Dijkstra(単一始点最短路) (graph/shortest-path/dijkstra.hpp)

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• Last update: 2024-05-01 23:44:47+09:00
• Include: #include "graph/shortest-path/dijkstra.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. 各地点でもっとも近い頂点から距離が確定していく. 距離順でソートされたヒープを用いると, 効率よく距離を確定していくことができる.

• dijkstra(g, s): 重み付きグラフ g で, 頂点 s から全点間の最短コストを求める. dist には最短コスト(到達できないとき型の最大値), from にはその頂点の直前に訪れた頂点(頂点 s または到達できない頂点は $-1$), id はその頂点の直前に使った辺番号が格納される.

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-0275.test.cpp
• test/verify/aoj-grl-1-a.test.cpp
• test/verify/yosupo-k-shortest-walk.test.cpp
• test/verify/yosupo-shortest-path.test.cpp

Code

#pragma once

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

/**
 * @brief Dijkstra(単一始点最短路)
 *
 */
template <typename T>
struct ShortestPath {
  vector<T> dist;
  vector<int> from, id;
};

template <typename T>
ShortestPath<T> dijkstra(const Graph<T> &g, int s) {
  const auto INF = numeric_limits<T>::max();
  vector<T> dist(g.size(), INF);
  vector<int> from(g.size(), -1), id(g.size(), -1);
  using Pi = pair<T, int>;
  priority_queue<Pi, vector<Pi>, greater<> > que;
  dist[s] = 0;
  que.emplace(dist[s], s);
  while (!que.empty()) {
    T cost;
    int idx;
    tie(cost, idx) = que.top();
    que.pop();
    if (dist[idx] < cost) continue;
    for (auto &e : g[idx]) {
      auto next_cost = cost + e.cost;
      if (dist[e.to] <= next_cost) continue;
      dist[e.to] = next_cost;
      from[e.to] = idx;
      id[e.to] = e.idx;
      que.emplace(dist[e.to], e.to);
    }
  }
  return {dist, from, id};
}

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

/**
 * @brief Dijkstra(単一始点最短路)
 *
 */
template <typename T>
struct ShortestPath {
  vector<T> dist;
  vector<int> from, id;
};

template <typename T>
ShortestPath<T> dijkstra(const Graph<T> &g, int s) {
  const auto INF = numeric_limits<T>::max();
  vector<T> dist(g.size(), INF);
  vector<int> from(g.size(), -1), id(g.size(), -1);
  using Pi = pair<T, int>;
  priority_queue<Pi, vector<Pi>, greater<> > que;
  dist[s] = 0;
  que.emplace(dist[s], s);
  while (!que.empty()) {
    T cost;
    int idx;
    tie(cost, idx) = que.top();
    que.pop();
    if (dist[idx] < cost) continue;
    for (auto &e : g[idx]) {
      auto next_cost = cost + e.cost;
      if (dist[e.to] <= next_cost) continue;
      dist[e.to] = next_cost;
      from[e.to] = idx;
      id[e.to] = e.idx;
      que.emplace(dist[e.to], e.to);
    }
  }
  return {dist, from, id};
}

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