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:heavy_check_mark: graph/find_path_through_specified.hpp

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Code

#include "graph/base.hpp"
#include "random/base.hpp"
#include "nt/GF2.hpp"

// (s,t) path であって need をすべて通るものの最小長さを求める.
// s=t でもよい. 最初の 1 歩目も求める.
// {length, first v}. なければ {-1,-1}.
// O(2^K(N+M)L), L = shortest path length
template <typename GT>
pair<int, int> find_path_through_specified(GT& G, int s, int t, vc<int> need) {
  static_assert(!GT::is_directed);
  int N = G.N;
  assert(0 <= s && s < N && 0 <= t && t < N);
  using F = GF2<64>;
  // frm, to, wt
  vc<tuple<int, int, F>> edges;
  vc<int> S, T;
  for (auto& e: G.edges) {
    int a = e.frm, b = e.to;
    if ((a == s && b == t) || (a == t && b == s)) {
      if (need.empty()) return {1, t};
      continue;
    }
    if (a == s || b == s) { S.eb(a ^ b ^ s); }
    if (a == t || b == t) { T.eb(a ^ b ^ t); }
    if (a != s && a != t && b != s && b != t) {
      F x = RNG_64();
      edges.eb(a, b, x), edges.eb(b, a, x);
    }
  }
  int K = len(need);
  vc<int> get(N);
  FOR(k, K) get[need[k]] = 1 << k;
  int M = len(edges);

  vv(F, dp_e, 1 << K, M);
  vv(F, dp_v, 1 << K, N);
  for (auto& v: T) dp_v[get[v]][v] = RNG_64();
  FOR(L, 1, N) {
    for (auto& v: S) {
      if (dp_v.back()[v] != F(0)) return {1 + L, v};
    }
    vv(F, newdp_e, 1 << K, M);
    vv(F, newdp_v, 1 << K, N);
    FOR(s, 1 << K) {
      FOR(m, M) {
        auto [a, b, c] = edges[m];
        int t = s | get[b];
        if (get[b] && s == t) continue;
        F x = (dp_v[s][a] + dp_e[s][m ^ 1]) * c;
        newdp_e[t][m] += x, newdp_v[t][b] += x;
      }
    }
    swap(dp_v, newdp_v), swap(dp_e, newdp_e);
  }
  return {-1, -1};
}
#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 2 "random/base.hpp"

u64 RNG_64() {
  static uint64_t x_
      = uint64_t(chrono::duration_cast<chrono::nanoseconds>(
                     chrono::high_resolution_clock::now().time_since_epoch())
                     .count())
        * 10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}

u64 RNG(u64 lim) { return RNG_64() % lim; }

ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 1 "nt/GF2.hpp"
#include <emmintrin.h>
#include <smmintrin.h>
#include <wmmintrin.h>

__attribute__((target("pclmul"))) inline __m128i myclmul(const __m128i &a,
                                                         const __m128i &b) {
  return _mm_clmulepi64_si128(a, b, 0);
}

// 2^n 元体
template <int K>
struct GF2 {
  // https://oeis.org/A344141
  // irreducible poly x^K + ...
  static constexpr int POLY[65]
      = {0,  0, 3,  3,   3,  5,   3,  3,  27,  3,  9,  5,   9, 27, 33, 3,   43,
         9,  9, 39, 9,   5,  3,   33, 27, 9,   27, 39, 3,   5, 3,  9,  141, 75,
         27, 5, 53, 63,  99, 17,  57, 9,  39,  89, 33, 27,  3, 33, 45, 113, 29,
         75, 9, 71, 125, 71, 149, 17, 99, 123, 3,  39, 105, 3, 27};

  static constexpr u64 mask() { return u64(-1) >> (64 - K); }

  __attribute__((target("sse4.2"))) static u64 mul(u64 a, u64 b) {
    static bool prepared = 0;
    static u64 MEMO[8][65536];
    if (!prepared) {
      prepared = 1;
      vc<u64> tmp(128);
      tmp[0] = 1;
      FOR(i, 127) {
        tmp[i + 1] = tmp[i] << 1;
        if (tmp[i] >> (K - 1) & 1) {
          tmp[i + 1] ^= POLY[K];
          tmp[i + 1] &= mask();
        }
      }
      FOR(k, 8) {
        MEMO[k][0] = 0;
        FOR(i, 16) {
          FOR(s, 1 << i) { MEMO[k][s | 1 << i] = MEMO[k][s] ^ tmp[16 * k + i]; }
        }
      }
    }
    const __m128i a_ = _mm_set_epi64x(0, a);
    const __m128i b_ = _mm_set_epi64x(0, b);
    const __m128i c_ = myclmul(a_, b_);
    u64 lo = _mm_extract_epi64(c_, 0);
    u64 hi = _mm_extract_epi64(c_, 1);
    u64 x = 0;
    x ^= MEMO[0][lo & 65535];
    x ^= MEMO[1][(lo >> 16) & 65535];
    x ^= MEMO[2][(lo >> 32) & 65535];
    x ^= MEMO[3][(lo >> 48) & 65535];
    x ^= MEMO[4][hi & 65535];
    x ^= MEMO[5][(hi >> 16) & 65535];
    x ^= MEMO[6][(hi >> 32) & 65535];
    x ^= MEMO[7][(hi >> 48) & 65535];
    return x;
  }

  u64 val;
  constexpr GF2(const u64 val = 0) noexcept : val(val & mask()) {}
  bool operator<(const GF2 &other) const {
    return val < other.val;
  } // To use std::map
  GF2 &operator+=(const GF2 &p) {
    val ^= p.val;
    return *this;
  }
  GF2 &operator-=(const GF2 &p) {
    val ^= p.val;
    return *this;
  }
  GF2 &operator*=(const GF2 &p) {
    val = mul(val, p.val);
    return *this;
  }

  GF2 &operator/=(const GF2 &p) {
    *this *= p.inverse();
    return *this;
  }
  GF2 operator-() const { return GF2(-val); }
  GF2 operator+(const GF2 &p) const { return GF2(*this) += p; }
  GF2 operator-(const GF2 &p) const { return GF2(*this) -= p; }
  GF2 operator*(const GF2 &p) const { return GF2(*this) *= p; }
  GF2 operator/(const GF2 &p) const { return GF2(*this) /= p; }
  bool operator==(const GF2 &p) const { return val == p.val; }
  bool operator!=(const GF2 &p) const { return val != p.val; }
  GF2 inverse() const { return pow((u64(1) << K) - 2); }
  GF2 pow(u64 n) const {
    GF2 ret(1), mul(val);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
};

#ifdef FASTIO
template <int K>
void rd(GF2<K> &x) {
  fastio::rd(x.val);
  x &= GF2<K>::mask;
}
template <int K>
void wt(GF2<K> x) {
  fastio::wt(x.val);
}
#endif
#line 4 "graph/find_path_through_specified.hpp"

// (s,t) path であって need をすべて通るものの最小長さを求める.
// s=t でもよい. 最初の 1 歩目も求める.
// {length, first v}. なければ {-1,-1}.
// O(2^K(N+M)L), L = shortest path length
template <typename GT>
pair<int, int> find_path_through_specified(GT& G, int s, int t, vc<int> need) {
  static_assert(!GT::is_directed);
  int N = G.N;
  assert(0 <= s && s < N && 0 <= t && t < N);
  using F = GF2<64>;
  // frm, to, wt
  vc<tuple<int, int, F>> edges;
  vc<int> S, T;
  for (auto& e: G.edges) {
    int a = e.frm, b = e.to;
    if ((a == s && b == t) || (a == t && b == s)) {
      if (need.empty()) return {1, t};
      continue;
    }
    if (a == s || b == s) { S.eb(a ^ b ^ s); }
    if (a == t || b == t) { T.eb(a ^ b ^ t); }
    if (a != s && a != t && b != s && b != t) {
      F x = RNG_64();
      edges.eb(a, b, x), edges.eb(b, a, x);
    }
  }
  int K = len(need);
  vc<int> get(N);
  FOR(k, K) get[need[k]] = 1 << k;
  int M = len(edges);

  vv(F, dp_e, 1 << K, M);
  vv(F, dp_v, 1 << K, N);
  for (auto& v: T) dp_v[get[v]][v] = RNG_64();
  FOR(L, 1, N) {
    for (auto& v: S) {
      if (dp_v.back()[v] != F(0)) return {1 + L, v};
    }
    vv(F, newdp_e, 1 << K, M);
    vv(F, newdp_v, 1 << K, N);
    FOR(s, 1 << K) {
      FOR(m, M) {
        auto [a, b, c] = edges[m];
        int t = s | get[b];
        if (get[b] && s == t) continue;
        F x = (dp_v[s][a] + dp_e[s][m ^ 1]) * c;
        newdp_e[t][m] += x, newdp_v[t][b] += x;
      }
    }
    swap(dp_v, newdp_v), swap(dp_e, newdp_e);
  }
  return {-1, -1};
}
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