Dijkstra (単一始点最短経路)
(graph/dijkstra.hpp)

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• Last update: 2022-07-07 14:50:02+09:00
• Include: #include "graph/dijkstra.hpp"

概要

• 負辺のないグラフで単一始点最短経路を求めるアルゴリズム
• ShortestPath の from にはその頂点に到達する直前の頂点が格納され、id には直前の辺の id が格納される。
• $O(E\log V)$

実装のヒント

• priority_que で距離が短い順に処理する
• すでに小さいコストで到達できる経路が確認できている場合はスキップする
• priority_que には、すでに短い距離で訪れている頂点を追加しないようにする

Depends on

• graph/graphTemplate.hpp

Verified with

• test/graph/dijkstra.aoj_GRL_1_A.test.cpp

Code

#pragma once
#include <limits>
#include <queue>
#include <vector>
#include <algorithm>

#include "./graphTemplate.hpp"
using namespace std;

template <typename T = int>
struct ShortestPath {
  vector<T> dist;
  vector<int> from, id;
};

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

#line 2 "graph/dijkstra.hpp"
#include <limits>
#include <queue>
#include <vector>
#include <algorithm>

#line 3 "graph/graphTemplate.hpp"
#include <iostream>
using namespace std;

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>
using Edges = vector<Edge<T>>;
#line 8 "graph/dijkstra.hpp"
using namespace std;

template <typename T = int>
struct ShortestPath {
  vector<T> dist;
  vector<int> from, id;
};

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

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