graph/st_numbering.hpp

• View this file on GitHub
• Last update: 2025-04-06 22:14:02+09:00
• Include: #include "graph/st_numbering.hpp"
• Link: https://en.wikipedia.org/wiki/Bipolar_orientation

Depends on

• ds/hashmap.hpp
• ds/unionfind/unionfind.hpp
• graph/base.hpp
• graph/block_cut.hpp
• graph/shortest_path/bfs01.hpp
• graph/shortest_path/restore_path.hpp

Verified with

• test/1_mytest/st_numbering.test.cpp
• test/2_library_checker/graph/st_numbering.test.cpp

Code

#include "graph/base.hpp"
#include "ds/unionfind/unionfind.hpp"
#include "graph/block_cut.hpp"
#include "graph/shortest_path/bfs01.hpp"
#include "graph/shortest_path/restore_path.hpp"

// https://en.wikipedia.org/wiki/Bipolar_orientation
// 順列 p を求める. p[s]=0,p[t]=n-1.
// p[u]<p[v] となる向きに辺を向き付けると任意の v に対して svt パスが存在.
// 存在条件：BCT で全部の成分を通る st パスがある 不可能ならば empty をかえす.
template <typename GT>
vc<int> st_numbering(GT &G, int s, int t) {
  static_assert(!GT::is_directed);
  assert(G.is_prepared());
  int N = G.N;
  if (N == 1) return {0};
  if (s == t) return {};
  vc<int> par(N, -1), pre(N, -1), low(N, -1);
  vc<int> V;

  auto dfs = [&](auto &dfs, int v) -> void {
    pre[v] = len(V), V.eb(v);
    low[v] = v;
    for (auto &e: G[v]) {
      int w = e.to;
      if (v == w) continue;
      if (pre[w] == -1) {
        dfs(dfs, w);
        par[w] = v;
        if (pre[low[w]] < pre[low[v]]) { low[v] = low[w]; }
      }
      elif (pre[w] < pre[low[v]]) { low[v] = w; }
    }
  };

  pre[s] = 0, V.eb(s);
  dfs(dfs, t);
  if (len(V) != N) return {};
  vc<int> nxt(N, -1), prev(N);
  nxt[s] = t, prev[t] = s;

  vc<int> sgn(N);
  sgn[s] = -1;
  FOR(i, 2, len(V)) {
    int v = V[i];
    int p = par[v];
    if (sgn[low[v]] == -1) {
      int q = prev[p];
      if (q == -1) return {};
      nxt[q] = v, nxt[v] = p;
      prev[v] = q, prev[p] = v;
      sgn[p] = 1;
    } else {
      int q = nxt[p];
      if (q == -1) return {};
      nxt[p] = v, nxt[v] = q;
      prev[v] = p, prev[q] = v;
      sgn[p] = -1;
    }
  }
  vc<int> A = {s};
  while (A.back() != t) { A.eb(nxt[A.back()]); }
  // 作れているか判定
  if (len(A) < N) return {};
  assert(A[0] == s && A.back() == t);
  vc<int> rk(N, -1);
  FOR(i, N) rk[A[i]] = i;
  assert(MIN(rk) != -1);
  FOR(i, N) {
    bool l = 0, r = 0;
    int v = A[i];
    for (auto &e: G[v]) {
      if (rk[e.to] < rk[v]) l = 1;
      if (rk[v] < rk[e.to]) r = 1;
    }
    if (i > 0 && !l) return {};
    if (i < N - 1 && !r) return {};
  }
  vc<int> res(N);
  FOR(i, N) res[A[i]] = i;
  return res;
}

bool check_st_numbering(Graph<int, 0> G, int s, int t) {
  int N = G.N;
  assert(N >= 2);
  if (s == t) return 0;

  UnionFind uf(N);
  for (auto &e: G.edges) uf.merge(e.frm, e.to);
  if (uf.n_comp >= 2) return 0; // disconnected

  // BCT において st パスがすべての block を通ることが必要
  auto BCT = block_cut(G);
  auto [dist, par] = bfs01<int>(BCT, s);
  vc<int> path = restore_path(par, t);

  vc<int> vis(BCT.N);
  for (auto &x: path) vis[x] = 1;

  FOR(i, N, BCT.N) {
    if (!vis[i]) return 0;
  }
  return 1;
}

#line 1 "graph/st_numbering.hpp"

#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() {
#ifdef LOCAL
    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
  }
#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 2 "ds/unionfind/unionfind.hpp"

struct UnionFind {
  int n, n_comp;
  vc<int> dat; // par or (-size)
  UnionFind(int n = 0) { build(n); }

  void build(int m) {
    n = m, n_comp = m;
    dat.assign(n, -1);
  }

  void reset() { build(n); }

  int operator[](int x) {
    while (dat[x] >= 0) {
      int pp = dat[dat[x]];
      if (pp < 0) { return dat[x]; }
      x = dat[x] = pp;
    }
    return x;
  }

  ll size(int x) {
    x = (*this)[x];
    return -dat[x];
  }

  bool merge(int x, int y) {
    x = (*this)[x], y = (*this)[y];
    if (x == y) return false;
    if (-dat[x] < -dat[y]) swap(x, y);
    dat[x] += dat[y], dat[y] = x, n_comp--;
    return true;
  }

  vc<int> get_all() {
    vc<int> A(n);
    FOR(i, n) A[i] = (*this)[i];
    return A;
  }
};
#line 2 "graph/block_cut.hpp"

/*
block-cut tree を、block に通常の頂点を隣接させて拡張しておく
https://twitter.com/noshi91/status/1529858538650374144?s=20&t=eznpFbuD9BDhfTb4PplFUg
[0, n)：もとの頂点 [n, n + n_block)：block
関節点：[0, n) のうちで、degree >= 2 を満たすもの
孤立点は、1 点だけからなる block
成分が欲しい場合：近傍を見ると点集合. 辺から成分を得るには tree.jump
と思ったが非連結なときに注意がいるな…
*/
template <typename GT>
Graph<int, 0> block_cut(GT& G) {
  int n = G.N;
  vc<int> low(n), ord(n), st;
  vc<bool> used(n);
  st.reserve(n);
  int nxt = n;
  int k = 0;
  vc<pair<int, int>> edges;

  auto dfs = [&](auto& dfs, int v, int p) -> void {
    st.eb(v);
    used[v] = 1;
    low[v] = ord[v] = k++;
    int child = 0;
    for (auto&& e: G[v]) {
      if (e.to == p) continue;
      if (!used[e.to]) {
        ++child;
        int s = len(st);
        dfs(dfs, e.to, v);
        chmin(low[v], low[e.to]);
        if ((p == -1 && child > 1) || (p != -1 && low[e.to] >= ord[v])) {
          edges.eb(nxt, v);
          while (len(st) > s) {
            edges.eb(nxt, st.back());
            st.pop_back();
          }
          ++nxt;
        }
      } else {
        chmin(low[v], ord[e.to]);
      }
    }
  };
  FOR(v, n) if (!used[v]) {
    dfs(dfs, v, -1);
    for (auto&& x: st) { edges.eb(nxt, x); }
    ++nxt;
    st.clear();
  }
  Graph<int, 0> BCT(nxt);
  for (auto&& [a, b]: edges) BCT.add(a, b);
  BCT.build();
  return BCT;
}
#line 3 "graph/shortest_path/bfs01.hpp"

template <typename T, typename GT>
pair<vc<T>, vc<int>> bfs01(GT& G, int v) {
  assert(G.is_prepared());
  int N = G.N;
  vc<T> dist(N, infty<T>);
  vc<int> par(N, -1);
  deque<int> que;

  dist[v] = 0;
  que.push_front(v);
  while (!que.empty()) {
    auto v = que.front();
    que.pop_front();
    for (auto&& e: G[v]) {
      if (dist[e.to] == infty<T> || dist[e.to] > dist[e.frm] + e.cost) {
        dist[e.to] = dist[e.frm] + e.cost;
        par[e.to] = e.frm;
        if (e.cost == 0)
          que.push_front(e.to);
        else
          que.push_back(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>> bfs01(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);
  deque<int> que;

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

  while (!que.empty()) {
    auto v = que.front();
    que.pop_front();
    for (auto&& e: G[v]) {
      if (dist[e.to] == infty<T> || dist[e.to] > dist[e.frm] + e.cost) {
        dist[e.to] = dist[e.frm] + e.cost;
        root[e.to] = root[e.frm];
        par[e.to] = e.frm;
        if (e.cost == 0)
          que.push_front(e.to);
        else
          que.push_back(e.to);
      }
    }
  }
  return {dist, par, root};
}
#line 1 "graph/shortest_path/restore_path.hpp"
vector<int> restore_path(vector<int> par, int t){
  vector<int> pth = {t};
  while (par[pth.back()] != -1) pth.eb(par[pth.back()]);
  reverse(all(pth));
  return pth;
}
#line 7 "graph/st_numbering.hpp"

// https://en.wikipedia.org/wiki/Bipolar_orientation
// 順列 p を求める. p[s]=0,p[t]=n-1.
// p[u]<p[v] となる向きに辺を向き付けると任意の v に対して svt パスが存在.
// 存在条件：BCT で全部の成分を通る st パスがある 不可能ならば empty をかえす.
template <typename GT>
vc<int> st_numbering(GT &G, int s, int t) {
  static_assert(!GT::is_directed);
  assert(G.is_prepared());
  int N = G.N;
  if (N == 1) return {0};
  if (s == t) return {};
  vc<int> par(N, -1), pre(N, -1), low(N, -1);
  vc<int> V;

  auto dfs = [&](auto &dfs, int v) -> void {
    pre[v] = len(V), V.eb(v);
    low[v] = v;
    for (auto &e: G[v]) {
      int w = e.to;
      if (v == w) continue;
      if (pre[w] == -1) {
        dfs(dfs, w);
        par[w] = v;
        if (pre[low[w]] < pre[low[v]]) { low[v] = low[w]; }
      }
      elif (pre[w] < pre[low[v]]) { low[v] = w; }
    }
  };

  pre[s] = 0, V.eb(s);
  dfs(dfs, t);
  if (len(V) != N) return {};
  vc<int> nxt(N, -1), prev(N);
  nxt[s] = t, prev[t] = s;

  vc<int> sgn(N);
  sgn[s] = -1;
  FOR(i, 2, len(V)) {
    int v = V[i];
    int p = par[v];
    if (sgn[low[v]] == -1) {
      int q = prev[p];
      if (q == -1) return {};
      nxt[q] = v, nxt[v] = p;
      prev[v] = q, prev[p] = v;
      sgn[p] = 1;
    } else {
      int q = nxt[p];
      if (q == -1) return {};
      nxt[p] = v, nxt[v] = q;
      prev[v] = p, prev[q] = v;
      sgn[p] = -1;
    }
  }
  vc<int> A = {s};
  while (A.back() != t) { A.eb(nxt[A.back()]); }
  // 作れているか判定
  if (len(A) < N) return {};
  assert(A[0] == s && A.back() == t);
  vc<int> rk(N, -1);
  FOR(i, N) rk[A[i]] = i;
  assert(MIN(rk) != -1);
  FOR(i, N) {
    bool l = 0, r = 0;
    int v = A[i];
    for (auto &e: G[v]) {
      if (rk[e.to] < rk[v]) l = 1;
      if (rk[v] < rk[e.to]) r = 1;
    }
    if (i > 0 && !l) return {};
    if (i < N - 1 && !r) return {};
  }
  vc<int> res(N);
  FOR(i, N) res[A[i]] = i;
  return res;
}

bool check_st_numbering(Graph<int, 0> G, int s, int t) {
  int N = G.N;
  assert(N >= 2);
  if (s == t) return 0;

  UnionFind uf(N);
  for (auto &e: G.edges) uf.merge(e.frm, e.to);
  if (uf.n_comp >= 2) return 0; // disconnected

  // BCT において st パスがすべての block を通ることが必要
  auto BCT = block_cut(G);
  auto [dist, par] = bfs01<int>(BCT, s);
  vc<int> path = restore_path(par, t);

  vc<int> vis(BCT.N);
  for (auto &x: path) vis[x] = 1;

  FOR(i, N, BCT.N) {
    if (!vis[i]) return 0;
  }
  return 1;
}

Back to top page