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:heavy_check_mark: other/connected_dp.hpp

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#include "ds/hashmap.hpp"
#include "random/hash_vector.hpp"

namespace connected_dp_squares {
// pair<新しい状態、今の成分 → 新しい成分>
vc<pair<vc<int>, vc<int>>> next_states(const vc<int>& now) {
  int N = len(now);
  vc<pair<vc<int>, vc<int>>> res;
  FOR(s, 1 << N) {
    vc<int> par(N + N);
    FOR(i, N) par[i] = (s & 1 << i ? i : -1);
    FOR(i, N) par[N + i] = (now[i] == -1 ? -1 : now[i] + N);
    auto find = [&](int x) -> int {
      while (par[x] != x) { x = par[x] = par[par[x]]; }
      return x;
    };
    auto merge = [&](int a, int b) -> void {
      a = find(a), b = find(b);
      if (a == b) return;
      if (a > b) swap(a, b);
      par[b] = a;
    };

    FOR(i, N - 1) if (par[i] != -1 && par[i + 1] != -1) merge(i, i + 1);
    FOR(i, N) if (par[i] != -1 && par[N + i] != -1) merge(i, N + i);
    FOR(i, N + N) if (par[i] != -1) par[i] = find(i);
    FOR(i, N, N + N) if (par[i] >= N) par[i] = -1;
    res.eb(vc<int>(par.begin(), par.begin() + N),
           vc<int>(par.begin() + N, par.end()));
  }
  return res;
}

vc<int> reverse_state(const vc<int>& now) {
  int N = len(now);
  vc<int> max_i(N, -1);
  FOR(i, N) if (now[i] != -1) max_i[now[i]] = i;
  vc<int> rev(N, -1);
  FOR(i, N) {
    if (now[i] == -1) continue;
    int x = max_i[now[i]];
    rev[N - 1 - i] = N - 1 - x;
  }
  return rev;
}

// 0, 1 ：空の列、領域の手前、後ろ
// 連結領域をひとつ作る。
// 状態：-1 が選んでいない。0,1,2,3 等は同じ成分には同じ値が入る。
// [states, edges]
pair<vvc<int>, vc<pair<int, int>>> connedted_dp_graph(int N,
                                                      bool merge_reverse) {
  static HashMap<int, 20, true> MP;
  MP.reset();
  vvc<int> states;
  vc<pair<int, int>> edges;

  states.eb(vc<int>(N, -1));
  states.eb(vc<int>(N, -1));
  MP[hash_vector<int>(states[0])] = 0;

  int p = -1;
  while (1) {
    if (++p == len(states)) break;
    if (p == 1) {
      edges.eb(1, 1);
      continue;
    }
    vc<int> now = states[p];
    for (auto&& [nxt, convert]: next_states(now)) {
      // 今の成分数、消える成分数
      int a = 0, b = 0;
      FOR(v, N) if (now[v] == v) {
        ++a;
        if (convert[v] == -1) ++b;
      }
      // 消える成分があってよいのは、終状態にいくときのみ
      if (b >= 2) continue;
      if (b == 1) {
        if (MAX(nxt) != -1) continue;
        edges.eb(p, 1);
        continue;
      }
      u64 h = hash_vector<int>(nxt);
      if (merge_reverse) { chmin(h, hash_vector<int>(reverse_state(nxt))); }
      if (!MP.count(h)) { MP[h] = len(states), states.eb(nxt); }
      edges.eb(p, MP[h]);
    }
  }
  return {states, edges};
}

// 0, 1 ：空の列、領域の手前、後ろ
// 多角形（空洞なし）をひとつ作る。
// 状態：-1 が選んでいない。0,1,2,3 等は同じ成分には同じ値が入る。
// [states, edges]
pair<vvc<int>, vc<pair<int, int>>> polygon_dp_graph(int N) {
  static HashMap<int, 20, true> MP;
  MP.reset();
  vvc<int> states;
  vc<pair<int, int>> edges;

  states.eb(vc<int>(N, -1));
  states.eb(vc<int>(N, -1));
  MP[hash_vector<int>(states[0])] = 0;

  int p = -1;
  while (1) {
    if (++p == len(states)) break;
    if (p == 1) {
      edges.eb(1, 1);
      continue;
    }
    vc<int> now = states[p];
    for (auto&& [nxt, convert]: next_states(now)) {
      // 今の成分数、消える成分数
      int a = 0, b = 0;
      FOR(v, N) if (now[v] == v) {
        ++a;
        if (convert[v] == -1) ++b;
      }
      // 消える成分があってよいのは、終状態にいくときのみ
      if (b >= 2) continue;
      if (b == 1) {
        if (MAX(nxt) != -1) continue;
        edges.eb(p, 1);
        continue;
      }
      bool ok = [&](vc<int>& now, vc<int>& nxt, vc<int>& convert) -> bool {
        // 頂点のみで接するのはダメ
        FOR(i, N - 1) {
          bool a1 = now[i] != -1, a2 = now[i + 1] != -1;
          bool b1 = nxt[i] != -1, b2 = nxt[i + 1] != -1;
          if (a1 && !a2 && !b1 && b2) return false;
          if (!a1 && a2 && b1 && !b2) return false;
        }
        // empty region を閉じることと、異なる連結成分がマージされることが同値
        int close = 0;
        int after = 0;
        vc<bool> is_new(N, 1);
        FOR(i, N) if (convert[i] != -1) is_new[convert[i]] = 0;
        FOR(i, N) if (nxt[i] == i && !is_new[i])++ after;
        vc<int> I;
        FOR(i, N) if (now[i] != -1) I.eb(i);
        FOR(k, len(I) - 1) {
          int i = I[k], j = I[k + 1];
          if (j == i + 1) continue;
          bool cl = 1;
          FOR(p, i + 1, j) if (nxt[p] == -1) cl = 0;
          if (cl) close++;
        }
        return a - close == after;
      }(now, nxt, convert);
      if (!ok) continue;
      u64 h = hash_vector<int>(nxt);
      int idx = MP.index(h);
      if (!MP.used[idx]) {
        MP.used[idx] = 1, MP.IDS.eb(idx), MP.key[idx] = h,
        MP.val[idx] = len(states);
        states.eb(nxt);
      }
      edges.eb(p, MP.val[idx]);
    }
  }
  return {states, edges};
}

} // namespace connected_dp_squares
#line 2 "ds/hashmap.hpp"

// u64 -> Val

template <typename Val, int LOG = 20, bool KEEP_IDS = false>
struct HashMap {
  static constexpr int N = (1 << LOG);
  u64* key;
  Val* val;
  vc<int> IDS;
  bitset<N> used;
  const int shift;
  const u64 r = 11995408973635179863ULL;
  HashMap() : key(new u64[N]), val(new Val[N]), shift(64 - LOG) {}
  u32 hash(u64 x) {
    static const u64 FIXED_RANDOM
        = std::chrono::steady_clock::now().time_since_epoch().count();
    return (u64(x + FIXED_RANDOM) * r) >> shift;
  }

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

  Val& operator[](const u64& k) {
    int i = index(k);
    if (!used[i]) {
      used[i] = 1, key[i] = k, val[i] = Val{};
      if constexpr (KEEP_IDS) IDS.eb(i);
    }
    return val[i];
  }

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

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

  void reset() {
    static_assert(KEEP_IDS);
    for (auto&& i: IDS) used[i] = 0;
    IDS.clear();
  }

  // f(key, val)

  template <typename F>
  void enumerate_all(F f) {
    static_assert(KEEP_IDS);
    for (auto&& i: IDS) f(key[i], val[i]);
  }
};
#line 2 "random/hash_vector.hpp"

#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 2 "mod/modint61.hpp"

struct modint61 {
  static constexpr u64 mod = (1ULL << 61) - 1;
  u64 val;
  constexpr modint61() : val(0ULL) {}
  constexpr modint61(u32 x) : val(x) {}
  constexpr modint61(u64 x) : val(x % mod) {}
  constexpr modint61(int x) : val((x < 0) ? (x + static_cast<ll>(mod)) : x) {}
  constexpr modint61(ll x)
      : val(((x %= static_cast<ll>(mod)) < 0) ? (x + static_cast<ll>(mod))
                                              : x) {}
  static constexpr u64 get_mod() { return mod; }
  modint61 &operator+=(const modint61 &a) {
    val = ((val += a.val) >= mod) ? (val - mod) : val;
    return *this;
  }
  modint61 &operator-=(const modint61 &a) {
    val = ((val -= a.val) >= mod) ? (val + mod) : val;
    return *this;
  }
  modint61 &operator*=(const modint61 &a) {
    const unsigned __int128 y = static_cast<unsigned __int128>(val) * a.val;
    val = (y >> 61) + (y & mod);
    val = (val >= mod) ? (val - mod) : val;
    return *this;
  }
  modint61 &operator/=(const modint61 &a) { return (*this *= a.inverse()); }
  modint61 operator+(const modint61 &p) const { return modint61(*this) += p; }
  modint61 operator-(const modint61 &p) const { return modint61(*this) -= p; }
  modint61 operator*(const modint61 &p) const { return modint61(*this) *= p; }
  modint61 operator/(const modint61 &p) const { return modint61(*this) /= p; }
  bool operator==(const modint61 &p) const { return val == p.val; }
  bool operator!=(const modint61 &p) const { return val != p.val; }
  modint61 inverse() const {
    ll a = val, b = mod, u = 1, v = 0, t;
    while (b > 0) {
      t = a / b;
      swap(a -= t * b, b), swap(u -= t * v, v);
    }
    return modint61(u);
  }
  modint61 pow(ll n) const {
    assert(n >= 0);
    modint61 ret(1), mul(val);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul, n >>= 1;
    }
    return ret;
  }
};

#ifdef FASTIO
void rd(modint61 &x) {
  fastio::rd(x.val);
  assert(0 <= x.val && x.val < modint61::mod);
}

void wt(modint61 x) { fastio::wt(x.val); }
#endif
#line 5 "random/hash_vector.hpp"

template <typename T>
u64 hash_vector(vc<T> X) {
  using mint = modint61;
  static vc<mint> hash_base;
  int n = len(X);
  while (len(hash_base) <= n) { hash_base.eb(RNG(mint::get_mod())); }
  mint H = 0;
  FOR(i, n) H += hash_base[i] * mint(X[i]);
  H += hash_base[n];
  return H.val;
}
#line 3 "other/connected_dp.hpp"

namespace connected_dp_squares {
// pair<新しい状態、今の成分 → 新しい成分>
vc<pair<vc<int>, vc<int>>> next_states(const vc<int>& now) {
  int N = len(now);
  vc<pair<vc<int>, vc<int>>> res;
  FOR(s, 1 << N) {
    vc<int> par(N + N);
    FOR(i, N) par[i] = (s & 1 << i ? i : -1);
    FOR(i, N) par[N + i] = (now[i] == -1 ? -1 : now[i] + N);
    auto find = [&](int x) -> int {
      while (par[x] != x) { x = par[x] = par[par[x]]; }
      return x;
    };
    auto merge = [&](int a, int b) -> void {
      a = find(a), b = find(b);
      if (a == b) return;
      if (a > b) swap(a, b);
      par[b] = a;
    };

    FOR(i, N - 1) if (par[i] != -1 && par[i + 1] != -1) merge(i, i + 1);
    FOR(i, N) if (par[i] != -1 && par[N + i] != -1) merge(i, N + i);
    FOR(i, N + N) if (par[i] != -1) par[i] = find(i);
    FOR(i, N, N + N) if (par[i] >= N) par[i] = -1;
    res.eb(vc<int>(par.begin(), par.begin() + N),
           vc<int>(par.begin() + N, par.end()));
  }
  return res;
}

vc<int> reverse_state(const vc<int>& now) {
  int N = len(now);
  vc<int> max_i(N, -1);
  FOR(i, N) if (now[i] != -1) max_i[now[i]] = i;
  vc<int> rev(N, -1);
  FOR(i, N) {
    if (now[i] == -1) continue;
    int x = max_i[now[i]];
    rev[N - 1 - i] = N - 1 - x;
  }
  return rev;
}

// 0, 1 ：空の列、領域の手前、後ろ
// 連結領域をひとつ作る。
// 状態：-1 が選んでいない。0,1,2,3 等は同じ成分には同じ値が入る。
// [states, edges]
pair<vvc<int>, vc<pair<int, int>>> connedted_dp_graph(int N,
                                                      bool merge_reverse) {
  static HashMap<int, 20, true> MP;
  MP.reset();
  vvc<int> states;
  vc<pair<int, int>> edges;

  states.eb(vc<int>(N, -1));
  states.eb(vc<int>(N, -1));
  MP[hash_vector<int>(states[0])] = 0;

  int p = -1;
  while (1) {
    if (++p == len(states)) break;
    if (p == 1) {
      edges.eb(1, 1);
      continue;
    }
    vc<int> now = states[p];
    for (auto&& [nxt, convert]: next_states(now)) {
      // 今の成分数、消える成分数
      int a = 0, b = 0;
      FOR(v, N) if (now[v] == v) {
        ++a;
        if (convert[v] == -1) ++b;
      }
      // 消える成分があってよいのは、終状態にいくときのみ
      if (b >= 2) continue;
      if (b == 1) {
        if (MAX(nxt) != -1) continue;
        edges.eb(p, 1);
        continue;
      }
      u64 h = hash_vector<int>(nxt);
      if (merge_reverse) { chmin(h, hash_vector<int>(reverse_state(nxt))); }
      if (!MP.count(h)) { MP[h] = len(states), states.eb(nxt); }
      edges.eb(p, MP[h]);
    }
  }
  return {states, edges};
}

// 0, 1 ：空の列、領域の手前、後ろ
// 多角形（空洞なし）をひとつ作る。
// 状態：-1 が選んでいない。0,1,2,3 等は同じ成分には同じ値が入る。
// [states, edges]
pair<vvc<int>, vc<pair<int, int>>> polygon_dp_graph(int N) {
  static HashMap<int, 20, true> MP;
  MP.reset();
  vvc<int> states;
  vc<pair<int, int>> edges;

  states.eb(vc<int>(N, -1));
  states.eb(vc<int>(N, -1));
  MP[hash_vector<int>(states[0])] = 0;

  int p = -1;
  while (1) {
    if (++p == len(states)) break;
    if (p == 1) {
      edges.eb(1, 1);
      continue;
    }
    vc<int> now = states[p];
    for (auto&& [nxt, convert]: next_states(now)) {
      // 今の成分数、消える成分数
      int a = 0, b = 0;
      FOR(v, N) if (now[v] == v) {
        ++a;
        if (convert[v] == -1) ++b;
      }
      // 消える成分があってよいのは、終状態にいくときのみ
      if (b >= 2) continue;
      if (b == 1) {
        if (MAX(nxt) != -1) continue;
        edges.eb(p, 1);
        continue;
      }
      bool ok = [&](vc<int>& now, vc<int>& nxt, vc<int>& convert) -> bool {
        // 頂点のみで接するのはダメ
        FOR(i, N - 1) {
          bool a1 = now[i] != -1, a2 = now[i + 1] != -1;
          bool b1 = nxt[i] != -1, b2 = nxt[i + 1] != -1;
          if (a1 && !a2 && !b1 && b2) return false;
          if (!a1 && a2 && b1 && !b2) return false;
        }
        // empty region を閉じることと、異なる連結成分がマージされることが同値
        int close = 0;
        int after = 0;
        vc<bool> is_new(N, 1);
        FOR(i, N) if (convert[i] != -1) is_new[convert[i]] = 0;
        FOR(i, N) if (nxt[i] == i && !is_new[i])++ after;
        vc<int> I;
        FOR(i, N) if (now[i] != -1) I.eb(i);
        FOR(k, len(I) - 1) {
          int i = I[k], j = I[k + 1];
          if (j == i + 1) continue;
          bool cl = 1;
          FOR(p, i + 1, j) if (nxt[p] == -1) cl = 0;
          if (cl) close++;
        }
        return a - close == after;
      }(now, nxt, convert);
      if (!ok) continue;
      u64 h = hash_vector<int>(nxt);
      int idx = MP.index(h);
      if (!MP.used[idx]) {
        MP.used[idx] = 1, MP.IDS.eb(idx), MP.key[idx] = h,
        MP.val[idx] = len(states);
        states.eb(nxt);
      }
      edges.eb(p, MP.val[idx]);
    }
  }
  return {states, edges};
}

} // namespace connected_dp_squares
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