graph/shortest_path/K_shortest_walk.hpp

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• Last update: 2023-11-24 15:10:29+09:00
• Include: #include "graph/shortest_path/K_shortest_walk.hpp"

Depends on

• ds/meldable_heap.hpp
• graph/base.hpp
• graph/reverse_graph.hpp
• graph/shortest_path/dijkstra.hpp

Verified with

• test/library_checker/graph/K_shortest_walk.test.cpp

Code

#include "ds/meldable_heap.hpp"
#include "graph/shortest_path/dijkstra.hpp"
#include "graph/reverse_graph.hpp"

// infty<T> 埋めして必ず長さ K にしたものをかえす。
template <typename T, typename GT, int NODES>
vc<T> K_shortest_walk(GT &G, int s, int t, int K) {
  static_assert(GT::is_directed);
  int N = G.N;
  auto RG = reverse_graph(G);
  auto [dist, par] = dijkstra<T, decltype(RG)>(RG, t);
  if (dist[s] == infty<T>) { return vc<T>(K, infty<T>); }

  using P = pair<T, int>;
  Meldable_Heap<P, true, NODES, true> X;
  using np = typename decltype(X)::np;
  vc<np> nodes(N, nullptr);

  vc<bool> vis(N);
  vc<int> st = {t};
  vis[t] = 1;
  while (len(st)) {
    int v = POP(st);
    bool done = 0;
    for (auto &&e: G[v]) {
      if (dist[e.to] == infty<T>) continue;
      if (!done && par[v] == e.to && dist[v] == dist[e.to] + e.cost) {
        done = 1;
        continue;
      }
      T cost = -dist[v] + e.cost + dist[e.to];
      nodes[v] = X.push(nodes[v], {cost, e.to});
    }
    for (auto &&e: RG[v]) {
      if (vis[e.to]) continue;
      if (par[e.to] == v) {
        nodes[e.to] = X.meld(nodes[e.to], nodes[v]);
        vis[e.to] = 1;
        st.eb(e.to);
      }
    }
  }

  T base = dist[s];
  vc<T> ANS = {base};
  if (nodes[s]) {
    using PAIR = pair<T, np>;
    auto comp = [](auto a, auto b) { return a.fi > b.fi; };
    priority_queue<PAIR, vc<PAIR>, decltype(comp)> que(comp);
    que.emplace(base + X.top(nodes[s]).fi, nodes[s]);
    while (len(ANS) < K && len(que)) {
      auto [d, n] = que.top();
      que.pop();
      ANS.eb(d);
      if (n->l) que.emplace(d + (n->l->x.fi) - (n->x.fi), n->l);
      if (n->r) que.emplace(d + (n->r->x.fi) - (n->x.fi), n->r);
      np m = nodes[n->x.se];
      if (m) { que.emplace(d + m->x.fi, m); }
    }
  }
  while (len(ANS) < K) ANS.eb(infty<T>);
  return ANS;
}

#line 1 "ds/meldable_heap.hpp"

template <typename VAL, bool PERSISTENT, int NODES, bool TOP_IS_MIN>
struct Meldable_Heap {
  struct Node {
    Node *l, *r;
    VAL x;
    u32 size, dist; // dist: leaf までの距離
  };
  Node *pool;
  int pid;
  using np = Node *;

  Meldable_Heap() : pid(0) { pool = new Node[NODES]; }

  np new_root() { return nullptr; }
  np new_node(const VAL &x) {
    pool[pid].l = pool[pid].r = nullptr;
    pool[pid].x = x, pool[pid].size = 1, pool[pid].dist = 1;
    return &(pool[pid++]);
  }
  np copy_node(np a) {
    if (!a || !PERSISTENT) return a;
    np b = new_node(a->x);
    b->l = a->l, b->r = a->r;
    b->size = a->size, b->dist = a->dist;
    return b;
  }
  np meld(np a, np b) {
    if (!a) return b;
    if (!b) return a;
    a = copy_node(a);
    b = copy_node(b);
    if constexpr (TOP_IS_MIN) {
      if ((a->x) > (b->x)) swap(a, b);
    } else {
      if ((a->x) < (b->x)) swap(a, b);
    }
    a->r = meld(a->r, b);
    if (!(a->l) || (a->l->dist < a->r->dist)) swap(a->l, a->r);
    a->dist = (a->r ? a->r->dist : 0) + 1;
    a->size = 1;
    if (a->l) a->size += a->l->size;
    if (a->r) a->size += a->r->size;
    return a;
  }
  np push(np a, VAL x) { return meld(a, new_node(x)); }
  np pop(np a) { return meld(a->l, a->r); }
  VAL top(np a) { return a->x; }
  vc<VAL> get_all(np a) {
    vc<VAL> A;
    auto dfs = [&](auto &dfs, np a) -> void {
      if (!a) return;
      A.eb(a->x), dfs(dfs, a->l), dfs(dfs, a->r);
    };
    dfs(dfs, a);
    sort(all(A));
    if (!TOP_IS_MIN) reverse(all(A));
    return A;
  }
};
#line 2 "graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  static constexpr bool is_directed = directed;
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

#ifdef FASTIO
  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }
#endif

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed)
        csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

#ifdef FASTIO
  void debug() {
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
  }
#endif

  vc<int> new_idx;
  vc<bool> used_e;

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  // {G, es}
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    if (len(used_e) != M) used_e.assign(M, 0);
    int n = len(V);
    FOR(i, n) new_idx[V[i]] = i;
    Graph<T, directed> G(n);
    vc<int> history;
    FOR(i, n) {
      for (auto&& e: (*this)[V[i]]) {
        if (used_e[e.id]) continue;
        int a = e.frm, b = e.to;
        if (new_idx[a] != -1 && new_idx[b] != -1) {
          history.eb(e.id);
          used_e[e.id] = 1;
          int eid = (keep_eid ? e.id : -1);
          G.add(new_idx[a], new_idx[b], e.cost, eid);
        }
      }
    }
    FOR(i, n) new_idx[V[i]] = -1;
    for (auto&& eid: history) used_e[eid] = 0;
    G.build();
    return G;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 3 "graph/shortest_path/dijkstra.hpp"

template <typename T, typename GT>
pair<vc<T>, vc<int>> dijkstra_dense(GT& G, int s) {
  const int N = G.N;
  vc<T> dist(N, infty<T>);
  vc<int> par(N, -1);
  vc<bool> done(N);
  dist[s] = 0;
  while (1) {
    int v = -1;
    T mi = infty<T>;
    FOR(i, N) {
      if (!done[i] && chmin(mi, dist[i])) v = i;
    }
    if (v == -1) break;
    done[v] = 1;
    for (auto&& e: G[v]) {
      if (chmin(dist[e.to], dist[v] + e.cost)) par[e.to] = v;
    }
  }
  return {dist, par};
}

template <typename T, typename GT, bool DENSE = false>
pair<vc<T>, vc<int>> dijkstra(GT& G, int v) {
  if (DENSE) return dijkstra_dense<T>(G, v);
  auto N = G.N;
  vector<T> dist(N, infty<T>);
  vector<int> par(N, -1);
  using P = pair<T, int>;

  priority_queue<P, vector<P>, greater<P>> que;

  dist[v] = 0;
  que.emplace(0, v);
  while (!que.empty()) {
    auto [dv, v] = que.top();
    que.pop();
    if (dv > dist[v]) continue;
    for (auto&& e: G[v]) {
      if (chmin(dist[e.to], dist[e.frm] + e.cost)) {
        par[e.to] = e.frm;
        que.emplace(dist[e.to], e.to);
      }
    }
  }
  return {dist, par};
}

// 多点スタート。[dist, par, root]
template <typename T, typename GT>
tuple<vc<T>, vc<int>, vc<int>> dijkstra(GT& G, vc<int> vs) {
  assert(G.is_prepared());
  int N = G.N;
  vc<T> dist(N, infty<T>);
  vc<int> par(N, -1);
  vc<int> root(N, -1);

  using P = pair<T, int>;

  priority_queue<P, vector<P>, greater<P>> que;

  for (auto&& v: vs) {
    dist[v] = 0;
    root[v] = v;
    que.emplace(T(0), v);
  }

  while (!que.empty()) {
    auto [dv, v] = que.top();
    que.pop();
    if (dv > dist[v]) continue;
    for (auto&& e: G[v]) {
      if (chmin(dist[e.to], dist[e.frm] + e.cost)) {
        root[e.to] = root[e.frm];
        par[e.to] = e.frm;
        que.push(mp(dist[e.to], e.to));
      }
    }
  }
  return {dist, par, root};
}
#line 2 "graph/reverse_graph.hpp"

template <typename GT>
GT reverse_graph(GT& G) {
  static_assert(GT::is_directed);
  GT G1(G.N);
  for (auto&& e: G.edges) { G1.add(e.to, e.frm, e.cost, e.id); }
  G1.build();
  return G1;
}
#line 4 "graph/shortest_path/K_shortest_walk.hpp"

// infty<T> 埋めして必ず長さ K にしたものをかえす。
template <typename T, typename GT, int NODES>
vc<T> K_shortest_walk(GT &G, int s, int t, int K) {
  static_assert(GT::is_directed);
  int N = G.N;
  auto RG = reverse_graph(G);
  auto [dist, par] = dijkstra<T, decltype(RG)>(RG, t);
  if (dist[s] == infty<T>) { return vc<T>(K, infty<T>); }

  using P = pair<T, int>;
  Meldable_Heap<P, true, NODES, true> X;
  using np = typename decltype(X)::np;
  vc<np> nodes(N, nullptr);

  vc<bool> vis(N);
  vc<int> st = {t};
  vis[t] = 1;
  while (len(st)) {
    int v = POP(st);
    bool done = 0;
    for (auto &&e: G[v]) {
      if (dist[e.to] == infty<T>) continue;
      if (!done && par[v] == e.to && dist[v] == dist[e.to] + e.cost) {
        done = 1;
        continue;
      }
      T cost = -dist[v] + e.cost + dist[e.to];
      nodes[v] = X.push(nodes[v], {cost, e.to});
    }
    for (auto &&e: RG[v]) {
      if (vis[e.to]) continue;
      if (par[e.to] == v) {
        nodes[e.to] = X.meld(nodes[e.to], nodes[v]);
        vis[e.to] = 1;
        st.eb(e.to);
      }
    }
  }

  T base = dist[s];
  vc<T> ANS = {base};
  if (nodes[s]) {
    using PAIR = pair<T, np>;
    auto comp = [](auto a, auto b) { return a.fi > b.fi; };
    priority_queue<PAIR, vc<PAIR>, decltype(comp)> que(comp);
    que.emplace(base + X.top(nodes[s]).fi, nodes[s]);
    while (len(ANS) < K && len(que)) {
      auto [d, n] = que.top();
      que.pop();
      ANS.eb(d);
      if (n->l) que.emplace(d + (n->l->x.fi) - (n->x.fi), n->l);
      if (n->r) que.emplace(d + (n->r->x.fi) - (n->x.fi), n->r);
      np m = nodes[n->x.se];
      if (m) { que.emplace(d + m->x.fi, m); }
    }
  }
  while (len(ANS) < K) ANS.eb(infty<T>);
  return ANS;
}

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