Offline Dynamic Connectivity（クエリ先読み）

graph/offline_dynamic_connectivity.hpp · GitHub で見る

共通ページ

無向グラフの辺の追加・削除と問い合わせを登録し、最後にまとめて計算します。Rollback Union Find を利用します。

#include "lib/graph/offline_dynamic_connectivity.hpp"
#include <cassert>

int main() {
	OfflineDynamicConnectivity graph(4);
	graph.link(0, 1);
	int a = graph.query_same(0, 1);
	graph.link(1, 2);
	int b = graph.query_size(0);
	graph.cut(1, 0);
	int c = graph.query_same(0, 2);
	int d = graph.query_component_count();
	auto answer = graph.solve();
	assert(answer[a] == 1 && answer[b] == 3);
	assert(answer[c] == 0 && answer[d] == 3);
}

頂点は [0,n) です。各 query_* の返り値は0始まりの問い合わせ ID で、solve() の配列をその ID で参照します。問い合わせは登録した位置での状態を調べます。辺の更新だけでは問い合わせ ID は増えません。

操作	内容・返り値	計算量
`OfflineDynamicConnectivity(n=0)`	頂点数だけを登録。初期状態では辺なし	\(O(1)\)
`reset(n)`	頂点数を変え、全ての更新・問い合わせを削除	\(O(M+T)\)
`reserve(updates,queries=0)`	辺の更新回数・問い合わせ回数を目安に領域を確保	\(O(M+T)\)
`vertex_count()` / `query_count()`	頂点数／登録した問い合わせ数	\(O(1)\)
`link(u,v)`	無向辺を1本追加する操作を登録	償却 \(O(1)\)
`cut(u,v)`	無向辺を1本削除する操作を登録。存在しなければ何もしない	償却 \(O(1)\)
`query_same(u,v)`	連結判定を登録し、問い合わせ ID を返す。答えは0または1	償却 \(O(1)\)
`query_size(v)`	`v` の成分サイズの問い合わせを登録	償却 \(O(1)\)
`query_component_count()`	成分数の問い合わせを登録	償却 \(O(1)\)
`solve()`	登録した問い合わせの答えを `vector<int>` で返す	\(O(n+M+(T+K)\log n)\)
`query()`	任意の問い合わせ位置だけを登録し、問い合わせ ID を返す	償却 \(O(1)\)
`run(visit)`	全問い合わせを `visit(id,const_uf)` で処理	\(O(n+M+T+K\log n)\)
`run(on_link,on_cut,visit)`	成功した併合・その取り消しの通知を受けながら問い合わせを処理	\(O(n+M+T+K\log n)\)

並行辺に対応します。同じ辺を2本追加すると、1本削除した後も連結関係に使われます。link(0,1) と link(1,0) は同じ辺として扱います。自己ループは連結性を変えないため、登録時に無視します。

solve・run は登録内容を変更しません。繰り返し計算でき、登録を追加してから再計算することもできます。その場合は以前の問い合わせも含めて全体を計算し直します。頂点数0でも query_component_count と query() は使えます。

任意の問い合わせ

複数の情報をまとめて調べたい場合は query() と run を使います。各問い合わせ位置の Rollback Union Find を読み取り専用で受け取れます。

OfflineDynamicConnectivity graph(3);
graph.link(0, 1);
graph.query();
graph.link(1, 2);
graph.query();
graph.run([&](int id, const RollbackUnionFind &uf) {
	assert(uf.size(0) == id + 2);
});

visit は問い合わせ ID の昇順に1回ずつ呼びます。渡された uf はその呼び出しの中だけで使ってください。leader・same・size・component_count・groups を利用できます。独自の問い合わせ位置を作る query() を使った場合は、solve() ではなく run で答えを処理してください。既定の query_* も run の呼び出し対象になります。

成分の情報を追加する

run(on_link,on_cut,visit) の on_link(root,child) は、異なる成分を併合した直後に呼びます。新しい代表元は root、その子になった元の代表元は child です。既に連結な場合は呼びません。

on_cut(root,child) は、この併合を取り消す直前に呼びます。同じ組の on_link に対応し、取り消す順序は後入れ先出しです。これは入力の cut(u,v) の通知ではありません。内部の時間区間探索で併合を取り消すたびに呼ばれます。最後は全ての併合を取り消すため、on_link と on_cut の回数は一致します。

例えば初期頂点値の成分和を調べる場合は、外部の sum を次のように更新できます。

std::vector<long long> sum = {1, 10, 100};
OfflineDynamicConnectivity graph(3);
graph.link(0, 1);
graph.query();
graph.run([&](int root, int child) {
	sum[root] += sum[child];
}, [&](int root, int child) {
	sum[root] -= sum[child];
}, [&](int id, const RollbackUnionFind &uf) {
	assert(id == 0 && sum[uf.leader(0)] == 11);
});

頂点値の加算も行う利用例は Dynamic Graph Vertex Add Component Sum の検証コードにあります。外部にも親を記録し、頂点への加算をその祖先まで反映することで、時刻順の加算と成分和を扱います。

独自の集計は、内部で選ばれる併合順に依存しないようにしてください。問い合わせの答えを次の辺更新の生成に使うことはできません。コールバックからこのグラフを変更しないでください。繰り返し run する場合、必要に応じて外部データも開始時の状態へ戻します。

辺の有効期間は、問い合わせがある時点だけに圧縮します。重複辺は本数が0と正の間で変わる区間にまとめ、問い合わせを含まない区間は捨てます。時間セグメント木の操作は1本の配列にまとめ、同じ成分を再度併合する処理とその履歴は省きます。ハッシュ表には依存せず、正規化した64 bitの辺キーを安定基数ソートすることで、同じ辺の登録順を保って有効期間を求めます。

頂点数を \(n\)、自己ループ以外の辺更新数を \(M\)、問い合わせ数を \(T\)、辺の有効区間数を \(I\leq M\)、時間セグメント木に置く辺 ID の総数を \(K=O(I\log T)\) とします。run は \(O(n+M+T+K\log n)\) 時間で、コールバックとその中で呼ぶ操作の計算量は別に加算します。solve は連結判定なども含めて \(O(n+M+(T+K)\log n)\) 時間です。使用中のデータは \(O(n+M+T+K)\) 空間です。保持する更新・問い合わせの数はそれぞれ INT_MAX 以下としてください。問い合わせがなければ run は何もしません。空の場合や長さ1の場合にも定数時間かかります。reset は確保済み領域を保持します。

依存ライブラリ

検証コード

test/offline_dynamic_connectivity.test.cpp

コード

#line 1 "graph/offline_dynamic_connectivity.hpp"

#line 1 "components/offline_time_segments.hpp"

#include <bit>
#include <cassert>
#include <cstddef>
#include <utility>
#include <vector>

class OfflineTimeSegments {
	int n_;
	std::size_t base_;
	std::vector<std::size_t> offset_;
	std::vector<int> items_;

public:
	struct Interval { int left, right, item; };
	OfflineTimeSegments(int n, const std::vector<Interval> &intervals) : n_(n), base_(1) {
		assert(n >= 0);
		if (!n || intervals.empty()) return;
		base_ = std::bit_ceil(static_cast<std::size_t>(n));
		offset_.resize(base_ * 2 + 1);
		auto cover = [&](const Interval &interval, auto &&visit) {
			assert(0 <= interval.left && interval.left <= interval.right && interval.right <= n);
			for (std::size_t l = base_ + interval.left, r = base_ + interval.right; l < r; l >>= 1, r >>= 1) {
				if (l & 1) visit(l++);
				if (r & 1) visit(--r);
			}
		};
		for (const auto &interval : intervals) cover(interval, [&](std::size_t v) { ++offset_[v]; });
		for (std::size_t v = 1; v < offset_.size(); ++v) offset_[v] += offset_[v - 1];
		items_.resize(offset_.back());
		for (const auto &interval : intervals)
			cover(interval, [&](std::size_t v) { items_[--offset_[v]] = interval.item; });
	}
	int size() const { return n_; }
	std::size_t entry_count() const { return items_.size(); }
	template <class State, class Apply, class Visit>
	void run(State &state, Apply &&apply, Visit &&visit) const {
		if (!n_) return;
		if (items_.empty()) {
			for (int t = 0; t < n_; ++t) visit(t, std::as_const(state));
			return;
		}
		auto dfs = [&](auto &&self, std::size_t v, std::size_t l, std::size_t r) -> void {
			auto saved = state.snapshot();
			for (std::size_t i = offset_[v]; i < offset_[v + 1]; ++i) apply(state, items_[i]);
			if (r - l == 1) visit(static_cast<int>(l), std::as_const(state));
			else {
				std::size_t m = (l + r) / 2;
				self(self, v * 2, l, m);
				if (m < static_cast<std::size_t>(n_)) self(self, v * 2 + 1, m, r);
			}
			state.rollback(saved);
		};
		dfs(dfs, 1, 0, base_);
	}
};

#line 1 "components/radix_sort.hpp"

#include <algorithm>
#include <array>
#line 7 "components/radix_sort.hpp"
#include <cstdint>
#include <type_traits>
#line 11 "components/radix_sort.hpp"

template <int Bits, class T, class Key>
void stable_radix_sort(std::vector<T> &values, Key &&key) {
	static_assert(1 <= Bits && Bits <= 16);
	constexpr unsigned buckets = 1U << Bits, mask = buckets - 1;
	using K = std::remove_cvref_t<decltype(key(std::declval<const T &>()))>;
	static_assert(std::is_integral_v<K> && std::is_unsigned_v<K> && sizeof(K) <= sizeof(std::uint64_t));
	if (values.size() < 2) return;
	if (values.size() < 256) {
		std::stable_sort(values.begin(), values.end(), [&](const T &a, const T &b) { return key(a) < key(b); });
		return;
	}
	std::uint64_t first = key(values.front()), previous = first, varying = 0;
	bool sorted = true;
	for (std::size_t i = 1; i < values.size(); ++i) {
		std::uint64_t current = key(values[i]);
		varying |= first ^ current;
		sorted &= previous <= current;
		previous = current;
	}
	if (sorted) return;
	std::vector<T> scratch(values.size());
	using Counts = std::conditional_t<(Bits <= 11), std::array<std::size_t, buckets>, std::vector<std::size_t>>;
	Counts count = [] {
		if constexpr (Bits <= 11) return Counts{};
		else return Counts(buckets);
	}();
	for (int shift = 0; shift < 64; shift += Bits) {
		if (!((varying >> shift) & mask)) continue;
		std::fill(count.begin(), count.end(), 0);
		for (const auto &value : values) ++count[(static_cast<std::uint64_t>(key(value)) >> shift) & mask];
		std::size_t sum = 0;
		for (auto &x : count) {
			auto n = x;
			x = sum, sum += n;
		}
		for (auto &&value : values) scratch[count[(static_cast<std::uint64_t>(key(value)) >> shift) & mask]++] = std::move(value);
		values.swap(scratch);
	}
}

template <class T, class Key>
void stable_radix_sort(std::vector<T> &values, Key key) {
	stable_radix_sort<8>(values, std::move(key));
}

namespace radix_sort_detail {
template <int Bits = 8, class T, class Key>
void sort_by_key(std::vector<T> &values, Key key) {
	using K = std::remove_cvref_t<decltype(key(std::declval<const T &>()))>;
	if constexpr (std::is_integral_v<K> && !std::is_same_v<K, bool> && sizeof(K) <= 8 && std::is_default_constructible_v<T>) {
		using U = std::make_unsigned_t<K>;
		constexpr U sign = std::is_signed_v<K> ? U(1) << (sizeof(K) * 8 - 1) : U(0);
		stable_radix_sort<Bits>(values, [&](const T &value) { return U(U(key(value)) ^ sign); });
	} else std::sort(values.begin(), values.end(), [&](const T &a, const T &b) { return key(a) < key(b); });
}
}

#line 1 "data_structure/rollback_union_find.hpp"

#line 8 "data_structure/rollback_union_find.hpp"

class RollbackUnionFind {
	struct Change { int child, size; };
	std::vector<int> parent_;
	std::vector<Change> history_;
	int components_ = 0;

public:
	explicit RollbackUnionFind(int n = 0) { reset(n); }
	void reset(int n) {
		assert(n >= 0);
		parent_.assign(n, -1), history_.clear();
		components_ = n;
	}
	void reserve(std::size_t updates) { history_.reserve(updates); }
	int vertex_count() const { return static_cast<int>(parent_.size()); }
	int component_count() const { return components_; }
	int leader(int x) const {
		assert(0 <= x && x < vertex_count());
		while (parent_[x] >= 0) x = parent_[x];
		return x;
	}
	bool same(int x, int y) const { return leader(x) == leader(y); }
	int size(int x) const { return -parent_[leader(x)]; }
	bool merge(int x, int y) {
		x = leader(x), y = leader(y);
		if (x == y) {
			history_.push_back({-1, 0});
			return false;
		}
		if (parent_[x] > parent_[y]) std::swap(x, y);
		history_.push_back({y, parent_[y]});
		parent_[x] += parent_[y], parent_[y] = x;
		--components_;
		return true;
	}
	std::size_t snapshot() const { return history_.size(); }
	void undo() {
		assert(!history_.empty());
		auto [child, size] = history_.back();
		history_.pop_back();
		if (child < 0) return;
		parent_[parent_[child]] -= size, parent_[child] = size;
		++components_;
	}
	void rollback(std::size_t state = 0) {
		assert(state <= history_.size());
		while (history_.size() > state) undo();
	}
	std::vector<std::vector<int>> groups() const {
		std::vector<std::vector<int>> result;
		result.reserve(components_);
		std::vector<int> index(vertex_count(), -1);
		for (int v = 0; v < vertex_count(); ++v) {
			int r = leader(v);
			if (index[r] == -1) {
				index[r] = static_cast<int>(result.size());
				result.emplace_back();
				result.back().reserve(-parent_[r]);
			}
			result[index[r]].push_back(v);
		}
		return result;
	}
};

#line 11 "graph/offline_dynamic_connectivity.hpp"
#include <limits>
#line 14 "graph/offline_dynamic_connectivity.hpp"

class OfflineDynamicConnectivity {
	struct Event { std::uint64_t key; int time; std::uint32_t order; };
	struct Query { int type, u, v; };
	int n_;
	std::vector<Event> events_;
	std::vector<Query> queries_;
	void check_([[maybe_unused]] int v) const { assert(0 <= v && v < n_); }
	void update_(int u, int v, bool remove) {
		check_(u), check_(v);
		if (u == v) return;
		assert(events_.size() < static_cast<std::size_t>(std::numeric_limits<int>::max()));
		if (u > v) std::swap(u, v);
		std::uint64_t key = (std::uint64_t(static_cast<unsigned>(u)) << 32) | static_cast<unsigned>(v);
		std::uint32_t order = static_cast<std::uint32_t>(events_.size()) | (std::uint32_t(remove) << 31);
		events_.push_back({key, query_count(), order});
	}
	int query_(int type, int u, int v) {
		assert(queries_.size() < static_cast<std::size_t>(std::numeric_limits<int>::max()));
		int id = query_count();
		queries_.push_back({type, u, v});
		return id;
	}
	template <bool Hooks, class Link, class Cut, class Visit>
	void run_(Link &link, Cut &cut, Visit &visit) const {
		if (queries_.empty()) return;
		auto events = events_;
		stable_radix_sort(events, [](const Event &event) { return event.key; });
		std::vector<OfflineTimeSegments::Interval> intervals;
		std::vector<std::pair<int, int>> edges;
		intervals.reserve(events.size() / 2), edges.reserve(events.size() / 2);
		for (std::size_t i = 0; i < events.size();) {
			std::uint64_t key = events[i].key;
			int count = 0, start = 0;
			auto append = [&](int end) {
				if (start == end) return;
				intervals.push_back({start, end, static_cast<int>(edges.size())});
				edges.emplace_back(static_cast<int>(key >> 32), static_cast<int>(key & 0xffffffffU));
			};
			do {
				const auto &event = events[i++];
				if (!(event.order >> 31)) {
					if (count++ == 0) start = event.time;
				} else if (count && --count == 0) append(event.time);
			} while (i < events.size() && events[i].key == key);
			if (count) append(query_count());
		}
		OfflineTimeSegments timeline(query_count(), intervals);
		struct State {
			RollbackUnionFind uf;
			std::vector<std::pair<int, int>> history;
			Link &link;
			Cut &cut;
			std::size_t snapshot() const { return uf.snapshot(); }
			void merge(int u, int v) {
				int a = uf.leader(u), b = uf.leader(v);
				if (a == b) return;
				if (uf.size(a) < uf.size(b)) std::swap(a, b);
				uf.merge(a, b);
				if constexpr (Hooks) {
					history.emplace_back(a, b);
					link(a, b);
				}
			}
			void rollback(std::size_t saved) {
				if constexpr (Hooks) {
					while (uf.snapshot() > saved) {
						auto [a, b] = history.back();
						cut(a, b);
						history.pop_back(), uf.undo();
					}
				} else uf.rollback(saved);
			}
		};
		State state{RollbackUnionFind(n_), {}, link, cut};
		std::size_t capacity = std::min<std::size_t>(n_ ? n_ - 1 : 0, edges.size());
		state.uf.reserve(capacity);
		if constexpr (Hooks) state.history.reserve(capacity);
		timeline.run(state, [&](State &s, int id) { s.merge(edges[id].first, edges[id].second); },
			[&](int id, const State &s) { visit(id, s.uf); });
	}

public:
	explicit OfflineDynamicConnectivity(int n = 0) : n_(n) { assert(n >= 0); }
	void reset(int n) { assert(n >= 0); n_ = n; events_.clear(), queries_.clear(); }
	void reserve(int updates, int queries = 0) {
		assert(updates >= 0 && queries >= 0);
		events_.reserve(updates), queries_.reserve(queries);
	}
	int vertex_count() const { return n_; }
	int query_count() const { return static_cast<int>(queries_.size()); }
	void link(int u, int v) { update_(u, v, false); }
	void cut(int u, int v) { update_(u, v, true); }
	int query_same(int u, int v) { check_(u), check_(v); return query_(0, u, v); }
	int query_size(int v) { check_(v); return query_(1, v, 0); }
	int query_component_count() { return query_(2, 0, 0); }
	int query() { return query_(3, 0, 0); }
	template <class Visit>
	void run(Visit &&visit) const {
		auto noop = [](int, int) {};
		run_<false>(noop, noop, visit);
	}
	template <class Link, class Cut, class Visit>
	void run(Link &&link, Cut &&cut, Visit &&visit) const { run_<true>(link, cut, visit); }
	std::vector<int> solve() const {
		std::vector<int> answers(queries_.size());
		run([&](int id, const RollbackUnionFind &uf) {
			const auto &q = queries_[id];
			if (q.type == 0) answers[id] = uf.same(q.u, q.v);
			else if (q.type == 1) answers[id] = uf.size(q.u);
			else { assert(q.type == 2); answers[id] = uf.component_count(); }
		});
		return answers;
	}
};

#ifndef OFFLINE_DYNAMIC_CONNECTIVITY_HPP
#define OFFLINE_DYNAMIC_CONNECTIVITY_HPP

#include "../components/offline_time_segments.hpp"
#include "../components/radix_sort.hpp"
#include "../data_structure/rollback_union_find.hpp"
#include <algorithm>
#include <cassert>
#include <cstddef>
#include <cstdint>
#include <limits>
#include <utility>
#include <vector>

class OfflineDynamicConnectivity {
	struct Event { std::uint64_t key; int time; std::uint32_t order; };
	struct Query { int type, u, v; };
	int n_;
	std::vector<Event> events_;
	std::vector<Query> queries_;
	void check_([[maybe_unused]] int v) const { assert(0 <= v && v < n_); }
	void update_(int u, int v, bool remove) {
		check_(u), check_(v);
		if (u == v) return;
		assert(events_.size() < static_cast<std::size_t>(std::numeric_limits<int>::max()));
		if (u > v) std::swap(u, v);
		std::uint64_t key = (std::uint64_t(static_cast<unsigned>(u)) << 32) | static_cast<unsigned>(v);
		std::uint32_t order = static_cast<std::uint32_t>(events_.size()) | (std::uint32_t(remove) << 31);
		events_.push_back({key, query_count(), order});
	}
	int query_(int type, int u, int v) {
		assert(queries_.size() < static_cast<std::size_t>(std::numeric_limits<int>::max()));
		int id = query_count();
		queries_.push_back({type, u, v});
		return id;
	}
	template <bool Hooks, class Link, class Cut, class Visit>
	void run_(Link &link, Cut &cut, Visit &visit) const {
		if (queries_.empty()) return;
		auto events = events_;
		stable_radix_sort(events, [](const Event &event) { return event.key; });
		std::vector<OfflineTimeSegments::Interval> intervals;
		std::vector<std::pair<int, int>> edges;
		intervals.reserve(events.size() / 2), edges.reserve(events.size() / 2);
		for (std::size_t i = 0; i < events.size();) {
			std::uint64_t key = events[i].key;
			int count = 0, start = 0;
			auto append = [&](int end) {
				if (start == end) return;
				intervals.push_back({start, end, static_cast<int>(edges.size())});
				edges.emplace_back(static_cast<int>(key >> 32), static_cast<int>(key & 0xffffffffU));
			};
			do {
				const auto &event = events[i++];
				if (!(event.order >> 31)) {
					if (count++ == 0) start = event.time;
				} else if (count && --count == 0) append(event.time);
			} while (i < events.size() && events[i].key == key);
			if (count) append(query_count());
		}
		OfflineTimeSegments timeline(query_count(), intervals);
		struct State {
			RollbackUnionFind uf;
			std::vector<std::pair<int, int>> history;
			Link &link;
			Cut &cut;
			std::size_t snapshot() const { return uf.snapshot(); }
			void merge(int u, int v) {
				int a = uf.leader(u), b = uf.leader(v);
				if (a == b) return;
				if (uf.size(a) < uf.size(b)) std::swap(a, b);
				uf.merge(a, b);
				if constexpr (Hooks) {
					history.emplace_back(a, b);
					link(a, b);
				}
			}
			void rollback(std::size_t saved) {
				if constexpr (Hooks) {
					while (uf.snapshot() > saved) {
						auto [a, b] = history.back();
						cut(a, b);
						history.pop_back(), uf.undo();
					}
				} else uf.rollback(saved);
			}
		};
		State state{RollbackUnionFind(n_), {}, link, cut};
		std::size_t capacity = std::min<std::size_t>(n_ ? n_ - 1 : 0, edges.size());
		state.uf.reserve(capacity);
		if constexpr (Hooks) state.history.reserve(capacity);
		timeline.run(state, [&](State &s, int id) { s.merge(edges[id].first, edges[id].second); },
			[&](int id, const State &s) { visit(id, s.uf); });
	}

  public:
	explicit OfflineDynamicConnectivity(int n = 0) : n_(n) { assert(n >= 0); }
	void reset(int n) { assert(n >= 0); n_ = n; events_.clear(), queries_.clear(); }
	void reserve(int updates, int queries = 0) {
		assert(updates >= 0 && queries >= 0);
		events_.reserve(updates), queries_.reserve(queries);
	}
	int vertex_count() const { return n_; }
	int query_count() const { return static_cast<int>(queries_.size()); }
	void link(int u, int v) { update_(u, v, false); }
	void cut(int u, int v) { update_(u, v, true); }
	int query_same(int u, int v) { check_(u), check_(v); return query_(0, u, v); }
	int query_size(int v) { check_(v); return query_(1, v, 0); }
	int query_component_count() { return query_(2, 0, 0); }
	int query() { return query_(3, 0, 0); }
	template <class Visit>
	void run(Visit &&visit) const {
		auto noop = [](int, int) {};
		run_<false>(noop, noop, visit);
	}
	template <class Link, class Cut, class Visit>
	void run(Link &&link, Cut &&cut, Visit &&visit) const { run_<true>(link, cut, visit); }
	std::vector<int> solve() const {
		std::vector<int> answers(queries_.size());
		run([&](int id, const RollbackUnionFind &uf) {
			const auto &q = queries_[id];
			if (q.type == 0) answers[id] = uf.same(q.u, q.v);
			else if (q.type == 1) answers[id] = uf.size(q.u);
			else { assert(q.type == 2); answers[id] = uf.component_count(); }
		});
		return answers;
	}
};

#endif

#line 1 "graph/offline_dynamic_connectivity.hpp"



#line 1 "components/offline_time_segments.hpp"



#include <bit>
#include <cassert>
#include <cstddef>
#include <utility>
#include <vector>

class OfflineTimeSegments {
	int n_;
	std::size_t base_;
	std::vector<std::size_t> offset_;
	std::vector<int> items_;

  public:
	struct Interval { int left, right, item; };
	OfflineTimeSegments(int n, const std::vector<Interval> &intervals) : n_(n), base_(1) {
		assert(n >= 0);
		if (!n || intervals.empty()) return;
		base_ = std::bit_ceil(static_cast<std::size_t>(n));
		offset_.resize(base_ * 2 + 1);
		auto cover = [&](const Interval &interval, auto &&visit) {
			assert(0 <= interval.left && interval.left <= interval.right && interval.right <= n);
			for (std::size_t l = base_ + interval.left, r = base_ + interval.right; l < r; l >>= 1, r >>= 1) {
				if (l & 1) visit(l++);
				if (r & 1) visit(--r);
			}
		};
		for (const auto &interval : intervals) cover(interval, [&](std::size_t v) { ++offset_[v]; });
		for (std::size_t v = 1; v < offset_.size(); ++v) offset_[v] += offset_[v - 1];
		items_.resize(offset_.back());
		for (const auto &interval : intervals)
			cover(interval, [&](std::size_t v) { items_[--offset_[v]] = interval.item; });
	}
	int size() const { return n_; }
	std::size_t entry_count() const { return items_.size(); }
	template <class State, class Apply, class Visit>
	void run(State &state, Apply &&apply, Visit &&visit) const {
		if (!n_) return;
		if (items_.empty()) {
			for (int t = 0; t < n_; ++t) visit(t, std::as_const(state));
			return;
		}
		auto dfs = [&](auto &&self, std::size_t v, std::size_t l, std::size_t r) -> void {
			auto saved = state.snapshot();
			for (std::size_t i = offset_[v]; i < offset_[v + 1]; ++i) apply(state, items_[i]);
			if (r - l == 1) visit(static_cast<int>(l), std::as_const(state));
			else {
				std::size_t m = (l + r) / 2;
				self(self, v * 2, l, m);
				if (m < static_cast<std::size_t>(n_)) self(self, v * 2 + 1, m, r);
			}
			state.rollback(saved);
		};
		dfs(dfs, 1, 0, base_);
	}
};


#line 1 "components/radix_sort.hpp"



#include <algorithm>
#include <array>
#line 7 "components/radix_sort.hpp"
#include <cstdint>
#include <type_traits>
#line 11 "components/radix_sort.hpp"

template <int Bits, class T, class Key>
void stable_radix_sort(std::vector<T> &values, Key &&key) {
	static_assert(1 <= Bits && Bits <= 16);
	constexpr unsigned buckets = 1U << Bits, mask = buckets - 1;
	using K = std::remove_cvref_t<decltype(key(std::declval<const T &>()))>;
	static_assert(std::is_integral_v<K> && std::is_unsigned_v<K> && sizeof(K) <= sizeof(std::uint64_t));
	if (values.size() < 2) return;
	if (values.size() < 256) {
		std::stable_sort(values.begin(), values.end(), [&](const T &a, const T &b) { return key(a) < key(b); });
		return;
	}
	std::uint64_t first = key(values.front()), previous = first, varying = 0;
	bool sorted = true;
	for (std::size_t i = 1; i < values.size(); ++i) {
		std::uint64_t current = key(values[i]);
		varying |= first ^ current;
		sorted &= previous <= current;
		previous = current;
	}
	if (sorted) return;
	std::vector<T> scratch(values.size());
	using Counts = std::conditional_t<(Bits <= 11), std::array<std::size_t, buckets>, std::vector<std::size_t>>;
	Counts count = [] {
		if constexpr (Bits <= 11) return Counts{};
		else return Counts(buckets);
	}();
	for (int shift = 0; shift < 64; shift += Bits) {
		if (!((varying >> shift) & mask)) continue;
		std::fill(count.begin(), count.end(), 0);
		for (const auto &value : values) ++count[(static_cast<std::uint64_t>(key(value)) >> shift) & mask];
		std::size_t sum = 0;
		for (auto &x : count) {
			auto n = x;
			x = sum, sum += n;
		}
		for (auto &&value : values) scratch[count[(static_cast<std::uint64_t>(key(value)) >> shift) & mask]++] = std::move(value);
		values.swap(scratch);
	}
}

template <class T, class Key>
void stable_radix_sort(std::vector<T> &values, Key key) {
	stable_radix_sort<8>(values, std::move(key));
}

namespace radix_sort_detail {
template <int Bits = 8, class T, class Key>
void sort_by_key(std::vector<T> &values, Key key) {
	using K = std::remove_cvref_t<decltype(key(std::declval<const T &>()))>;
	if constexpr (std::is_integral_v<K> && !std::is_same_v<K, bool> && sizeof(K) <= 8 && std::is_default_constructible_v<T>) {
		using U = std::make_unsigned_t<K>;
		constexpr U sign = std::is_signed_v<K> ? U(1) << (sizeof(K) * 8 - 1) : U(0);
		stable_radix_sort<Bits>(values, [&](const T &value) { return U(U(key(value)) ^ sign); });
	} else std::sort(values.begin(), values.end(), [&](const T &a, const T &b) { return key(a) < key(b); });
}
}


#line 1 "data_structure/rollback_union_find.hpp"



#line 8 "data_structure/rollback_union_find.hpp"

class RollbackUnionFind {
	struct Change { int child, size; };
	std::vector<int> parent_;
	std::vector<Change> history_;
	int components_ = 0;

  public:
	explicit RollbackUnionFind(int n = 0) { reset(n); }
	void reset(int n) {
		assert(n >= 0);
		parent_.assign(n, -1), history_.clear();
		components_ = n;
	}
	void reserve(std::size_t updates) { history_.reserve(updates); }
	int vertex_count() const { return static_cast<int>(parent_.size()); }
	int component_count() const { return components_; }
	int leader(int x) const {
		assert(0 <= x && x < vertex_count());
		while (parent_[x] >= 0) x = parent_[x];
		return x;
	}
	bool same(int x, int y) const { return leader(x) == leader(y); }
	int size(int x) const { return -parent_[leader(x)]; }
	bool merge(int x, int y) {
		x = leader(x), y = leader(y);
		if (x == y) {
			history_.push_back({-1, 0});
			return false;
		}
		if (parent_[x] > parent_[y]) std::swap(x, y);
		history_.push_back({y, parent_[y]});
		parent_[x] += parent_[y], parent_[y] = x;
		--components_;
		return true;
	}
	std::size_t snapshot() const { return history_.size(); }
	void undo() {
		assert(!history_.empty());
		auto [child, size] = history_.back();
		history_.pop_back();
		if (child < 0) return;
		parent_[parent_[child]] -= size, parent_[child] = size;
		++components_;
	}
	void rollback(std::size_t state = 0) {
		assert(state <= history_.size());
		while (history_.size() > state) undo();
	}
	std::vector<std::vector<int>> groups() const {
		std::vector<std::vector<int>> result;
		result.reserve(components_);
		std::vector<int> index(vertex_count(), -1);
		for (int v = 0; v < vertex_count(); ++v) {
			int r = leader(v);
			if (index[r] == -1) {
				index[r] = static_cast<int>(result.size());
				result.emplace_back();
				result.back().reserve(-parent_[r]);
			}
			result[index[r]].push_back(v);
		}
		return result;
	}
};


#line 11 "graph/offline_dynamic_connectivity.hpp"
#include <limits>
#line 14 "graph/offline_dynamic_connectivity.hpp"

class OfflineDynamicConnectivity {
	struct Event { std::uint64_t key; int time; std::uint32_t order; };
	struct Query { int type, u, v; };
	int n_;
	std::vector<Event> events_;
	std::vector<Query> queries_;
	void check_([[maybe_unused]] int v) const { assert(0 <= v && v < n_); }
	void update_(int u, int v, bool remove) {
		check_(u), check_(v);
		if (u == v) return;
		assert(events_.size() < static_cast<std::size_t>(std::numeric_limits<int>::max()));
		if (u > v) std::swap(u, v);
		std::uint64_t key = (std::uint64_t(static_cast<unsigned>(u)) << 32) | static_cast<unsigned>(v);
		std::uint32_t order = static_cast<std::uint32_t>(events_.size()) | (std::uint32_t(remove) << 31);
		events_.push_back({key, query_count(), order});
	}
	int query_(int type, int u, int v) {
		assert(queries_.size() < static_cast<std::size_t>(std::numeric_limits<int>::max()));
		int id = query_count();
		queries_.push_back({type, u, v});
		return id;
	}
	template <bool Hooks, class Link, class Cut, class Visit>
	void run_(Link &link, Cut &cut, Visit &visit) const {
		if (queries_.empty()) return;
		auto events = events_;
		stable_radix_sort(events, [](const Event &event) { return event.key; });
		std::vector<OfflineTimeSegments::Interval> intervals;
		std::vector<std::pair<int, int>> edges;
		intervals.reserve(events.size() / 2), edges.reserve(events.size() / 2);
		for (std::size_t i = 0; i < events.size();) {
			std::uint64_t key = events[i].key;
			int count = 0, start = 0;
			auto append = [&](int end) {
				if (start == end) return;
				intervals.push_back({start, end, static_cast<int>(edges.size())});
				edges.emplace_back(static_cast<int>(key >> 32), static_cast<int>(key & 0xffffffffU));
			};
			do {
				const auto &event = events[i++];
				if (!(event.order >> 31)) {
					if (count++ == 0) start = event.time;
				} else if (count && --count == 0) append(event.time);
			} while (i < events.size() && events[i].key == key);
			if (count) append(query_count());
		}
		OfflineTimeSegments timeline(query_count(), intervals);
		struct State {
			RollbackUnionFind uf;
			std::vector<std::pair<int, int>> history;
			Link &link;
			Cut &cut;
			std::size_t snapshot() const { return uf.snapshot(); }
			void merge(int u, int v) {
				int a = uf.leader(u), b = uf.leader(v);
				if (a == b) return;
				if (uf.size(a) < uf.size(b)) std::swap(a, b);
				uf.merge(a, b);
				if constexpr (Hooks) {
					history.emplace_back(a, b);
					link(a, b);
				}
			}
			void rollback(std::size_t saved) {
				if constexpr (Hooks) {
					while (uf.snapshot() > saved) {
						auto [a, b] = history.back();
						cut(a, b);
						history.pop_back(), uf.undo();
					}
				} else uf.rollback(saved);
			}
		};
		State state{RollbackUnionFind(n_), {}, link, cut};
		std::size_t capacity = std::min<std::size_t>(n_ ? n_ - 1 : 0, edges.size());
		state.uf.reserve(capacity);
		if constexpr (Hooks) state.history.reserve(capacity);
		timeline.run(state, [&](State &s, int id) { s.merge(edges[id].first, edges[id].second); },
			[&](int id, const State &s) { visit(id, s.uf); });
	}

  public:
	explicit OfflineDynamicConnectivity(int n = 0) : n_(n) { assert(n >= 0); }
	void reset(int n) { assert(n >= 0); n_ = n; events_.clear(), queries_.clear(); }
	void reserve(int updates, int queries = 0) {
		assert(updates >= 0 && queries >= 0);
		events_.reserve(updates), queries_.reserve(queries);
	}
	int vertex_count() const { return n_; }
	int query_count() const { return static_cast<int>(queries_.size()); }
	void link(int u, int v) { update_(u, v, false); }
	void cut(int u, int v) { update_(u, v, true); }
	int query_same(int u, int v) { check_(u), check_(v); return query_(0, u, v); }
	int query_size(int v) { check_(v); return query_(1, v, 0); }
	int query_component_count() { return query_(2, 0, 0); }
	int query() { return query_(3, 0, 0); }
	template <class Visit>
	void run(Visit &&visit) const {
		auto noop = [](int, int) {};
		run_<false>(noop, noop, visit);
	}
	template <class Link, class Cut, class Visit>
	void run(Link &&link, Cut &&cut, Visit &&visit) const { run_<true>(link, cut, visit); }
	std::vector<int> solve() const {
		std::vector<int> answers(queries_.size());
		run([&](int id, const RollbackUnionFind &uf) {
			const auto &q = queries_[id];
			if (q.type == 0) answers[id] = uf.same(q.u, q.v);
			else if (q.type == 1) answers[id] = uf.size(q.u);
			else { assert(q.type == 2); answers[id] = uf.component_count(); }
		});
		return answers;
	}
};