。

通过观察可知，这个东西等于所有白点的 LCA 和 $x$ 的 LCA 的点权。

注意到这是个点集 LCA 的问题，只需要把 DFS 序最大和最小的两个点弄出来即可，线段树维护。

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cmath>
#include <cstring>
#include <string>
#include <vector>
#include <queue>
using namespace std;

const int N = 6e5 + 5;

vector<int> G[N];
int val[N];
int n, q;
int nfa[N][21], dep[N];

class Union_Find
{
public:
	int idx;
	int fa[N];
	void Init()
	{
		for (int i = 0; i < N; i++) fa[i] = i;
		idx = n;
	}
	int find(int u)
	{
		return (fa[u] == u ? u : fa[u] = find(fa[u]));
	}
	void merge(int u, int v, int w)
	{
		if ((u = find(u)) == (v = find(v))) return;
		idx++;
		val[idx] = w;
		G[idx].emplace_back(u), G[idx].emplace_back(v);
		fa[u] = fa[v] = idx;
		nfa[u][0] = nfa[v][0] = idx;
	}
}uf;

struct Edge
{
	int u, v, w;

	Edge() = default;

	Edge(int u, int v, int w)
		: u(u), v(v), w(w)
	{
	}

	bool operator==(const Edge& other) const = default;
}p[N];

int dfn[N], idx, rid[N];

void dfs(int u, int f)
{
	dep[u] = dep[f] + 1;
	dfn[u] = ++idx;
	rid[idx] = u;
	for (auto& j : G[u]) dfs(j, u);
}

class SegmentTree
{
public:
	struct Node
	{
		int l, r, maxn, minn;
		int tag;
		int realmax, realmin;
	}tr[N << 2];
	void pushup(int u)
	{
		tr[u].maxn = max(tr[u << 1].maxn, tr[u << 1 | 1].maxn);
		tr[u].minn = min(tr[u << 1].minn, tr[u << 1 | 1].minn);
		tr[u].realmax = max(tr[u << 1].realmax, tr[u << 1 | 1].realmax);
		tr[u].realmin = min(tr[u << 1].realmin, tr[u << 1 | 1].realmin);
	}
	void pushdown(int u)
	{
		if (tr[u].tag == -1) return;
		if (tr[u].tag)
		{
			tr[u << 1].tag = 1;
			tr[u << 1 | 1].tag = 1;
			tr[u << 1].realmax = 0, tr[u << 1].realmin = N;
			tr[u << 1 | 1].realmax = 0, tr[u << 1 | 1].realmin = N;
		}
		else
		{
			tr[u << 1].tag = 0;
			tr[u << 1 | 1].tag = 0;
			tr[u << 1].realmax = tr[u << 1].maxn, tr[u << 1].realmin = tr[u << 1].minn;
			tr[u << 1 | 1].realmax = tr[u << 1 | 1].maxn, tr[u << 1 | 1].realmin = tr[u << 1 | 1].minn;
		}
		tr[u].tag = -1;
	}
	void build(int u, int l, int r)
	{
		tr[u] = { l, r, dfn[l], dfn[l], -1, 0, N };
		if (l == r) return;
		int mid = l + r >> 1;
		build(u << 1, l, mid);
		build(u << 1 | 1, mid + 1, r);
		pushup(u);
	}
	void update(int u, int l, int r, int v)
	{
		if (tr[u].l >= l and tr[u].r <= r)
		{
			if (!v)
			{
				tr[u].tag = 0;
				tr[u].realmax = tr[u].maxn, tr[u].realmin = tr[u].minn;
			}
			else
			{
				tr[u].tag = 1;
				tr[u].realmax = 0, tr[u].realmin = N;
			}
			return;
		}
		pushdown(u);
		int mid = tr[u].l + tr[u].r >> 1;
		if (l <= mid) update(u << 1, l, r, v);
		if (r > mid) update(u << 1 | 1, l, r, v);
		pushup(u);
	}
	int qmax(int u, int l, int r)
	{
		if (tr[u].l >= l and tr[u].r <= r) return tr[u].realmax;
		pushdown(u);
		int mid = tr[u].l + tr[u].r >> 1, res = 0;
		if (l <= mid) res = qmax(u << 1, l, r);
		if (r > mid) res = max(res, qmax(u << 1 | 1, l, r));
		return res;
	}
	int qmin(int u, int l, int r)
	{
		if (tr[u].l >= l and tr[u].r <= r) return tr[u].realmin;
		pushdown(u);
		int mid = tr[u].l + tr[u].r >> 1, res = N;
		if (l <= mid) res = qmin(u << 1, l, r);
		if (r > mid) res = min(res, qmin(u << 1 | 1, l, r));
		return res;
	}
}sgt;

int LCA(int u, int v)
{
	if (u == v) return u;
	if (dep[u] < dep[v]) swap(u, v);
	int k = dep[u] - dep[v], c = 0;
	while (k)
	{
		if (k & 1) u = nfa[u][c];
		c++;
		k >>= 1;
	}
	if (u == v) return u;
	for (int i = 20; i >= 0; i--)
	{
		if (nfa[u][i] != nfa[v][i])
		{
			u = nfa[u][i], v = nfa[v][i];
		}
	}
	return nfa[u][0];
}

int main()
{
	ios::sync_with_stdio(0), cin.tie(0);
	cin >> n >> q;
	uf.Init();
	for (int i = 1; i < n; i++)
	{
		int u, v, w;
		cin >> u >> v >> w;
		p[i] = Edge(u, v, w);
	}
	sort(p + 1, p + n, [&](const Edge& x, const Edge& y) {return x.w < y.w; });
	for (int i = 1; i < n; i++) uf.merge(p[i].u, p[i].v, p[i].w);
	dfs(2 * n - 1, 2 * n - 1);
	sgt.build(1, 1, n);
	for (int i = 1; i <= 20; i++)
	{
		for (int j = 1; j <= 2 * n - 1; j++) nfa[j][i] = nfa[nfa[j][i - 1]][i - 1];
	}
	while (q--)
	{
		int op;
		cin >> op;
		if (op == 1)
		{
			int l, r;
			cin >> l >> r;
			sgt.update(1, l, r, 0);
		}
		else if (op == 2)
		{
			int l, r;
			cin >> l >> r;
			sgt.update(1, l, r, 1);
		}
		else
		{
			int x;
			cin >> x;
			int maxid = sgt.qmax(1, 1, n);
			int minid = sgt.qmin(1, 1, n);
			if (maxid == 0 || minid == N || (maxid == minid && maxid == dfn[x])) cout << "-1\n";
			else
			{
				cout << val[LCA(x, LCA(rid[maxid], rid[minid]))] << "\n";
			}
		}
	}
	return 0;
}