Directed MST(最小有向全域木) (graph/mst/directed-mst.hpp)

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• Last update: 2024-04-24 02:34:04+09:00
• Include: #include "graph/mst/directed-mst.hpp"

概要

有向グラフが与えられたとき, ある頂点を根とする最小有向全域木を求める. 頂点が指定されない場合は, 超頂点を追加して各頂点に重み $0$ の辺を張れば同じ問題に帰着される.

基本的には, 各頂点について, その頂点に入ってくる辺のうち最小の重みのものを選べば良い. このときに, サイクルが生えると困るので, 賢いことをしている.

使い方

• directed_mst(V, root, edges): V 頂点のグラフ edges の root を頂点とする最小有向全域木を返す. cost はそのときの辺のコストの和, edges は用いた辺集合が格納される. 頂点 root から全ての頂点へ到達可能であることを仮定している.

計算量

• $O(E \log V)$

$V$: 頂点数, $E$: 辺の本数

Depends on

• Graph Template(グラフテンプレート) (graph/graph-template.hpp)
• Skew-Heap (structure/heap/skew-heap.hpp)

Verified with

• test/verify/aoj-grl-2-b.test.cpp
• test/verify/yosupo-directedmst.test.cpp

Code

#pragma once

#include "../graph-template.hpp"
#include "../../structure/heap/skew-heap.hpp"

/**
 * @brief Directed MST(最小有向全域木)
 * 
 */
template< typename T >
struct MinimumSpanningTree {
  T cost;
  Edges< T > edges;
};

template< typename T >
MinimumSpanningTree< T > directed_mst(int V, int root, Edges< T > edges) {
  for(int i = 0; i < V; ++i) {
    if(i != root) edges.emplace_back(i, root, 0);
  }

  int x = 0;
  vector< int > par(2 * V, -1), vis(par), link(par);

  using Heap = SkewHeap< T, true >;
  using Node = typename Heap::Node;

  Heap heap;
  vector< Node * > ins(2 * V, heap.make_root());

  for(int i = 0; i < (int) edges.size(); i++) {
    auto &e = edges[i];
    ins[e.to] = heap.push(ins[e.to], e.cost, i);
  }
  vector< int > st;
  auto go = [&](int x) {
    x = edges[ins[x]->idx].from;
    while(link[x] != -1) {
      st.emplace_back(x);
      x = link[x];
    }
    for(auto &p : st) {
      link[p] = x;
    }
    st.clear();
    return x;
  };
  for(int i = V; ins[x]; i++) {
    for(; vis[x] == -1; x = go(x)) vis[x] = 0;
    for(; x != i; x = go(x)) {
      auto w = ins[x]->key;
      auto v = heap.pop(ins[x]);
      v = heap.add(v, -w);
      ins[i] = heap.meld(ins[i], v);
      par[x] = i;
      link[x] = i;
    }
    for(; ins[x] && go(x) == x; ins[x] = heap.pop(ins[x]));
  }
  T cost = 0;
  Edges< T > ans;
  for(int i = root; i != -1; i = par[i]) {
    vis[i] = 1;
  }
  for(int i = x; i >= 0; i--) {
    if(vis[i] == 1) continue;
    cost += edges[ins[i]->idx].cost;
    ans.emplace_back(edges[ins[i]->idx]);
    for(int j = edges[ins[i]->idx].to; j != -1 && vis[j] == 0; j = par[j]) {
      vis[j] = 1;
    }
  }
  return {cost, ans};
}

#line 2 "graph/mst/directed-mst.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/heap/skew-heap.hpp"
/**
 * @brief Skew-Heap
 */
template< typename T, bool isMin = true >
struct SkewHeap {
  struct Node {
    T key, lazy;
    Node *l, *r;
    int idx;

    explicit Node(const T &key, int idx) : key(key), idx(idx), lazy(0), l(nullptr), r(nullptr) {}
  };

  SkewHeap() = default;

  Node *alloc(const T &key, int idx = -1) {
    return new Node(key, idx);
  }

  Node *propagate(Node *t) {
    if(t && t->lazy != 0) {
      if(t->l) t->l->lazy += t->lazy;
      if(t->r) t->r->lazy += t->lazy;
      t->key += t->lazy;
      t->lazy = 0;
    }
    return t;
  }

  Node *meld(Node *x, Node *y) {
    propagate(x), propagate(y);
    if(!x || !y) return x ? x : y;
    if((x->key < y->key) ^ isMin) swap(x, y);
    x->r = meld(y, x->r);
    swap(x->l, x->r);
    return x;
  }

  Node *push(Node *t, const T &key, int idx = -1) {
    return meld(t, alloc(key, idx));
  }

  Node *pop(Node *t) {
    assert(t != nullptr);
    return meld(t->l, t->r);
  }

  Node *add(Node *t, const T &lazy) {
    if(t) {
      t->lazy += lazy;
      propagate(t);
    }
    return t;
  }

  Node *make_root() {
    return nullptr;
  }
};
#line 5 "graph/mst/directed-mst.hpp"

/**
 * @brief Directed MST(最小有向全域木)
 * 
 */
template< typename T >
struct MinimumSpanningTree {
  T cost;
  Edges< T > edges;
};

template< typename T >
MinimumSpanningTree< T > directed_mst(int V, int root, Edges< T > edges) {
  for(int i = 0; i < V; ++i) {
    if(i != root) edges.emplace_back(i, root, 0);
  }

  int x = 0;
  vector< int > par(2 * V, -1), vis(par), link(par);

  using Heap = SkewHeap< T, true >;
  using Node = typename Heap::Node;

  Heap heap;
  vector< Node * > ins(2 * V, heap.make_root());

  for(int i = 0; i < (int) edges.size(); i++) {
    auto &e = edges[i];
    ins[e.to] = heap.push(ins[e.to], e.cost, i);
  }
  vector< int > st;
  auto go = [&](int x) {
    x = edges[ins[x]->idx].from;
    while(link[x] != -1) {
      st.emplace_back(x);
      x = link[x];
    }
    for(auto &p : st) {
      link[p] = x;
    }
    st.clear();
    return x;
  };
  for(int i = V; ins[x]; i++) {
    for(; vis[x] == -1; x = go(x)) vis[x] = 0;
    for(; x != i; x = go(x)) {
      auto w = ins[x]->key;
      auto v = heap.pop(ins[x]);
      v = heap.add(v, -w);
      ins[i] = heap.meld(ins[i], v);
      par[x] = i;
      link[x] = i;
    }
    for(; ins[x] && go(x) == x; ins[x] = heap.pop(ins[x]));
  }
  T cost = 0;
  Edges< T > ans;
  for(int i = root; i != -1; i = par[i]) {
    vis[i] = 1;
  }
  for(int i = x; i >= 0; i--) {
    if(vis[i] == 1) continue;
    cost += edges[ins[i]->idx].cost;
    ans.emplace_back(edges[ins[i]->idx]);
    for(int j = edges[ins[i]->idx].to; j != -1 && vis[j] == 0; j = par[j]) {
      vis[j] = 1;
    }
  }
  return {cost, ans};
}

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