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:heavy_check_mark: Dijkstra-Fibonacchi-Heap(単一始点最短路) (graph/shortest-path/dijkstra-fibonacchi-heap.hpp)

概要

負辺のないグラフで単一始点全点間最短路を求めるアルゴリズム.

通常の dijkstra 法では std::priority_queue を使用していたが, これをフィボナッチヒープにすることで計算量を落とせる(実用上早くなるかは知らない).

計算量

Depends on

Verified with

Code

#pragma once

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

/**
 * @brief Dijkstra-Fibonacchi-Heap(単一始点最短路)
 *
 */
template <typename T>
vector<T> dijkstra_fibonacchi_heap(Graph<T> &g, int s) {
  const auto INF = numeric_limits<T>::max();
  using Heap = FibonacchiHeap<T, int>;
  using Node = typename Heap::Node;

  Heap heap;
  vector<Node *> keep(g.size(), nullptr);
  vector<T> dist;
  dist.assign(g.size(), INF);
  dist[s] = 0;
  keep[s] = heap.push(dist[s], s);

  while (!heap.empty()) {
    T cost;
    int idx;
    tie(cost, idx) = heap.pop();
    if (dist[idx] < cost) continue;
    for (auto &e : g[idx]) {
      auto next_cost = cost + e.cost;
      if (dist[e.to] <= next_cost) continue;
      if (keep[e.to] == nullptr) {
        dist[e.to] = next_cost;
        keep[e.to] = heap.push(dist[e.to], e.to);
      } else {
        T d = dist[e.to] - next_cost;
        heap.decrease_key(keep[e.to], d);
        dist[e.to] -= d;
      }
    }
  }
  return dist;
}

#line 2 "graph/shortest-path/dijkstra-fibonacchi-heap.hpp"

#line 1 "structure/heap/fibonacchi-heap.hpp"
/**
 * @brief Fibonacchi-Heap(フィボナッチヒープ)
 * @see https://www.cs.princeton.edu/~wayne/teaching/fibonacci-heap.pdf
 */
template <typename key_t, typename val_t>
struct FibonacchiHeap {
  struct Node {
    key_t key;
    val_t val;
    Node *left, *right, *child, *par;
    int sz;
    bool mark;

    Node(const key_t &key, const val_t &val)
        : key(key),
          val(val),
          left(this),
          right(this),
          par(nullptr),
          child(nullptr),
          sz(0),
          mark(false) {}
  };

  Node *root;
  size_t sz;
  vector<Node *> rank;

  FibonacchiHeap() : root(nullptr), sz(0) {}

  size_t size() const { return sz; }

  bool empty() const { return sz == 0; }

  void update_min(Node *t) {
    if (!root || t->key < root->key) {
      root = t;
    }
  }

  void concat(Node *&r, Node *t) {
    if (!r) {
      r = t;
    } else {
      t->left->right = r->right;
      r->right->left = t->left;
      t->left = r;
      r->right = t;
    }
  }

  void delete_node(Node *t) {
    t->left->right = t->right;
    t->right->left = t->left;
    t->left = t;
    t->right = t;
  }

  Node *push(const key_t &key, const val_t &val) {
    ++sz;
    auto node = new Node(key, val);
    concat(root, node);
    update_min(node);
    return node;
  }

  Node *consolidate(Node *s, Node *t) {
    if (root == s || s->key < t->key) {
      delete_node(t);
      ++s->sz;
      t->par = s;
      concat(s->child, t);
      return s;
    } else {
      delete_node(s);
      ++t->sz;
      s->par = t;
      concat(t->child, s);
      return t;
    }
  }

  pair<key_t, val_t> pop() {
    --sz;

    Node *rem = root;

    auto ret = make_pair(rem->key, rem->val);

    {
      root = root->left == root ? nullptr : root->left;
      delete_node(rem);
    }

    if (rem->child) {
      concat(root, rem->child);
    }

    if (root) {
      {
        Node *base = root, *cur = base;
        do {
          cur->par = nullptr;
          update_min(cur);
          cur = cur->right;
        } while (cur != base);
      }

      {
        Node *base = root;
        int last = -1;
        do {
          Node *nxt = base->right;
          while (base->sz < rank.size() && rank[base->sz]) {
            Node *u = rank[base->sz];
            rank[base->sz] = nullptr;
            base = consolidate(u, base);
          }
          if (base->sz >= rank.size()) rank.resize(base->sz + 1);
          last = max(last, base->sz);
          rank[base->sz] = base;
          base = nxt;
        } while (base != root);

        for (int i = last; i >= 0; i--) rank[i] = nullptr;
      }
    }

    return ret;
  }

  inline void mark_dfs(Node *t) {
    if (!t->par) {
      t->mark = false;
    } else if (t->mark) {
      mark_dfs(t->par);
      t->par->child = t->left == t ? nullptr : t->left;
      delete_node(t);
      t->sz--;
      t->mark = false;
      t->par = nullptr;
      concat(root, t);
    } else {
      t->mark = true;
      t->sz--;
    }
  }

  void decrease_key(Node *t, const key_t &d) {
    t->key -= d;

    if (!t->par) {
      update_min(t);
      return;
    }

    if (t->par->key <= t->key) {
      return;
    }

    t->sz++;
    t->mark = true;
    mark_dfs(t);
    update_min(t);
  }
};
#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/shortest-path/dijkstra-fibonacchi-heap.hpp"

/**
 * @brief Dijkstra-Fibonacchi-Heap(単一始点最短路)
 *
 */
template <typename T>
vector<T> dijkstra_fibonacchi_heap(Graph<T> &g, int s) {
  const auto INF = numeric_limits<T>::max();
  using Heap = FibonacchiHeap<T, int>;
  using Node = typename Heap::Node;

  Heap heap;
  vector<Node *> keep(g.size(), nullptr);
  vector<T> dist;
  dist.assign(g.size(), INF);
  dist[s] = 0;
  keep[s] = heap.push(dist[s], s);

  while (!heap.empty()) {
    T cost;
    int idx;
    tie(cost, idx) = heap.pop();
    if (dist[idx] < cost) continue;
    for (auto &e : g[idx]) {
      auto next_cost = cost + e.cost;
      if (dist[e.to] <= next_cost) continue;
      if (keep[e.to] == nullptr) {
        dist[e.to] = next_cost;
        keep[e.to] = heap.push(dist[e.to], e.to);
      } else {
        T d = dist[e.to] - next_cost;
        heap.decrease_key(keep[e.to], d);
        dist[e.to] -= d;
      }
    }
  }
  return dist;
}
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