Luzhiled's Library

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:heavy_check_mark: RMQ-Lowest-Common-Ancestor(最小共通祖先)
(graph/tree/rmq-lowest-common-ancestor.hpp)

概要

オイラーツアーとスパーステーブル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. 区間の最小値なのでスパーステーブルにより前計算しておくと, クエリあたり $O(1)$ で処理できる.

使い方

計算量

Depends on

Verified with

Code

#pragma once

#include "../graph-template.hpp"
#include "../../structure/others/sparse-table.hpp"

/**
 * @brief RMQ-Lowest-Common-Ancestor(最小共通祖先)
 * @docs docs/rmq-lowest-common-ancestor.md
 **/
template< typename T = int >
struct RMQLowestCommonAncestor : 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);
    F f = [&](int a, int b) { return dep[a] < dep[b] ? a : b; };
    st = get_sparse_table(vs, f);
  }

  int lca(int x, int y) const {
    if(in[x] > in[y]) swap(x, y);
    return x == y ? x : ord[st.fold(in[x], in[y])];
  }

private:
  vector< int > ord, dep, in;
  SparseTable< int, F > 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/rmq-lowest-common-ancestor.hpp"

#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 1 "structure/others/sparse-table.hpp"
/**
 * @brief Sparse-Table(スパーステーブル)
 * @docs docs/sparse-table.md
 */
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 5 "graph/tree/rmq-lowest-common-ancestor.hpp"

/**
 * @brief RMQ-Lowest-Common-Ancestor(最小共通祖先)
 * @docs docs/rmq-lowest-common-ancestor.md
 **/
template< typename T = int >
struct RMQLowestCommonAncestor : 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);
    F f = [&](int a, int b) { return dep[a] < dep[b] ? a : b; };
    st = get_sparse_table(vs, f);
  }

  int lca(int x, int y) const {
    if(in[x] > in[y]) swap(x, y);
    return x == y ? x : ord[st.fold(in[x], in[y])];
  }

private:
  vector< int > ord, dep, in;
  SparseTable< int, F > 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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