LOJ-2001-Luogu-3703-BZOJ-4817-「SDOI2017」树点涂色

题目

https://loj.ac/problem/2001

https://www.luogu.org/problemnew/show/P3703

题意

略。

题解

  • 如果把操作 1 看作是 LCT 的 access 操作,那么路径的权值就是路径上的虚边条数加一。
  • 用线段树维护每个结点到根的虚边条数,access 中轻重边的切换就对应于子树的虚边条数区间加减。
  • 操作 3 就是区间最大值,操作 2 就是单点查询 + LCA。
  • 复杂度 $O(n\log ^2 n)$。
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#include <bits/stdc++.h>
using namespace std;
using LL = long long;
#define FOR(i, x, y) for (decay<decltype(y)>::type i = (x), _##i = (y); i < _##i; ++i)
#define FORD(i, x, y) for (decay<decltype(x)>::type i = (x), _##i = (y); i > _##i; --i)
#ifdef zerol
#define dbg(args...) do { cout << "\033[32;1m" << #args << " -> "; err(args); } while (0)
#else
#define dbg(...)
#endif
void err() { cout << "\033[39;0m" << endl; }
template<template<typename...> class T, typename t, typename... Args>
void err(T<t> a, Args... args) { for (auto x: a) cout << x << ' '; err(args...); }
template<typename T, typename... Args>
void err(T a, Args... args) { cout << a << ' '; err(args...); }
// -----------------------------------------------------------------------------
const int maxn = 1E5 + 100;
int n, Q;
vector<int> G[maxn];

namespace sg {
struct Q {
int add;
Q(int add = 0): add(add) {}
void operator += (Q& q) { add += q.add; }
};
struct P {
int max;
explicit P(int max = 0): max(max) {}
void up(Q& q) { max += q.add; }
};
template<typename T>
P operator & (T&& a, T&& b) {
return P(max(a.max, b.max));
}
P p[maxn << 2];
Q q[maxn << 2];
#define lson o * 2, l, (l + r) / 2
#define rson o * 2 + 1, (l + r) / 2 + 1, r
void maintain(int o, int l, int r) {
if (l == r) p[o] = P();
else p[o] = p[o * 2] & p[o * 2 + 1];
p[o].up(q[o]);
}
void pushdown(int o, int l, int r) {
q[o * 2] += q[o]; q[o * 2 + 1] += q[o];
q[o] = Q();
maintain(lson); maintain(rson);
}
template<typename T>
void build(T f, int o = 1, int l = 1, int r = n) {
if (l == r) q[o] = f(l);
else { build(f, lson); build(f, rson); }
maintain(o, l, r);
}
P query(int ql, int qr, int o = 1, int l = 1, int r = n) {
if (ql > r || l > qr) return P();
if (ql <= l && r <= qr) return p[o];
pushdown(o, l, r);
return query(ql, qr, lson) & query(ql, qr, rson);
}
void update(int ql, int qr, Q v, int o = 1, int l = 1, int r = n) {
if (ql > r || l > qr) return;
if (ql <= l && r <= qr) q[o] += v;
else {
pushdown(o, l, r);
update(ql, qr, v, lson); update(ql, qr, v, rson);
}
maintain(o, l, r);
}
}

int fa[maxn], dep[maxn], idx[maxn], out[maxn], ridx[maxn];
namespace hlc {
int sz[maxn], son[maxn], top[maxn], clk;
void predfs(int u, int d) {
dep[u] = d; sz[u] = 1;
int& maxs = son[u] = -1;
for (int& v: G[u]) {
if (v == fa[u]) continue;
fa[v] = u;
predfs(v, d + 1);
sz[u] += sz[v];
if (maxs == -1 || sz[v] > sz[maxs]) maxs = v;
}
}
void dfs(int u, int tp) {
top[u] = tp; idx[u] = ++clk; ridx[clk] = u;
if (son[u] != -1) dfs(son[u], tp);
for (int& v: G[u])
if (v != fa[u] && v != son[u]) dfs(v, v);
out[u] = clk;
}
int go(int u, int v) {
int uu = top[u], vv = top[v];
while (uu != vv) {
if (dep[uu] < dep[vv]) { swap(uu, vv); swap(u, v); }
u = fa[uu]; uu = top[u];
}
if (dep[u] < dep[v]) swap(u, v);
return v;
}
}

namespace lct {
struct P {
P *fa, *ls, *rs;
int idx;
bool has_fa() { return fa->ls == this || fa->rs == this; }
bool d() { return fa->ls == this; }
P*& c(bool x) { return x ? ls : rs; }
} *null = new P{nullptr, nullptr, nullptr}, pool[maxn], *pit = pool;
P* rt[maxn];
void rot(P* o) {
bool dd = o->d();
P *f = o->fa, *t = o->c(!dd);
if (f->has_fa()) f->fa->c(f->d()) = o; o->fa = f->fa;
if (t != null) t->fa = f; f->c(dd) = t;
o->c(!dd) = f; f->fa = o;
}
void splay(P* o) {
while (o->has_fa()) {
if (o->fa->has_fa()) rot(o->d() xor o->fa->d() ? o : o->fa);
rot(o);
}
}
void calc(P* o, int v) {
if (o == null) return;
while (o->ls != null) o = o->ls;
int x = o->idx;
sg::update(idx[x], out[x], v);
}
void access(P* u, P* v = null) {
if (u == null) return;
splay(u);
P* t = u->rs;
calc(v, -1);
u->rs = v;
calc(t, 1);
access(u->fa, u);
}
}


int main() {
cin >> n >> Q;
FOR (_, 1, n) {
int u, v; scanf("%d%d", &u, &v);
G[u].push_back(v); G[v].push_back(u);
}
hlc::predfs(1, 0); hlc::dfs(1, 1);
FOR (i, 1, n + 1) {
using namespace lct;
rt[i] = new (pit++) P{i == 1 ? null: rt[fa[i]], null, null, i};
}
sg::build([](int x)->sg::Q { return sg::Q(dep[ridx[x]]); });
while (Q--) {
int op; scanf("%d", &op);
if (op == 1) {
int u; scanf("%d", &u);
lct::access(lct::rt[u]);
} else if (op == 2) {
int u, v; scanf("%d%d", &u, &v);
int lca = hlc::go(u, v);
u = idx[u]; v = idx[v]; lca = idx[lca];
int ans = sg::query(u, u).max + sg::query(v, v).max - 2 * sg::query(lca, lca).max;
printf("%d\n", ans + 1);
} else {
int u; scanf("%d", &u);
printf("%d\n", sg::query(idx[u], out[u]).max + 1);
}
}
}