PMORMQ-Lowest-Common-Ancestor(最小共通祖先) (graph/tree/pmormq-lowest-common-ancestor.hpp)

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• Last update: 2024-04-27 01:27:28+09:00
• Include: #include "graph/tree/pmormq-lowest-common-ancestor.hpp"

概要

オイラーツアーと±1RMQ によって最小共通祖先を求める.

辺属性のオイラーツアーをする. すべての頂点について, その頂点 $k$ に最初に到達した時刻 $in[k]$ と深さ $dep[k]$ を求めておく. 頂点 $u, v$ の最小共通祖先は区間 $[in[u], in[v]]$ の要素のうち深さが最小となる頂点である.

スパーステーブルを用いると前計算 $O(V \log V)$, クエリ $O(1)$ となるが, $dep[k]$ 隣接する要素の差がちょうど $1$ であることを利用した ±1RMQ を用いると前計算 $O(V)$, クエリ $O(1)$ となる.

使い方

• build(): 構築する.
• lca(u, v): 頂点 u, v の最小共通祖先(LCA)を返す.

計算量

• build(): $O(V)$
• lca(): $O(1)$

Depends on

• Graph Template(グラフテンプレート) (graph/graph-template.hpp)
• Plus-Minus-One-RMQ (structure/others/plus-minus-one-rmq.hpp)
• Sparse-Table(スパーステーブル) (structure/others/sparse-table.hpp)

Verified with

• test/verify/aoj-grl-5-c-4.test.cpp
• test/verify/yosupo-lca.test.cpp
• test/verify/yosupo-staticrmq-4.test.cpp

Code

#pragma once

#include "../../structure/others/plus-minus-one-rmq.hpp"
#include "../graph-template.hpp"

/**
 * @brief PMORMQ-Lowest-Common-Ancestor(最小共通祖先)
 *
 **/
template <typename T = int>
struct PMORMQLowestCommonAncestor : Graph<T> {
 public:
  using Graph<T>::Graph;
  using Graph<T>::g;
  using F = function<int(int, int)>;

  void build(int root = 0) {
    ord.reserve(g.size() * 2 - 1);
    dep.reserve(g.size() * 2 - 1);
    in.resize(g.size());
    dfs(root, -1, 0);
    vector<int> vs(g.size() * 2 - 1);
    iota(begin(vs), end(vs), 0);
    st = PlusMinusOneRMQ<int>(dep);
  }

  int lca(int x, int y) const {
    if (in[x] > in[y]) swap(x, y);
    return ord[st.fold(in[x], in[y] + 1).second];
  }

 private:
  vector<int> ord, dep, in;
  PlusMinusOneRMQ<int> st;

  void dfs(int idx, int par, int d) {
    in[idx] = (int)ord.size();
    ord.emplace_back(idx);
    dep.emplace_back(d);
    for (auto &to : g[idx]) {
      if (to != par) {
        dfs(to, idx, d + 1);
        ord.emplace_back(idx);
        dep.emplace_back(d);
      }
    }
  }
};

#line 2 "graph/tree/pmormq-lowest-common-ancestor.hpp"

#line 1 "structure/others/sparse-table.hpp"
/**
 * @brief Sparse-Table(スパーステーブル)
 *
 */
template <typename T, typename F>
struct SparseTable {
  F f;
  vector<vector<T> > st;
  vector<int> lookup;

  SparseTable() = default;

  explicit SparseTable(const vector<T> &v, const F &f) : f(f) {
    const int n = (int)v.size();
    const int b = 32 - __builtin_clz(n);
    st.assign(b, vector<T>(n));
    for (int i = 0; i < v.size(); i++) {
      st[0][i] = v[i];
    }
    for (int i = 1; i < b; i++) {
      for (int j = 0; j + (1 << i) <= n; j++) {
        st[i][j] = f(st[i - 1][j], st[i - 1][j + (1 << (i - 1))]);
      }
    }
    lookup.resize(v.size() + 1);
    for (int i = 2; i < lookup.size(); i++) {
      lookup[i] = lookup[i >> 1] + 1;
    }
  }

  inline T fold(int l, int r) const {
    int b = lookup[r - l];
    return f(st[b][l], st[b][r - (1 << b)]);
  }
};

template <typename T, typename F>
SparseTable<T, F> get_sparse_table(const vector<T> &v, const F &f) {
  return SparseTable<T, F>(v, f);
}
#line 2 "structure/others/plus-minus-one-rmq.hpp"

/**
 * @brief Plus-Minus-One-RMQ
 **/
template <typename T>
struct PlusMinusOneRMQ {
  using F = function<int(int, int)>;

  int backet;
  vector<T> vs;
  vector<int> bidx, bbit;
  SparseTable<int, F> st;
  vector<vector<vector<int> > > lookup;

  explicit PlusMinusOneRMQ() = default;

  explicit PlusMinusOneRMQ(const vector<T> &vs) : vs(vs) {
    int n = (int)vs.size();
    backet = max(1, (31 - __builtin_clz(n)) / 2);
    int sz = (n + backet - 1) / backet;
    bidx.assign(sz, -1);
    bbit.assign(sz, 0);
    for (int i = 0; i < sz; i++) {
      int l = i * backet;
      int r = min(l + backet, n);
      bidx[i] = l;
      for (int j = l + 1; j < r; j++) {
        if (vs[j] < vs[bidx[i]]) bidx[i] = j;
        if (vs[j - 1] < vs[j]) bbit[i] |= 1 << (j - l - 1);
      }
    }
    F f = [&](int a, int b) { return vs[a] < vs[b] ? a : b; };
    st = get_sparse_table(bidx, f);
    lookup.assign(1 << (backet - 1),
                  vector<vector<int> >(backet, vector<int>(backet + 1)));
    for (int i = 0; i < (1 << (backet - 1)); i++) {
      for (int j = 0; j < backet; j++) {
        int sum = 0, ret = 0, pos = j;
        for (int k = j + 1; k <= backet; k++) {
          lookup[i][j][k] = pos;
          if (i & (1 << (k - 1)))
            ++sum;
          else
            --sum;
          if (sum < ret) {
            pos = k;
            ret = sum;
          }
        }
      }
    }
  }

  pair<T, int> fold(int l, int r) const {
    int lb = l / backet;
    int rb = r / backet;
    if (lb == rb) {
      int pos = lb * backet + lookup[bbit[lb]][l % backet][r % backet];
      return {vs[pos], pos};
    }
    int pos = lb * backet + lookup[bbit[lb]][l % backet][backet];
    if (r % backet > 0) {
      int sub = rb * backet + lookup[bbit[rb]][0][r % backet];
      if (vs[sub] < vs[pos]) pos = sub;
    }
    if (lb + 1 == rb) {
      return {vs[pos], pos};
    } else {
      int sub = st.fold(lb + 1, rb);
      if (vs[sub] < vs[pos]) pos = sub;
      return {vs[pos], pos};
    }
  }
};
#line 2 "graph/graph-template.hpp"

/**
 * @brief Graph Template(グラフテンプレート)
 */
template <typename T = int>
struct Edge {
  int from, to;
  T cost;
  int idx;

  Edge() = default;

  Edge(int from, int to, T cost = 1, int idx = -1)
      : from(from), to(to), cost(cost), idx(idx) {}

  operator int() const { return to; }
};

template <typename T = int>
struct Graph {
  vector<vector<Edge<T> > > g;
  int es;

  Graph() = default;

  explicit Graph(int n) : g(n), es(0) {}

  size_t size() const { return g.size(); }

  void add_directed_edge(int from, int to, T cost = 1) {
    g[from].emplace_back(from, to, cost, es++);
  }

  void add_edge(int from, int to, T cost = 1) {
    g[from].emplace_back(from, to, cost, es);
    g[to].emplace_back(to, from, cost, es++);
  }

  void read(int M, int padding = -1, bool weighted = false,
            bool directed = false) {
    for (int i = 0; i < M; i++) {
      int a, b;
      cin >> a >> b;
      a += padding;
      b += padding;
      T c = T(1);
      if (weighted) cin >> c;
      if (directed)
        add_directed_edge(a, b, c);
      else
        add_edge(a, b, c);
    }
  }

  inline vector<Edge<T> > &operator[](const int &k) { return g[k]; }

  inline const vector<Edge<T> > &operator[](const int &k) const { return g[k]; }
};

template <typename T = int>
using Edges = vector<Edge<T> >;
#line 5 "graph/tree/pmormq-lowest-common-ancestor.hpp"

/**
 * @brief PMORMQ-Lowest-Common-Ancestor(最小共通祖先)
 *
 **/
template <typename T = int>
struct PMORMQLowestCommonAncestor : Graph<T> {
 public:
  using Graph<T>::Graph;
  using Graph<T>::g;
  using F = function<int(int, int)>;

  void build(int root = 0) {
    ord.reserve(g.size() * 2 - 1);
    dep.reserve(g.size() * 2 - 1);
    in.resize(g.size());
    dfs(root, -1, 0);
    vector<int> vs(g.size() * 2 - 1);
    iota(begin(vs), end(vs), 0);
    st = PlusMinusOneRMQ<int>(dep);
  }

  int lca(int x, int y) const {
    if (in[x] > in[y]) swap(x, y);
    return ord[st.fold(in[x], in[y] + 1).second];
  }

 private:
  vector<int> ord, dep, in;
  PlusMinusOneRMQ<int> st;

  void dfs(int idx, int par, int d) {
    in[idx] = (int)ord.size();
    ord.emplace_back(idx);
    dep.emplace_back(d);
    for (auto &to : g[idx]) {
      if (to != par) {
        dfs(to, idx, d + 1);
        ord.emplace_back(idx);
        dep.emplace_back(d);
      }
    }
  }
};

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