Dominator Tree
(graph/others/dominator-tree.hpp)

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• Last update: 2022-09-11 00:53:50+09:00
• Include: #include "graph/others/dominator-tree.hpp"
• Link: http://sigma425.hatenablog.com/entry/2015/12/25/224053

概要

有向グラフが与えられたとき, 頂点 root を根とする Dominator tree を求める.

もとの有向グラフで, 頂点 root からある頂点 i へ向かうパスを考える. Dominator tree 上の頂点 root から i までのパス上にある頂点は, 頂点 i へ到達するために必ず通る必要のある頂点である(雰囲気的には関節点の有向グラフ版). 特に頂点 i の親は直接支配節 idom(i) と呼び, 必ず通る必要のある頂点のうち最も w に近い頂点を指す.

使い方

• build(root): 頂点 root を根とする Dominator tree を返す. 各要素にはその要素の親の idx が格納される. ただし root には root が格納される. 頂点 root からその頂点に到達できない場合 -1.

計算量

$O(E \log V)$

Depends on

• Graph Template(グラフテンプレート) (graph/graph-template.hpp)

Verified with

• test/verify/aoj-0294.test.cpp
• test/verify/yosupo-dominatortree.test.cpp

Code

#pragma once

#include "../graph-template.hpp"

/**
 * @brief Dominator Tree
 * @docs docs/dominator-tree.md
 * @see http://sigma425.hatenablog.com/entry/2015/12/25/224053
 */
template< typename T = int >
struct DominatorTree : Graph< T > {
public:
  using Graph< T >::Graph;
  using Graph< T >::g;

  void build(int root) {
    rg = Graph< T >(g.size());
    par.assign(g.size(), 0);
    idom.assign(g.size(), -1);
    semi.assign(g.size(), -1);
    ord.reserve(g.size());
    UnionFind uf(semi);

    const int N = (int) g.size();
    dfs(root);
    for(int i = 0; i < N; i++) {
      for(auto &to : g[i]) {
        if(~semi[i]) rg.add_directed_edge(to, i);
      }
    }

    vector< vector< int > > bucket(N);
    vector< int > U(N);
    for(int i = (int) ord.size() - 1; i >= 0; i--) {
      int x = ord[i];
      for(int v : rg[x]) {
        v = uf.eval(v);
        if(semi[x] > semi[v]) semi[x] = semi[v];
      }
      bucket[ord[semi[x]]].emplace_back(x);
      for(int v : bucket[par[x]]) U[v] = uf.eval(v);
      bucket[par[x]].clear();
      uf.link(par[x], x);
    }
    for(int i = 1; i < (int)ord.size(); i++) {
      int x = ord[i], u = U[x];
      idom[x] = semi[x] == semi[u] ? semi[x] : idom[u];
    }
    for(int i = 1; i < (int)ord.size(); i++) {
      int x = ord[i];
      idom[x] = ord[idom[x]];
    }
    idom[root] = root;
  }

  int operator[](const int &k) const {
    return idom[k];
  }

private:
  Graph< T > rg;

  struct UnionFind {
    const vector< int > &semi;
    vector< int > par, m;

    explicit UnionFind(const vector< int > &semi) : semi(semi), par(semi.size()), m(semi.size()) {
      iota(begin(par), end(par), 0);
      iota(begin(m), end(m), 0);
    }

    int find(int v) {
      if(par[v] == v) return v;
      int r = find(par[v]);
      if(semi[m[v]] > semi[m[par[v]]]) m[v] = m[par[v]];
      return par[v] = r;
    }

    int eval(int v) {
      find(v);
      return m[v];
    }

    void link(int p, int c) {
      par[c] = p;
    }
  };

  vector< int > ord, par;
  vector< int > idom, semi;

  void dfs(int idx) {
    semi[idx] = (int) ord.size();
    ord.emplace_back(idx);
    for(auto &to : g[idx]) {
      if(~semi[to]) continue;
      dfs(to);
      par[to] = idx;
    }
  }
};

#line 2 "graph/others/dominator-tree.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 4 "graph/others/dominator-tree.hpp"

/**
 * @brief Dominator Tree
 * @docs docs/dominator-tree.md
 * @see http://sigma425.hatenablog.com/entry/2015/12/25/224053
 */
template< typename T = int >
struct DominatorTree : Graph< T > {
public:
  using Graph< T >::Graph;
  using Graph< T >::g;

  void build(int root) {
    rg = Graph< T >(g.size());
    par.assign(g.size(), 0);
    idom.assign(g.size(), -1);
    semi.assign(g.size(), -1);
    ord.reserve(g.size());
    UnionFind uf(semi);

    const int N = (int) g.size();
    dfs(root);
    for(int i = 0; i < N; i++) {
      for(auto &to : g[i]) {
        if(~semi[i]) rg.add_directed_edge(to, i);
      }
    }

    vector< vector< int > > bucket(N);
    vector< int > U(N);
    for(int i = (int) ord.size() - 1; i >= 0; i--) {
      int x = ord[i];
      for(int v : rg[x]) {
        v = uf.eval(v);
        if(semi[x] > semi[v]) semi[x] = semi[v];
      }
      bucket[ord[semi[x]]].emplace_back(x);
      for(int v : bucket[par[x]]) U[v] = uf.eval(v);
      bucket[par[x]].clear();
      uf.link(par[x], x);
    }
    for(int i = 1; i < (int)ord.size(); i++) {
      int x = ord[i], u = U[x];
      idom[x] = semi[x] == semi[u] ? semi[x] : idom[u];
    }
    for(int i = 1; i < (int)ord.size(); i++) {
      int x = ord[i];
      idom[x] = ord[idom[x]];
    }
    idom[root] = root;
  }

  int operator[](const int &k) const {
    return idom[k];
  }

private:
  Graph< T > rg;

  struct UnionFind {
    const vector< int > &semi;
    vector< int > par, m;

    explicit UnionFind(const vector< int > &semi) : semi(semi), par(semi.size()), m(semi.size()) {
      iota(begin(par), end(par), 0);
      iota(begin(m), end(m), 0);
    }

    int find(int v) {
      if(par[v] == v) return v;
      int r = find(par[v]);
      if(semi[m[v]] > semi[m[par[v]]]) m[v] = m[par[v]];
      return par[v] = r;
    }

    int eval(int v) {
      find(v);
      return m[v];
    }

    void link(int p, int c) {
      par[c] = p;
    }
  };

  vector< int > ord, par;
  vector< int > idom, semi;

  void dfs(int idx) {
    semi[idx] = (int) ord.size();
    ord.emplace_back(idx);
    for(auto &to : g[idx]) {
      if(~semi[to]) continue;
      dfs(to);
      par[to] = idx;
    }
  }
};

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