最小共通祖先 (lowest common ancestor) Euler tour technique 版
(include/emthrm/graph/tree/lowest_common_ancestor_by_euler_tour_technique.hpp)

View this file on GitHub
Last update: 2023-05-12 15:57:02+09:00
Include: #include "emthrm/graph/tree/lowest_common_ancestor_by_euler_tour_technique.hpp"

最小共通祖先 (lowest common ancestor)

根付き木のある2頂点に対して最も深い共通祖先である。

時間計算量

	時間計算量
ダブリング版	$\langle O(\lvert V \rvert \log{\lvert V \rvert}), O(\log{\lvert V \rvert}) \rangle$
Euler tour technique 版	$\langle O(\lvert V \rvert \log{\lvert V \rvert}), O(1) \rangle$

仕様

ダブリング版

template <typename CostType>
struct LowestCommonAncestorByDoubling;

CostType：辺のコストを表す型

メンバ変数

名前	説明	備考
`std::vector<int> depth`	`depth[i]` は頂点 $i$ の深さを表す。
`std::vector<CostType> dist`	`dist[i]` は根と頂点 $i$ の間の距離を表す。	cost-free 版は `depth` に同じである。

メンバ関数

名前	効果・戻り値	要件
`explicit LowestCommonAncestorByDoubling(const std::vector<std::vector<Edge<CostType>>>& graph);`	木 $\mathrm{graph}$ に対してオブジェクトを構築する。
`void build(const int root = 0);`	根を $\mathrm{root}$ として構築する。
`int query(int u, int v) const;`	頂点 $u, v$ の最小共通祖先
`CostType distance(const int u, const int v) const;`	頂点 $u, v$ 間の距離
`int level_ancestor(int v, const int d) const;`	level ancestor problem $\mathrm{LA}(v, d)$。ただし $d \leq \mathrm{depth}(v)$ でなければ $-1$ を返す。
`int jump(const int u, const int v, const int d) const;`	頂点 $u$ から頂点 $v$ まで距離 $d$ だけ進んだときの頂点。ただし $d > \mathrm{dist}(u, v)$ を満たすときは $-1$ を返す。	cost-free 版

Euler tour technique版

template <typename CostType>
struct LowestCommonAncestor : EulerTourTechnique<CostType>;

CostType：辺のコストを表す型

メンバ関数

名前	効果・戻り値
`explicit LowestCommonAncestor(const std::vector<std::vector<Edge<CostType>>>& graph, const int root = 0);`	根を $\mathrm{root}$ とする木 $\mathrm{graph}$ に対してオブジェクトを構築する。
`int query(int u, int v) const;`	頂点 $u, v$ の最小共通祖先

heavy-light decomposition 版

参考文献

秋葉拓哉，岩田陽一，北川宜稔：プログラミングコンテストチャレンジブック [第2版]，pp.292-295，マイナビ出版（2012）
https://yukicoder.me/wiki/lowest_common_ancestor

level ancestor problem

https://en.wikipedia.org/wiki/Level_ancestor_problem

Euler tour technique 版

Omer Berkman and Uzi Vishkin: Recursive Star-Tree Parallel Data Structure, SIAM Journal on Computing, Vol. 22, No. 2, pp. 221–242 (1993). https://doi.org/10.1137/0222017
https://github.com/drken1215/algorithm/blob/efb8cf052b095e49e70135a6fb628308d06f49b2/DataStructureOnTree/euler_tour_on_nodes.cpp

TODO

Tarjan’s off-line lowest common ancestors algorithm
- ~~http://www.prefield.com/algorithm/graph/least_common_ancestor.html~~
- https://github.com/spaghetti-source/algorithm/blob/master/graph/least_common_ancestor_tarjan.cc
- http://monyone.github.io/teihen_library/#OfflineLCA
level ancestor problem $\langle O(\lvert V \rvert), O(1) \rangle$
- https://www2.compute.dtu.dk/courses/02282/2018/levelancestor/levelancestor1x1.pdf
- https://37zigen.com/level-ancestor-problem/
- https://hdbn.hatenadiary.org/entry/20111125/1322194487
- https://drive.google.com/drive/folders/1atQRO6Y9bHgLDH-YLq3obDwMxIuk7–h
- https://noshi91.hatenablog.com/entry/2019/09/22/114149
- https://suisen-kyopro.hatenablog.com/entry/2022/04/04/043452
- https://twitter.com/GauravML/status/1415073033522319367
- https://twitter.com/keijak/status/1510953079872663555
- https://github.com/noshi91/n91lib_rs/blob/c9bd9cf36cbf4637884a049a4fe44f45b06ff71f/src/data_structure/level_ancestor.rs

Submissons

Depends on

Verified with

グラフ/木/最小共通祖先 Euler tour technique 版 (test/graph/tree/lowest_common_ancestor_by_euler_tour.test.cpp)

Code

#ifndef EMTHRM_GRAPH_TREE_LOWEST_COMMON_ANCESTOR_BY_EULER_TOUR_TECHNIQUE_HPP_
#define EMTHRM_GRAPH_TREE_LOWEST_COMMON_ANCESTOR_BY_EULER_TOUR_TECHNIQUE_HPP_

#include <algorithm>
#include <utility>
#include <vector>

#include "emthrm/data_structure/sparse_table.hpp"
#include "emthrm/graph/edge.hpp"
#include "emthrm/graph/tree/euler_tour_technique.hpp"

namespace emthrm {

template <typename CostType>
struct LowestCommonAncestor : EulerTourTechnique<CostType> {
  explicit LowestCommonAncestor(
      const std::vector<std::vector<Edge<CostType>>>& graph,
      const int root = 0)
      : EulerTourTechnique<CostType>(graph, root) {
    const int n = this->preorder.size();
    std::vector<std::pair<int, int>> nodes(n);
    for (int i = 0; i < n; ++i) {
      nodes[i] = {this->depth[i], this->preorder[i]};
    }
    sparse_table.init(
        nodes,
        [](const std::pair<int, int>& a, const std::pair<int, int>& b)
            -> std::pair<int, int> {
          return std::min(a, b);
        });
  }

  int query(int u, int v) const {
    u = this->left[u];
    v = this->left[v];
    if (u > v) std::swap(u, v);
    return sparse_table.query(u, v + 1).second;
  }

 private:
  SparseTable<std::pair<int, int>> sparse_table;
};

}  // namespace emthrm

#endif  // EMTHRM_GRAPH_TREE_LOWEST_COMMON_ANCESTOR_BY_EULER_TOUR_TECHNIQUE_HPP_

#line 1 "include/emthrm/graph/tree/lowest_common_ancestor_by_euler_tour_technique.hpp"



#include <algorithm>
#include <utility>
#include <vector>

#line 1 "include/emthrm/data_structure/sparse_table.hpp"



#line 5 "include/emthrm/data_structure/sparse_table.hpp"
#include <bit>
#include <cassert>
#include <functional>
#line 9 "include/emthrm/data_structure/sparse_table.hpp"

namespace emthrm {

template <typename Band>
struct SparseTable {
  using BinOp = std::function<Band(Band, Band)>;

  SparseTable() = default;

  explicit SparseTable(const std::vector<Band>& a, const BinOp bin_op) {
    init(a, bin_op);
  }

  void init(const std::vector<Band>& a, const BinOp bin_op_) {
    bin_op = bin_op_;
    const int n = a.size();
    assert(n > 0);
    lg.assign(n + 1, 0);
    for (int i = 2; i <= n; ++i) {
      lg[i] = lg[i >> 1] + 1;
    }
    const int table_h = std::countr_zero(std::bit_floor(a.size())) + 1;
    data.assign(table_h, std::vector<Band>(n));
    std::copy(a.begin(), a.end(), data.front().begin());
    for (int i = 1; i < table_h; ++i) {
      for (int j = 0; j + (1 << i) <= n; ++j) {
        data[i][j] = bin_op(data[i - 1][j], data[i - 1][j + (1 << (i - 1))]);
      }
    }
  }

  Band query(const int left, const int right) const {
    assert(left < right);
    const int h = lg[right - left];
    return bin_op(data[h][left], data[h][right - (1 << h)]);
  }

 private:
  BinOp bin_op;
  std::vector<int> lg;
  std::vector<std::vector<Band>> data;
};

}  // namespace emthrm


#line 1 "include/emthrm/graph/edge.hpp"
/**
 * @title 辺
 */

#ifndef EMTHRM_GRAPH_EDGE_HPP_
#define EMTHRM_GRAPH_EDGE_HPP_

#include <compare>

namespace emthrm {

template <typename CostType>
struct Edge {
  CostType cost;
  int src, dst;

  explicit Edge(const int src, const int dst, const CostType cost = 0)
      : cost(cost), src(src), dst(dst) {}

  auto operator<=>(const Edge& x) const = default;
};

}  // namespace emthrm

#endif  // EMTHRM_GRAPH_EDGE_HPP_
#line 1 "include/emthrm/graph/tree/euler_tour_technique.hpp"



#line 5 "include/emthrm/graph/tree/euler_tour_technique.hpp"

#line 1 "include/emthrm/graph/edge.hpp"
/**
 * @title 辺
 */

#ifndef EMTHRM_GRAPH_EDGE_HPP_
#define EMTHRM_GRAPH_EDGE_HPP_

#include <compare>

namespace emthrm {

template <typename CostType>
struct Edge {
  CostType cost;
  int src, dst;

  explicit Edge(const int src, const int dst, const CostType cost = 0)
      : cost(cost), src(src), dst(dst) {}

  auto operator<=>(const Edge& x) const = default;
};

}  // namespace emthrm

#endif  // EMTHRM_GRAPH_EDGE_HPP_
#line 7 "include/emthrm/graph/tree/euler_tour_technique.hpp"

namespace emthrm {

template <typename CostType>
struct EulerTourTechnique {
  std::vector<int> preorder, depth, left, right, down, up;
  std::vector<CostType> tour;

  explicit EulerTourTechnique(
      const std::vector<std::vector<Edge<CostType>>> &graph, const int root = 0)
      : graph(graph) {
    const int n = graph.size();
    left.resize(n);
    right.resize(n);
    down.assign(n, -1);
    up.assign(n, (n - 1) << 1);
    dfs(-1, root, 0);
  }

  template <typename Fn>
  void update_v(const int ver, const Fn f) const {
    f(left[ver], right[ver] + 1);
  }

  template <typename T, typename Fn>
  T query_v(const int ver, const Fn f) const {
    return f(left[ver], right[ver] + 1);
  }

  template <typename T, typename Fn>
  T query_e(const int u, const int v, const Fn f) const {
    return f(down[u] + 1, down[v] + 1);
  }

  template <typename Fn>
  void update_subtree_e(const int ver, const Fn f) const {
    f(down[ver] + 1, up[ver]);
  }

  template <typename T, typename Fn>
  T query_subtree_e(const int ver, const Fn f) const {
    return f(down[ver] + 1, up[ver]);
  }

 private:
  const std::vector<std::vector<Edge<CostType>>> graph;

  void dfs(const int par, const int ver, const int cur_depth) {
    left[ver] = preorder.size();
    preorder.emplace_back(ver);
    depth.emplace_back(cur_depth);
    for (const Edge<CostType>& e : graph[ver]) {
      if (e.dst != par) {
        down[e.dst] = tour.size();
        tour.emplace_back(e.cost);
        dfs(ver, e.dst, cur_depth + 1);
        preorder.emplace_back(ver);
        depth.emplace_back(cur_depth);
        up[e.dst] = tour.size();
        tour.emplace_back(-e.cost);
      }
    }
    right[ver] = preorder.size() - 1;
  }
};

}  // namespace emthrm


#line 11 "include/emthrm/graph/tree/lowest_common_ancestor_by_euler_tour_technique.hpp"

namespace emthrm {

template <typename CostType>
struct LowestCommonAncestor : EulerTourTechnique<CostType> {
  explicit LowestCommonAncestor(
      const std::vector<std::vector<Edge<CostType>>>& graph,
      const int root = 0)
      : EulerTourTechnique<CostType>(graph, root) {
    const int n = this->preorder.size();
    std::vector<std::pair<int, int>> nodes(n);
    for (int i = 0; i < n; ++i) {
      nodes[i] = {this->depth[i], this->preorder[i]};
    }
    sparse_table.init(
        nodes,
        [](const std::pair<int, int>& a, const std::pair<int, int>& b)
            -> std::pair<int, int> {
          return std::min(a, b);
        });
  }

  int query(int u, int v) const {
    u = this->left[u];
    v = this->left[v];
    if (u > v) std::swap(u, v);
    return sparse_table.query(u, v + 1).second;
  }

 private:
  SparseTable<std::pair<int, int>> sparse_table;
};

}  // namespace emthrm

最小共通祖先 (lowest common ancestor) Euler tour technique 版 (include/emthrm/graph/tree/lowest_common_ancestor_by_euler_tour_technique.hpp)

最小共通祖先 (lowest common ancestor)

時間計算量

仕様

ダブリング版

メンバ変数

メンバ関数

Euler tour technique版

メンバ関数

heavy-light decomposition 版

参考文献

TODO

Submissons

Depends on

Verified with

Code

最小共通祖先 (lowest common ancestor) Euler tour technique 版
(include/emthrm/graph/tree/lowest_common_ancestor_by_euler_tour_technique.hpp)