結果

問題 No.569 3 x N グリッドのパスの数
コンテスト
ユーザー pekempey
提出日時 2017-09-16 18:27:38
言語 C++14
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 11 ms / 2,000 ms
コード長 5,393 bytes
コンパイル時間 1,794 ms
コンパイル使用メモリ 107,732 KB
実行使用メモリ 5,248 KB
最終ジャッジ日時 2024-11-08 01:36:18
合計ジャッジ時間 3,950 ms
ジャッジサーバーID
（参考情報） 		judge5 / judge1
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ファイルパターン 結果
other AC * 60
権限があれば一括ダウンロードができます

ソースコード

diff # 

#include <iostream>
#include <algorithm>
#include <vector>
#include <cmath>
#include <array>

using namespace std;

constexpr int mod = 1e9 + 7;

struct modint {
  int n;
  modint(int n = 0) : n(n) {}
};

modint operator+(modint a, modint b) { return modint((a.n += b.n) >= mod ? a.n - mod : a.n); }
modint operator-(modint a, modint b) { return modint((a.n -= b.n) < 0 ? a.n + mod : a.n); }
modint operator*(modint a, modint b) { return modint(1LL * a.n * b.n % mod); }
bool operator<(modint a, modint b) { return a.n < b.n; }
modint &operator+=(modint &a, modint b) { return a = a + b; }
modint &operator-=(modint &a, modint b) { return a = a - b; }
modint &operator*=(modint &a, modint b) { return a = a * b; }

modint modinv(modint n) {
  if (n.n == 1) return 1;
  return modinv(mod % n.n) * (mod - mod / n.n);
}
modint operator/(modint a, modint b) { return a * modinv(b); }

modint simpath(int n) {
  constexpr int H = 4;
  auto setval = [&](vector<int> &d, array<int, H> &a, int k, int v) {
    for (int i = 0; i < d.size(); i++) {
      if (d[i] == k) a[i] = v;
    }
  };

  auto getval = [&](vector<int> &d, array<int, H> &a, int k) -> int {
    for (int i = 0; i < d.size(); i++) {
      if (d[i] == k) return a[i];
    }
    return k;
  };

  vector<pair<int, int>> es;
  for (int x = 0; x < n; x++) {
    for (int y = 0; y < H; y++) {
      if (y + 1 < H) es.emplace_back(x * H + y, x * H + y + 1);
      if (x + 1 < n) es.emplace_back(x * H + y, x * H + y + H);
    }
  }
  vector<int> deg(n * H);
  for (auto e : es) {
    deg[e.first]++;
    deg[e.second]++;
  }

  vector<pair<array<int, H>, modint>> dp, val;
  vector<int> fr = { 0 };
  dp.push_back(make_pair(array<int, H> {}, 1));

  vector<int> fr1;

  for (auto e : es) {
    deg[e.first]--;
    deg[e.second]--;
    fr1.clear();
    for (int u : fr) if (deg[u] > 0) fr1.push_back(u);
    fr1.push_back(e.second);
    sort(fr1.begin(), fr1.end());
    fr1.erase(unique(fr1.begin(), fr1.end()), fr1.end());

    val.clear();

    for (auto kv : dp) {
      auto mate = kv.first;
      array<int, H> mate1 = {};
      for (int i = 0; i < fr1.size(); i++) mate1[i] = fr1[i];
      for (int i = 0; i < fr.size(); i++) setval(fr1, mate1, fr[i], mate[i]);
      val.emplace_back(mate1, kv.second);
      int mateu = getval(fr, mate, e.first);
      int matev = getval(fr, mate, e.second);
      if (mateu == e.second && matev == e.first) continue;
      if (mateu == -1 || matev == -1) continue;
      setval(fr1, mate1, e.first, -1);
      setval(fr1, mate1, e.second, -1);
      setval(fr1, mate1, mateu, matev);
      setval(fr1, mate1, matev, mateu);
      val.emplace_back(mate1, kv.second);
    }

    sort(val.begin(), val.end());
    int i = 0;
    dp.clear();
    while (i < val.size()) {
      int j = i;
      modint sum = 0;
      while (i < val.size() && val[i].first == val[j].first) {
        sum += val[i].second;
        i++;
      }
      bool ok = true;
      bool has0 = false;
      for (int u : fr1) {
        int v = getval(fr1, val[j].first, u);
        if (v == 0) has0 = true;
        if (v != 0 && v != -1 && u != v && deg[v] == 0) ok = false;
      }
      if (ok && has0) dp.emplace_back(val[j].first, sum);
    }
    swap(fr, fr1);
  }

  for (auto kv : dp) {
    auto mate = kv.first;
    if (getval(fr, mate, H * n - 1) == 0) {
      return kv.second;
    }
  }
  // don't come here
  abort();
}

// ref: https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm 
vector<modint> berlekamp_massey(vector<modint> s) {
  const int N = s.size();
  vector<modint> C(N);
  vector<modint> B(N);
  C[0] = 1;
  B[0] = 1;
  int L = 0;
  int m = 1;
  modint b = 1;
  for (int n = 0; n < N; n++) {
    modint d = s[n];
    for (int i = 1; i <= L; i++) d += C[i] * s[n - i];
    if (d.n == 0) {
      m++;
    } else if (2 * L <= n) {
      auto T = C;
      for (int i = 0; i + m < N; i++) C[i + m] -= B[i] * (d / b);
      L = n + 1 - L;
      B = T;
      b = d;
      m = 1;
    } else {
      for (int i = 0; i + m < N; i++) C[i + m] -= B[i] * (d / b);
      m++;
    }
  }
  C.resize(L + 1);
  reverse(C.begin(), C.end());
  return C;
}

vector<modint> poly_mod(vector<modint> a, const vector<modint> &m) {
  const int n = m.size();
  for (int i = a.size() - 1; i >= m.size(); i--) {
    for (int j = 0; j < m.size(); j++) {
      a[i - n + j] += a[i] * m[j];
    }
  }
  a.resize(m.size());
  return a;
}

// a*b mod m
vector<modint> poly_mul(const vector<modint> &a, const vector<modint> &b, const vector<modint> &m) {
  vector<modint> ret(a.size() + b.size() - 1);
  for (int i = 0; i < a.size(); i++) {
    for (int j = 0; j < b.size(); j++) {
      ret[i + j] += a[i] * b[j];
    }
  }
  return poly_mod(ret, m);
}

// x^n mod m
vector<modint> nth_power(long long n, const vector<modint> &m) {
  vector<modint> ret(1);
  vector<modint> x(2);
  ret[0] = x[1] = 1;
  while (n > 0) {
    if (n & 1) ret = poly_mul(ret, x, m);
    x = poly_mul(x, x, m);
    n /= 2;
  }
  return poly_mod(ret, m);
}

int main() {
  vector<modint> a(30);
  for (int i = 0; i < a.size(); i++) {
    a[i] = simpath(i + 1);
  }

  vector<modint> m = berlekamp_massey(a);
  m.pop_back();
  for (int i = 0; i < m.size(); i++) {
    m[i] *= mod - 1;
  }

  long long n;
  cin >> n;
  auto x = nth_power(n, m);

  modint ans;
  for (int i = 0; i < x.size(); i++) {
    ans += x[i] * a[i];
  }
  cout << ans.n << endl;
}
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