K-Shortest-Path (graph/shortest-path/k-shortest-path.hpp)

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• Last update: 2024-05-01 23:44:47+09:00
• Include: #include "graph/shortest-path/k-shortest-path.hpp"
• Link: https://qiita.com/nariaki3551/items/821dc6ffdc552d3d5f22

概要

頂点 $s$ から $t$ へのパス(Path) のうち, 昇順 $k$ 個のパスの長さを Yen’s Algorithm により求める.

パス(Path, 道) は同じ頂点を通らない経路である.

verify が甘いため合っているかかなり不安です

• k_shotest_path(g, s, t, k): 重み付き有向グラフ g の頂点 s から t へのパスのうち, 昇順 k 個のパスの長さとそのパスの辺番号の列を返す(パスの個数が k 個に満たないとき全てを返す).

計算量

• $O(KV ((E + v) \log V))$

Depends on

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

Verified with

• test/verify/yukicoder-1069.test.cpp

Code

#pragma once

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

/**
 * @brief K-Shortest-Path
 *
 * @see https://qiita.com/nariaki3551/items/821dc6ffdc552d3d5f22
 */
template <typename T>
vector<pair<T, vector<int> > > k_shortest_path(const Graph<T> &g, int s, int t,
                                               int k) {
  assert(s != t);
  int N = (int)g.size();
  int M = 0;
  for (int i = 0; i < N; i++) M += (int)g[i].size();
  vector<int> latte(M), malta(M);
  vector<T> cost(M);
  for (int i = 0; i < N; i++) {
    for (auto &e : g[i]) {
      latte[e.idx] = i;
      malta[e.idx] = e.to;
      cost[e.idx] = e.cost;
    }
  }
  const auto INF = numeric_limits<T>::max();
  vector<int> dame(M, -1);
  int timestamp = 0;
  auto shortest_path = [&](vector<T> &dist, vector<int> &from, vector<int> &id,
                           int st) {
    using Pi = pair<T, int>;
    priority_queue<Pi, vector<Pi>, greater<> > que;
    que.emplace(dist[st], st);
    while (!que.empty()) {
      T cost;
      int idx;
      tie(cost, idx) = que.top();
      que.pop();
      if (dist[idx] < cost) continue;
      if (idx == t) return;
      for (auto &e : g[idx]) {
        auto next_cost = cost + e.cost;
        if (dist[e.to] <= next_cost) continue;
        if (dame[e.idx] == timestamp) continue;
        dist[e.to] = next_cost;
        from[e.to] = idx;
        id[e.to] = e.idx;
        que.emplace(dist[e.to], e.to);
      }
    }
  };
  auto restore = [](const vector<int> &es, const vector<int> &vs, int from,
                    int to) {
    vector<int> tap;
    while (to != from) {
      tap.emplace_back(es[to]);
      to = vs[to];
    }
    reverse(begin(tap), end(tap));
    return tap;
  };

  vector<T> dist(g.size(), INF);
  vector<int> from(g.size(), -1), id(g.size(), -1);
  dist[s] = 0;
  shortest_path(dist, from, id, s);
  if (dist[t] == INF) return {};

  vector<pair<T, vector<int> > > A;
  set<pair<T, vector<int> > > B;
  A.emplace_back(dist[t], restore(id, from, s, t));

  for (int i = 1; i < k; i++) {
    dist.assign(g.size(), INF);
    from.assign(g.size(), -1);
    id.assign(g.size(), -1);
    dist[s] = 0;
    vector<int> candidate(A.size());
    iota(begin(candidate), end(candidate), 0);
    auto &last_path = A.back().second;
    int cur = s;
    for (int j = 0; j < last_path.size(); j++) {
      for (auto &k : candidate) {
        if (j < A[k].second.size()) dame[A[k].second[j]] = timestamp;
      }
      vector<T> dist2{dist};
      vector<int> from2{from}, id2{id};
      shortest_path(dist2, from2, id2, cur);
      ++timestamp;
      if (dist2[t] != INF) {
        auto path = restore(id2, from2, s, t);
        bool ok = true;
        for (auto &p : candidate) {
          if (path == A[p].second) {
            ok = false;
            break;
          }
        }
        if (ok) B.emplace(dist2[t], path);
      }
      vector<int> accept;
      for (auto &k : candidate) {
        if (j < A[k].second.size() && A[k].second[j] == last_path[j]) {
          accept.emplace_back(k);
        }
      }
      dist[malta[last_path[j]]] =
          dist[latte[last_path[j]]] + cost[last_path[j]];
      from[malta[last_path[j]]] = latte[last_path[j]];
      id[malta[last_path[j]]] = last_path[j];
      cur = malta[last_path[j]];
      candidate = move(accept);
    }
    if (B.size()) {
      A.emplace_back(*B.begin());
      B.erase(B.begin());
    }
  }
  return A;
}

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

/**
 * @brief K-Shortest-Path
 *
 * @see https://qiita.com/nariaki3551/items/821dc6ffdc552d3d5f22
 */
template <typename T>
vector<pair<T, vector<int> > > k_shortest_path(const Graph<T> &g, int s, int t,
                                               int k) {
  assert(s != t);
  int N = (int)g.size();
  int M = 0;
  for (int i = 0; i < N; i++) M += (int)g[i].size();
  vector<int> latte(M), malta(M);
  vector<T> cost(M);
  for (int i = 0; i < N; i++) {
    for (auto &e : g[i]) {
      latte[e.idx] = i;
      malta[e.idx] = e.to;
      cost[e.idx] = e.cost;
    }
  }
  const auto INF = numeric_limits<T>::max();
  vector<int> dame(M, -1);
  int timestamp = 0;
  auto shortest_path = [&](vector<T> &dist, vector<int> &from, vector<int> &id,
                           int st) {
    using Pi = pair<T, int>;
    priority_queue<Pi, vector<Pi>, greater<> > que;
    que.emplace(dist[st], st);
    while (!que.empty()) {
      T cost;
      int idx;
      tie(cost, idx) = que.top();
      que.pop();
      if (dist[idx] < cost) continue;
      if (idx == t) return;
      for (auto &e : g[idx]) {
        auto next_cost = cost + e.cost;
        if (dist[e.to] <= next_cost) continue;
        if (dame[e.idx] == timestamp) continue;
        dist[e.to] = next_cost;
        from[e.to] = idx;
        id[e.to] = e.idx;
        que.emplace(dist[e.to], e.to);
      }
    }
  };
  auto restore = [](const vector<int> &es, const vector<int> &vs, int from,
                    int to) {
    vector<int> tap;
    while (to != from) {
      tap.emplace_back(es[to]);
      to = vs[to];
    }
    reverse(begin(tap), end(tap));
    return tap;
  };

  vector<T> dist(g.size(), INF);
  vector<int> from(g.size(), -1), id(g.size(), -1);
  dist[s] = 0;
  shortest_path(dist, from, id, s);
  if (dist[t] == INF) return {};

  vector<pair<T, vector<int> > > A;
  set<pair<T, vector<int> > > B;
  A.emplace_back(dist[t], restore(id, from, s, t));

  for (int i = 1; i < k; i++) {
    dist.assign(g.size(), INF);
    from.assign(g.size(), -1);
    id.assign(g.size(), -1);
    dist[s] = 0;
    vector<int> candidate(A.size());
    iota(begin(candidate), end(candidate), 0);
    auto &last_path = A.back().second;
    int cur = s;
    for (int j = 0; j < last_path.size(); j++) {
      for (auto &k : candidate) {
        if (j < A[k].second.size()) dame[A[k].second[j]] = timestamp;
      }
      vector<T> dist2{dist};
      vector<int> from2{from}, id2{id};
      shortest_path(dist2, from2, id2, cur);
      ++timestamp;
      if (dist2[t] != INF) {
        auto path = restore(id2, from2, s, t);
        bool ok = true;
        for (auto &p : candidate) {
          if (path == A[p].second) {
            ok = false;
            break;
          }
        }
        if (ok) B.emplace(dist2[t], path);
      }
      vector<int> accept;
      for (auto &k : candidate) {
        if (j < A[k].second.size() && A[k].second[j] == last_path[j]) {
          accept.emplace_back(k);
        }
      }
      dist[malta[last_path[j]]] =
          dist[latte[last_path[j]]] + cost[last_path[j]];
      from[malta[last_path[j]]] = latte[last_path[j]];
      id[malta[last_path[j]]] = last_path[j];
      cur = malta[last_path[j]];
      candidate = move(accept);
    }
    if (B.size()) {
      A.emplace_back(*B.begin());
      B.erase(B.begin());
    }
  }
  return A;
}

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