オフライン最小共通祖先 (Offline Lowest Common Ancestor)
(src/graph/tree/offline-query/offline-query-lowest-common-ancestor.hpp)

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• Last update: 2023-07-30 00:54:52+09:00
• Include: #include "src/graph/tree/offline-query/offline-query-lowest-common-ancestor.hpp"

Tarjan’s Offline Algorithm によって実装されているオフライン最小共通祖先

Appendix

テンプレートパラメータに渡すグラフ $G$ の仕様について

コンストラクタ

OfflineLCAQuery(const G &g)

最小共通祖先を求める木 g を渡す。

add_query

void add_query(usize u, usize v)

頂点 $u$ と $v$ の最小共通祖先 $lca(u, v)$ を求めるクエリを追加する。

後述する build をした後、lca(u, v) によって $lca(u, v)$ を得ることができる。

制約

• $0 \leq u, v < |V|$

計算量

• $O(1)$

build

void build(usize root)

頂点 root を根とする根付き木上での最小共通祖先を計算する。

制約

• $0 \leq root < |V|$

計算量

• $O((|V| + Q) \alpha (|V|))$
  • 追加されたクエリの数を $Q$ とする
  • アッカーマン関数の逆関数を $\alpha(n)$ とする

lca

usize lca(usize u, usize v)

$lca(u, v)$ を返す。根は直近で呼ばれた build(root) に依存する。

制約

• add_query(u, v) が呼ばれており、かつその後に build(root) が呼ばれていること。

計算量

• $O(1)$

Depends on

• rep 構造体 (src/cpp-template/header/rep.hpp)
• size alias (src/cpp-template/header/size-alias.hpp)
• Disjoint Set Union (Union Find) (src/data-structure/disjoint-set-union.hpp)
• std::pair の Hash (src/utility/pair-hash.hpp)

Required by

• (offine) 木のパス $u-v$ 上の $k$ 番目の頂点 (Offline Jump On Tree) (src/graph/tree/offline-query/offline-query-jump-on-tree.hpp)

Verified with

• test/aoj/grl_5_c/grl_5_c.test.cpp
• test/library-checker/jump_on_tree.test.cpp

Code

#pragma once

#include "src/cpp-template/header/rep.hpp"
#include "src/cpp-template/header/size-alias.hpp"
#include "src/data-structure/disjoint-set-union.hpp"
#include "src/utility/pair-hash.hpp"

#include <cassert>
#include <unordered_map>
#include <utility>
#include <vector>

namespace luz {

  template < class G >
  class OfflineLCAQuery {
    using graph     = G;
    using cost_type = typename G::cost_type;
    usize g_size;
    graph g;

    usize query_count;
    std::vector< std::vector< usize > > qs;

    DisjointSetUnion dsu;
    std::vector< bool > visited;
    std::vector< usize > ancestors;

    using query_type = std::pair< usize, usize >;
    std::unordered_map< query_type, usize, PairHash > results;

    void bound_check(usize v) const {
      assert(v < g_size);
    }

    void dfs(usize v) {
      visited[v]   = true;
      ancestors[v] = v;

      for (const auto &e: g[v]) {
        if (visited[e.to]) continue;
        dfs(e.to);
        dsu.merge(v, e.to);
        ancestors[dsu.leader(v)] = v;
      }

      for (const auto &u: qs[v]) {
        if (not visited[u]) continue;
        results[query_type(u, v)] = results[query_type(v, u)] =
            ancestors[dsu.leader(u)];
      }
    }

   public:
    using Queries = std::vector< std::pair< usize, usize > >;

    OfflineLCAQuery(G &g)
        : g_size(g.size()),
          g(g),
          query_count(0),
          qs(g_size),
          dsu(g_size),
          visited(g_size, false),
          ancestors(g_size) {}

    void add_query(usize u, usize v) {
      bound_check(u);
      bound_check(v);
      qs[u].emplace_back(v);
      qs[v].emplace_back(u);
      query_count++;
    }

    void build(usize root) {
      bound_check(root);
      results.reserve(2 * query_count);
      dfs(root);
    }

    usize lca(usize u, usize v) {
      bound_check(u);
      bound_check(v);
      query_type qi(u, v);
      assert(results.count(qi));
      return results[qi];
    }
  };

} // namespace luz

#line 2 "src/graph/tree/offline-query/offline-query-lowest-common-ancestor.hpp"

#line 2 "src/cpp-template/header/rep.hpp"

#line 2 "src/cpp-template/header/size-alias.hpp"

#include <cstddef>

namespace luz {

  using isize = std::ptrdiff_t;
  using usize = std::size_t;

} // namespace luz
#line 4 "src/cpp-template/header/rep.hpp"

#include <algorithm>

namespace luz {

  struct rep {
    struct itr {
      usize i;
      constexpr itr(const usize i) noexcept: i(i) {}
      void operator++() noexcept {
        ++i;
      }
      constexpr usize operator*() const noexcept {
        return i;
      }
      constexpr bool operator!=(const itr x) const noexcept {
        return i != x.i;
      }
    };
    const itr f, l;
    constexpr rep(const usize f, const usize l) noexcept
        : f(std::min(f, l)),
          l(l) {}
    constexpr auto begin() const noexcept {
      return f;
    }
    constexpr auto end() const noexcept {
      return l;
    }
  };

  struct rrep {
    struct itr {
      usize i;
      constexpr itr(const usize i) noexcept: i(i) {}
      void operator++() noexcept {
        --i;
      }
      constexpr usize operator*() const noexcept {
        return i;
      }
      constexpr bool operator!=(const itr x) const noexcept {
        return i != x.i;
      }
    };
    const itr f, l;
    constexpr rrep(const usize f, const usize l) noexcept
        : f(l - 1),
          l(std::min(f, l) - 1) {}
    constexpr auto begin() const noexcept {
      return f;
    }
    constexpr auto end() const noexcept {
      return l;
    }
  };

} // namespace luz
#line 2 "src/data-structure/disjoint-set-union.hpp"

#line 5 "src/data-structure/disjoint-set-union.hpp"

#line 7 "src/data-structure/disjoint-set-union.hpp"
#include <cassert>
#include <vector>

namespace luz {

  class DisjointSetUnion {
    usize n;

    // vals[v] :=
    //   if v is root node: -1 * component size
    //   otherwise: parent node
    std::vector< isize > vals;

    void bound_check(usize v) const {
      assert(v < n);
    }

    usize impl_leader(usize v) {
      if (vals[v] < 0) return v;
      return vals[v] = leader(vals[v]);
    }

   public:
    DisjointSetUnion() = default;
    explicit DisjointSetUnion(usize n): n(n), vals(n, -1) {}

    usize size() const {
      return n;
    }

    usize leader(usize v) {
      bound_check(v);
      return impl_leader(v);
    }

    bool same(usize u, usize v) {
      bound_check(u), bound_check(v);
      return impl_leader(u) == impl_leader(v);
    }

    usize merge(usize u, usize v) {
      bound_check(u);
      bound_check(v);

      isize x = impl_leader(u);
      isize y = impl_leader(v);
      if (x == y) return x;
      if (-vals[x] < -vals[y]) std::swap(x, y);
      vals[x] += vals[y];
      vals[y] = x;
      return x;
    }

    usize group_size(usize v) {
      bound_check(v);
      return -vals[impl_leader(v)];
    }

    std::vector< std::vector< usize > > groups() {
      std::vector< std::vector< usize > > result(n);

      std::vector< usize > leaders(n), g_sizes(n);
      for (usize v: rep(0, n)) {
        leaders[v] = impl_leader(v);
        g_sizes[leaders[v]]++;
      }
      for (usize v: rep(0, n)) {
        result[v].reserve(g_sizes[v]);
      }
      for (usize v: rep(0, n)) {
        result[leaders[v]].emplace_back(v);
      }

      auto empty_check = [](const std::vector< usize > &vs) {
        return vs.empty();
      };
      result.erase(
          std::remove_if(result.begin(), result.end(), empty_check),
          result.end());

      return result;
    }
  };

} // namespace luz
#line 2 "src/utility/pair-hash.hpp"

#line 4 "src/utility/pair-hash.hpp"

#include <functional>
#include <utility>

namespace luz {

  class PairHash {
    template < typename T >
    usize hash_combine(usize hr, const T &x) const {
      usize h = std::hash< T >()(x);
      return hr ^ (h + (hr << 11) + (hr >> 13));
    }

   public:
    template < typename F, typename S >
    usize operator()(const std::pair< F, S > &p) const {
      return hash_combine(hash_combine(0, p.first), p.second);
    }
  };

} // namespace luz
#line 7 "src/graph/tree/offline-query/offline-query-lowest-common-ancestor.hpp"

#line 9 "src/graph/tree/offline-query/offline-query-lowest-common-ancestor.hpp"
#include <unordered_map>
#line 12 "src/graph/tree/offline-query/offline-query-lowest-common-ancestor.hpp"

namespace luz {

  template < class G >
  class OfflineLCAQuery {
    using graph     = G;
    using cost_type = typename G::cost_type;
    usize g_size;
    graph g;

    usize query_count;
    std::vector< std::vector< usize > > qs;

    DisjointSetUnion dsu;
    std::vector< bool > visited;
    std::vector< usize > ancestors;

    using query_type = std::pair< usize, usize >;
    std::unordered_map< query_type, usize, PairHash > results;

    void bound_check(usize v) const {
      assert(v < g_size);
    }

    void dfs(usize v) {
      visited[v]   = true;
      ancestors[v] = v;

      for (const auto &e: g[v]) {
        if (visited[e.to]) continue;
        dfs(e.to);
        dsu.merge(v, e.to);
        ancestors[dsu.leader(v)] = v;
      }

      for (const auto &u: qs[v]) {
        if (not visited[u]) continue;
        results[query_type(u, v)] = results[query_type(v, u)] =
            ancestors[dsu.leader(u)];
      }
    }

   public:
    using Queries = std::vector< std::pair< usize, usize > >;

    OfflineLCAQuery(G &g)
        : g_size(g.size()),
          g(g),
          query_count(0),
          qs(g_size),
          dsu(g_size),
          visited(g_size, false),
          ancestors(g_size) {}

    void add_query(usize u, usize v) {
      bound_check(u);
      bound_check(v);
      qs[u].emplace_back(v);
      qs[v].emplace_back(u);
      query_count++;
    }

    void build(usize root) {
      bound_check(root);
      results.reserve(2 * query_count);
      dfs(root);
    }

    usize lca(usize u, usize v) {
      bound_check(u);
      bound_check(v);
      query_type qi(u, v);
      assert(results.count(qi));
      return results[qi];
    }
  };

} // namespace luz

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