graph/shortest_path/K_shortest_walk.hpp

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• Last update: 2024-12-25 20:50:37+09:00
• Include: #include "graph/shortest_path/K_shortest_walk.hpp"

Depends on

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

Verified with

• test/2_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>
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, true> X(20 * G.M);
  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, 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;
  const int NODES;
  int pid;
  using np = Node *;

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

// 全体加算ができるようにする
// path sum が実際の値となるようにすれば追加フィールドなしに実現できる
// https://qoj.ac/contest/1699/problem/8518
template <typename VAL, bool PERSISTENT, bool TOP_IS_MIN>
struct Lazy_Meldable_Heap {
  struct Node {
    Node *l, *r;
    VAL x;
    u32 size;
  };
  Node *pool;
  const int NODES;
  int pid;
  using np = Node *;

  Lazy_Meldable_Heap(int NODES) : NODES(NODES), pid(0) { pool = new Node[NODES]; }
  ~Lazy_Meldable_Heap() { delete[] pool; }
  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;
    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;
    return b;
  }
  np apply(np a, VAL x) {
    if (!a) return a;
    a = copy_node(a);
    a->x += x;
    return a;
  }
  np meld(np a, np b, VAL add_a = 0, VAL add_b = 0) {
    if (!a) { return (add_b == 0 ? b : apply(b, add_b)); }
    if (!b) { return (add_a == 0 ? a : apply(a, add_a)); }
    if constexpr (TOP_IS_MIN) {
      if ((a->x + add_a) > (b->x + add_b)) swap(a, b), swap(add_a, add_b);
    } else {
      if ((a->x + add_a) < (b->x + add_b)) swap(a, b), swap(add_a, add_b);
    }
    a = copy_node(a);
    a->x += add_a;
    a->r = meld(a->r, b, 0, -a->x + add_b);
    if (!(a->l) || (a->l->size < a->r->size)) swap(a->l, a->r);
    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, a->x, a->x); }
  VAL top(np a) { return a->x; }
  vc<VAL> get_all(np a) {
    vc<VAL> A;
    auto dfs = [&](auto &dfs, np a, VAL lazy) -> void {
      if (!a) return;
      A.eb(a->x + lazy);
      lazy += a->x;
      dfs(dfs, a->l, lazy), dfs(dfs, a->r, lazy);
    };
    dfs(dfs, a, 0);
    sort(all(A));
    if (!TOP_IS_MIN) reverse(all(A));
    return A;
  }
};
#line 2 "ds/hashmap.hpp"

// u64 -> Val

template <typename Val>
struct HashMap {
  // n は入れたいものの個数で ok

  HashMap(u32 n = 0) { build(n); }
  void build(u32 n) {
    u32 k = 8;
    while (k < n * 2) k *= 2;
    cap = k / 2, mask = k - 1;
    key.resize(k), val.resize(k), used.assign(k, 0);
  }

  // size を保ったまま. size=0 にするときは build すること.

  void clear() {
    used.assign(len(used), 0);
    cap = (mask + 1) / 2;
  }
  int size() { return len(used) / 2 - cap; }

  int index(const u64& k) {
    int i = 0;
    for (i = hash(k); used[i] && key[i] != k; i = (i + 1) & mask) {}
    return i;
  }

  Val& operator[](const u64& k) {
    if (cap == 0) extend();
    int i = index(k);
    if (!used[i]) { used[i] = 1, key[i] = k, val[i] = Val{}, --cap; }
    return val[i];
  }

  Val get(const u64& k, Val default_value) {
    int i = index(k);
    return (used[i] ? val[i] : default_value);
  }

  bool count(const u64& k) {
    int i = index(k);
    return used[i] && key[i] == k;
  }

  // f(key, val)

  template <typename F>
  void enumerate_all(F f) {
    FOR(i, len(used)) if (used[i]) f(key[i], val[i]);
  }

private:
  u32 cap, mask;
  vc<u64> key;
  vc<Val> val;
  vc<bool> used;

  u64 hash(u64 x) {
    static const u64 FIXED_RANDOM = std::chrono::steady_clock::now().time_since_epoch().count();
    x += FIXED_RANDOM;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return (x ^ (x >> 31)) & mask;
  }

  void extend() {
    vc<pair<u64, Val>> dat;
    dat.reserve(len(used) / 2 - cap);
    FOR(i, len(used)) {
      if (used[i]) dat.eb(key[i], val[i]);
    }
    build(2 * len(dat));
    for (auto& [a, b]: dat) (*this)[a] = b;
  }
};
#line 3 "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}
  // sum(deg(v)) の計算量になっていて、
  // 新しいグラフの n+m より大きい可能性があるので注意
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    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 (len(used_e) <= e.id) used_e.resize(e.id + 1);
        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;
  }

  Graph<T, true> to_directed_tree(int root = -1) {
    if (root == -1) root = 0;
    assert(!is_directed && prepared && M == N - 1);
    Graph<T, true> G1(N);
    vc<int> par(N, -1);
    auto dfs = [&](auto& dfs, int v) -> void {
      for (auto& e: (*this)[v]) {
        if (e.to == par[v]) continue;
        par[e.to] = v, dfs(dfs, e.to);
      }
    };
    dfs(dfs, root);
    for (auto& e: edges) {
      int a = e.frm, b = e.to;
      if (par[a] == b) swap(a, b);
      assert(par[b] == a);
      G1.add(a, b, e.cost);
    }
    G1.build();
    return G1;
  }

  HashMap<int> MP_FOR_EID;

  int get_eid(u64 a, u64 b) {
    if (len(MP_FOR_EID) == 0) {
      MP_FOR_EID.build(N - 1);
      for (auto& e: edges) {
        u64 a = e.frm, b = e.to;
        u64 k = to_eid_key(a, b);
        MP_FOR_EID[k] = e.id;
      }
    }
    return MP_FOR_EID.get(to_eid_key(a, b), -1);
  }

  u64 to_eid_key(u64 a, u64 b) {
    if (!directed && a > b) swap(a, b);
    return N * a + b;
  }

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>
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, true> X(20 * G.M);
  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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