指定された頂点たちの最小共通祖先関係を保って木を圧縮してできる補助的な木
(include/emthrm/graph/tree/auxiliary_tree.hpp)

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• Last update: 2024-08-10 05:09:34+09:00
• Include: #include "emthrm/graph/tree/auxiliary_tree.hpp"

時間計算量

指定する頂点の数を $K$ とおくと $\langle O(\lvert V \rvert \log{\lvert V \rvert}), O(\lvert K \rvert \log{\lvert K \rvert}) \rangle$

仕様

template <typename CostType>
struct AuxiliaryTree;

• CostType：辺のコストを表す型

メンバ変数

名前	説明
LowestCommonAncestor<CostType> euler_tour	Euler tour technique

メンバ関数

名前	効果
explicit AuxiliaryTree(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。
std::pair<std::vector<std::vector<Edge<CostType>>>, std::vector<int>> query(std::vector<int> vertices) const;	頂点集合 $\mathrm{vertices}$ に対して構築した木

参考文献

• https://noshi91.github.io/algorithm-encyclopedia/auxiliary-tree
• https://www.slideshare.net/iwiwi/2-12188845/7
• https://sotanishy.github.io/cp-library-cpp/tree/auxiliary_tree.cpp

Submissons

https://atcoder.jp/contests/typical90/submissions/56497303

Depends on

• sparse table (include/emthrm/data_structure/sparse_table.hpp)
• 辺 (include/emthrm/graph/edge.hpp)
• Euler tour technique (include/emthrm/graph/tree/euler_tour_technique.hpp)
• 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)

Verified with

• グラフ/木/指定された頂点たちの最小共通祖先関係を保って木を圧縮してできる補助的な木 (test/graph/tree/auxiliary_tree.test.cpp)

Code

#ifndef EMTHRM_GRAPH_TREE_AUXILIARY_TREE_HPP_
#define EMTHRM_GRAPH_TREE_AUXILIARY_TREE_HPP_

#include <algorithm>
#include <iterator>
#include <ranges>
#include <utility>
#include <vector>

#include "emthrm/graph/edge.hpp"
#include "emthrm/graph/tree/lowest_common_ancestor_by_euler_tour_technique.hpp"

namespace emthrm {

template <typename CostType>
struct AuxiliaryTree {
  LowestCommonAncestor<CostType> euler_tour;

  explicit AuxiliaryTree(
      const std::vector<std::vector<Edge<CostType>>>& graph, const int root = 0)
      : euler_tour(graph, root), depth(graph.size(), 0) {
    const auto dfs = [this, &graph](
        auto dfs, const int par, const int ver) -> void {
      for (const Edge<CostType>& e : graph[ver]) {
        if (e.dst == par) continue;
        depth[e.dst] = depth[ver] + e.cost;
        dfs(dfs, ver, e.dst);
      }
    };
    dfs(dfs, -1, root);
  }

  std::pair<std::vector<std::vector<Edge<CostType>>>,
            std::vector<int>> query(std::vector<int> vertices) const {
    if (vertices.empty()) [[unlikely]] return {};
    static std::vector<int> mp(depth.size(), -1);
    std::ranges::sort(vertices, {},
                      [&](const int v) -> int { return euler_tour.left[v]; });
    std::vector<std::vector<Edge<CostType>>> graph;
    std::vector<int> inv;
    const auto add_vertex = [this, &graph, &inv](const int v) -> void {
      mp[v] = graph.size();
      graph.emplace_back();
      inv.emplace_back(v);
    };
    const auto add_edge = [this, &graph](int u, int v) -> void {
      const CostType cost = depth[v] - depth[u];
      u = mp[u];
      v = mp[v];
      graph[u].emplace_back(u, v, cost);
      graph[v].emplace_back(v, u, cost);
    };
    add_vertex(vertices.front());
    std::vector<int> stck{vertices.front()};
    for (const int i : std::views::iota(1, std::ssize(vertices))) {
      const int l = euler_tour.query(vertices[i - 1], vertices[i]);
      if (mp[l] == -1) add_vertex(l);
      if (l != vertices[i - 1]) {
        const int depth_l = euler_tour.depth[euler_tour.left[l]];
        int cur = stck.back();
        stck.pop_back();
        while (!stck.empty() &&
               euler_tour.depth[euler_tour.left[stck.back()]] > depth_l) {
          add_edge(stck.back(), cur);
          cur = stck.back();
          stck.pop_back();
        }
        add_edge(l, cur);
        if (stck.empty() || stck.back() != l) stck.emplace_back(l);
      }
      add_vertex(vertices[i]);
      stck.emplace_back(vertices[i]);
    }
    for (; stck.size() >= 2; stck.pop_back()) {
      add_edge(stck.end()[-2], stck.back());
    }
    for (const int v : inv) mp[v] = -1;
    return {graph, inv};
  }

 private:
  std::vector<CostType> depth;
};

}  // namespace emthrm

#endif  // EMTHRM_GRAPH_TREE_AUXILIARY_TREE_HPP_

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



#include <algorithm>
#include <iterator>
#include <ranges>
#include <utility>
#include <vector>

#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/lowest_common_ancestor_by_euler_tour_technique.hpp"



#line 7 "include/emthrm/graph/tree/lowest_common_ancestor_by_euler_tour_technique.hpp"

#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


#line 12 "include/emthrm/graph/tree/auxiliary_tree.hpp"

namespace emthrm {

template <typename CostType>
struct AuxiliaryTree {
  LowestCommonAncestor<CostType> euler_tour;

  explicit AuxiliaryTree(
      const std::vector<std::vector<Edge<CostType>>>& graph, const int root = 0)
      : euler_tour(graph, root), depth(graph.size(), 0) {
    const auto dfs = [this, &graph](
        auto dfs, const int par, const int ver) -> void {
      for (const Edge<CostType>& e : graph[ver]) {
        if (e.dst == par) continue;
        depth[e.dst] = depth[ver] + e.cost;
        dfs(dfs, ver, e.dst);
      }
    };
    dfs(dfs, -1, root);
  }

  std::pair<std::vector<std::vector<Edge<CostType>>>,
            std::vector<int>> query(std::vector<int> vertices) const {
    if (vertices.empty()) [[unlikely]] return {};
    static std::vector<int> mp(depth.size(), -1);
    std::ranges::sort(vertices, {},
                      [&](const int v) -> int { return euler_tour.left[v]; });
    std::vector<std::vector<Edge<CostType>>> graph;
    std::vector<int> inv;
    const auto add_vertex = [this, &graph, &inv](const int v) -> void {
      mp[v] = graph.size();
      graph.emplace_back();
      inv.emplace_back(v);
    };
    const auto add_edge = [this, &graph](int u, int v) -> void {
      const CostType cost = depth[v] - depth[u];
      u = mp[u];
      v = mp[v];
      graph[u].emplace_back(u, v, cost);
      graph[v].emplace_back(v, u, cost);
    };
    add_vertex(vertices.front());
    std::vector<int> stck{vertices.front()};
    for (const int i : std::views::iota(1, std::ssize(vertices))) {
      const int l = euler_tour.query(vertices[i - 1], vertices[i]);
      if (mp[l] == -1) add_vertex(l);
      if (l != vertices[i - 1]) {
        const int depth_l = euler_tour.depth[euler_tour.left[l]];
        int cur = stck.back();
        stck.pop_back();
        while (!stck.empty() &&
               euler_tour.depth[euler_tour.left[stck.back()]] > depth_l) {
          add_edge(stck.back(), cur);
          cur = stck.back();
          stck.pop_back();
        }
        add_edge(l, cur);
        if (stck.empty() || stck.back() != l) stck.emplace_back(l);
      }
      add_vertex(vertices[i]);
      stck.emplace_back(vertices[i]);
    }
    for (; stck.size() >= 2; stck.pop_back()) {
      add_edge(stck.end()[-2], stck.back());
    }
    for (const int v : inv) mp[v] = -1;
    return {graph, inv};
  }

 private:
  std::vector<CostType> depth;
};

}  // namespace emthrm

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