Bellman-Ford (単一始点最短路) (graph/shortest-path/bellman-ford.hpp)

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• Last update: 2024-06-11 23:47:26+09:00
• Include: #include "graph/shortest-path/bellman-ford.hpp"

単一始点全点間最短路を求めるアルゴリズムです。負辺があっても動作します。経路上に負閉路がある場合はそれを検出します。

bellman_ford

template <typename T>
vector<T> bellman_ford(const Edges<T> &edges, int n, int s)

頂点数 $n$ 、辺集合が edges からなる有向グラフについて、始点 $s$ から各頂点への最短路の重みを求め、それを返します。

ただし、始点 $s$ からその頂点に到達できない場合は T の最大値、その頂点までの経路上に負閉路が存在する場合は T の最小値が格納されます。

制約

• $0 \leq s \lt n$

計算量

• $O(V E)$

Depends on

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

Verified with

• test/verify/aoj-0304.test.cpp
• test/verify/aoj-grl-1-b.test.cpp

Code

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

template <typename T>
vector<T> bellman_ford(const Edges<T> &edges, int n, int s) {
  const auto INF = numeric_limits<T>::max();
  const auto M_INF = numeric_limits<T>::min();
  vector<T> dist(n, INF);
  dist[s] = 0;
  for (int i = 0; i < n - 1; i++) {
    for (auto &e : edges) {
      if (dist[e.from] == INF) continue;
      dist[e.to] = min(dist[e.to], dist[e.from] + e.cost);
    }
  }
  vector<bool> negative(n);
  for (int i = 0; i < n; i++) {
    for (auto &e : edges) {
      if (dist[e.from] == INF) continue;
      if (dist[e.from] + e.cost < dist[e.to]) {
        dist[e.to] = dist[e.from] + e.cost;
        negative[e.to] = true;
      }
      if (negative[e.from]) {
        negative[e.to] = true;
      }
    }
  }
  for (int i = 0; i < n; i++) {
    if (negative[i]) dist[i] = M_INF;
  }
  return dist;
}

#line 2 "graph/graph-template.hpp"

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 2 "graph/shortest-path/bellman-ford.hpp"

template <typename T>
vector<T> bellman_ford(const Edges<T> &edges, int n, int s) {
  const auto INF = numeric_limits<T>::max();
  const auto M_INF = numeric_limits<T>::min();
  vector<T> dist(n, INF);
  dist[s] = 0;
  for (int i = 0; i < n - 1; i++) {
    for (auto &e : edges) {
      if (dist[e.from] == INF) continue;
      dist[e.to] = min(dist[e.to], dist[e.from] + e.cost);
    }
  }
  vector<bool> negative(n);
  for (int i = 0; i < n; i++) {
    for (auto &e : edges) {
      if (dist[e.from] == INF) continue;
      if (dist[e.from] + e.cost < dist[e.to]) {
        dist[e.to] = dist[e.from] + e.cost;
        negative[e.to] = true;
      }
      if (negative[e.from]) {
        negative[e.to] = true;
      }
    }
  }
  for (int i = 0; i < n; i++) {
    if (negative[i]) dist[i] = M_INF;
  }
  return dist;
}

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