cp-library

C++ Library for Competitive Programming

View the Project on GitHub emthrm/cp-library

:warning: 動的計画法/オフライン・オンライン変換
(test/dynamic_programming/convert_online_dp_to_offline_dp.test.cpp)

Depends on

Code

/*
 * @title 動的計画法/オフライン・オンライン変換
 *
 * verification-helper: IGNORE
 * verification-helper: PROBLEM https://atcoder.jp/contests/abc213/tasks/abc213_h
 */

#include <algorithm>
#include <functional>
#include <iostream>
#include <iterator>
#include <vector>

#include "emthrm/dynamic_programming/convert_online_dp_to_offline_dp.hpp"
#include "emthrm/math/convolution/number_theoretic_transform.hpp"
#include "emthrm/math/modint.hpp"

int main() {
  constexpr int MOD = 998244353;
  using ModInt = emthrm::MInt<MOD>;
  int n, m, t;
  std::cin >> n >> m >> t;
  std::vector<int> a(m), b(m);
  std::vector p(m, std::vector(t + 1, ModInt(0)));
  for (int i = 0; i < m; ++i) {
    std::cin >> a[i] >> b[i];
    --a[i]; --b[i];
    for (int j = 1; j <= t; ++j) {
      std::cin >> p[i][j];
    }
  }
  std::vector dp(n, std::vector(t + 1, ModInt(0)));
  dp[0][0] = 1;
  const std::function<void(int, int, int)> induce =
      [m, &a, &b, &p, &dp](const int l, const int mid, const int r) -> void {
        static emthrm::NumberTheoreticTransform<MOD> ntt;
        for (int id = 0; id < m; ++id) {
          std::vector<ModInt> dp_id(mid - l), p_id(r - l);
          std::copy(std::next(dp[a[id]].begin(), l),
                    std::next(dp[a[id]].begin(), mid), dp_id.begin());
          std::copy(p[id].begin(), std::next(p[id].begin(), r - l),
                    p_id.begin());
          std::vector<ModInt> c = ntt.convolution(dp_id, p_id);
          for (int i = mid; i < r; ++i) {
            dp[b[id]][i] += c[i - l];
          }
          std::copy(std::next(dp[b[id]].begin(), l),
                    std::next(dp[b[id]].begin(), mid), dp_id.begin());
          c = ntt.convolution(dp_id, p_id);
          for (int i = mid; i < r; ++i) {
            dp[a[id]][i] += c[i - l];
          }
        }
      };
  emthrm::convert_online_dp_to_offline_dp(t + 1, induce);
  std::cout << dp[0][t] << '\n';
  return 0;
}
#line 1 "test/dynamic_programming/convert_online_dp_to_offline_dp.test.cpp"
/*
 * @title 動的計画法/オフライン・オンライン変換
 *
 * verification-helper: IGNORE
 * verification-helper: PROBLEM https://atcoder.jp/contests/abc213/tasks/abc213_h
 */

#include <algorithm>
#include <functional>
#include <iostream>
#include <iterator>
#include <vector>

#line 1 "include/emthrm/dynamic_programming/convert_online_dp_to_offline_dp.hpp"



#line 5 "include/emthrm/dynamic_programming/convert_online_dp_to_offline_dp.hpp"
#include <numeric>

namespace emthrm {

void convert_online_dp_to_offline_dp(
    const int n, const std::function<void(int, int, int)> induce) {
  const auto solve = [induce](auto solve, const int l, const int r) -> void {
    if (l + 1 == r) {
      // dp(l) <- dp(l) ･ b_l
      return;
    }
    const int m = std::midpoint(l, r);
    solve(solve, l, m);
    induce(l, m, r);
    solve(solve, m, r);
  };
  if (n > 0) [[likely]] solve(solve, 0, n);
}

}  // namespace emthrm


#line 1 "include/emthrm/math/convolution/number_theoretic_transform.hpp"



#if __has_include(<atcoder/convolution>)
# include <atcoder/convolution>
# include <atcoder/modint>
#else
#line 9 "include/emthrm/math/convolution/number_theoretic_transform.hpp"
# include <bit>
# include <cassert>
#line 12 "include/emthrm/math/convolution/number_theoretic_transform.hpp"
# include <map>
# include <utility>
#endif
#line 16 "include/emthrm/math/convolution/number_theoretic_transform.hpp"

#line 1 "include/emthrm/math/modint.hpp"



#ifndef ARBITRARY_MODINT
# include <cassert>
#endif
#include <compare>
#line 9 "include/emthrm/math/modint.hpp"
// #include <numeric>
#include <utility>
#line 12 "include/emthrm/math/modint.hpp"

namespace emthrm {

#ifndef ARBITRARY_MODINT
template <unsigned int M>
struct MInt {
  unsigned int v;

  constexpr MInt() : v(0) {}
  constexpr MInt(const long long x) : v(x >= 0 ? x % M : x % M + M) {}
  static constexpr MInt raw(const int x) {
    MInt x_;
    x_.v = x;
    return x_;
  }

  static constexpr int get_mod() { return M; }
  static constexpr void set_mod(const int divisor) {
    assert(std::cmp_equal(divisor, M));
  }

  static void init(const int x) {
    inv<true>(x);
    fact(x);
    fact_inv(x);
  }

  template <bool MEMOIZES = false>
  static MInt inv(const int n) {
    // assert(0 <= n && n < M && std::gcd(n, M) == 1);
    static std::vector<MInt> inverse{0, 1};
    const int prev = inverse.size();
    if (n < prev) return inverse[n];
    if constexpr (MEMOIZES) {
      // "n!" and "M" must be disjoint.
      inverse.resize(n + 1);
      for (int i = prev; i <= n; ++i) {
        inverse[i] = -inverse[M % i] * raw(M / i);
      }
      return inverse[n];
    }
    int u = 1, v = 0;
    for (unsigned int a = n, b = M; b;) {
      const unsigned int q = a / b;
      std::swap(a -= q * b, b);
      std::swap(u -= q * v, v);
    }
    return u;
  }

  static MInt fact(const int n) {
    static std::vector<MInt> factorial{1};
    if (const int prev = factorial.size(); n >= prev) {
      factorial.resize(n + 1);
      for (int i = prev; i <= n; ++i) {
        factorial[i] = factorial[i - 1] * i;
      }
    }
    return factorial[n];
  }

  static MInt fact_inv(const int n) {
    static std::vector<MInt> f_inv{1};
    if (const int prev = f_inv.size(); n >= prev) {
      f_inv.resize(n + 1);
      f_inv[n] = inv(fact(n).v);
      for (int i = n; i > prev; --i) {
        f_inv[i - 1] = f_inv[i] * i;
      }
    }
    return f_inv[n];
  }

  static MInt nCk(const int n, const int k) {
    if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
    return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) :
                                  fact_inv(n - k) * fact_inv(k));
  }
  static MInt nPk(const int n, const int k) {
    return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k);
  }
  static MInt nHk(const int n, const int k) {
    return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k));
  }

  static MInt large_nCk(long long n, const int k) {
    if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
    inv<true>(k);
    MInt res = 1;
    for (int i = 1; i <= k; ++i) {
      res *= inv(i) * n--;
    }
    return res;
  }

  constexpr MInt pow(long long exponent) const {
    MInt res = 1, tmp = *this;
    for (; exponent > 0; exponent >>= 1) {
      if (exponent & 1) res *= tmp;
      tmp *= tmp;
    }
    return res;
  }

  constexpr MInt& operator+=(const MInt& x) {
    if ((v += x.v) >= M) v -= M;
    return *this;
  }
  constexpr MInt& operator-=(const MInt& x) {
    if ((v += M - x.v) >= M) v -= M;
    return *this;
  }
  constexpr MInt& operator*=(const MInt& x) {
    v = (unsigned long long){v} * x.v % M;
    return *this;
  }
  MInt& operator/=(const MInt& x) { return *this *= inv(x.v); }

  constexpr auto operator<=>(const MInt& x) const = default;

  constexpr MInt& operator++() {
    if (++v == M) [[unlikely]] v = 0;
    return *this;
  }
  constexpr MInt operator++(int) {
    const MInt res = *this;
    ++*this;
    return res;
  }
  constexpr MInt& operator--() {
    v = (v == 0 ? M - 1 : v - 1);
    return *this;
  }
  constexpr MInt operator--(int) {
    const MInt res = *this;
    --*this;
    return res;
  }

  constexpr MInt operator+() const { return *this; }
  constexpr MInt operator-() const { return raw(v ? M - v : 0); }

  constexpr MInt operator+(const MInt& x) const { return MInt(*this) += x; }
  constexpr MInt operator-(const MInt& x) const { return MInt(*this) -= x; }
  constexpr MInt operator*(const MInt& x) const { return MInt(*this) *= x; }
  MInt operator/(const MInt& x) const { return MInt(*this) /= x; }

  friend std::ostream& operator<<(std::ostream& os, const MInt& x) {
    return os << x.v;
  }
  friend std::istream& operator>>(std::istream& is, MInt& x) {
    long long v;
    is >> v;
    x = MInt(v);
    return is;
  }
};
#else  // ARBITRARY_MODINT
template <int ID>
struct MInt {
  unsigned int v;

  constexpr MInt() : v(0) {}
  MInt(const long long x) : v(x >= 0 ? x % mod() : x % mod() + mod()) {}
  static constexpr MInt raw(const int x) {
    MInt x_;
    x_.v = x;
    return x_;
  }

  static int get_mod() { return mod(); }
  static void set_mod(const unsigned int divisor) { mod() = divisor; }

  static void init(const int x) {
    inv<true>(x);
    fact(x);
    fact_inv(x);
  }

  template <bool MEMOIZES = false>
  static MInt inv(const int n) {
    // assert(0 <= n && n < mod() && std::gcd(x, mod()) == 1);
    static std::vector<MInt> inverse{0, 1};
    const int prev = inverse.size();
    if (n < prev) return inverse[n];
    if constexpr (MEMOIZES) {
      // "n!" and "M" must be disjoint.
      inverse.resize(n + 1);
      for (int i = prev; i <= n; ++i) {
        inverse[i] = -inverse[mod() % i] * raw(mod() / i);
      }
      return inverse[n];
    }
    int u = 1, v = 0;
    for (unsigned int a = n, b = mod(); b;) {
      const unsigned int q = a / b;
      std::swap(a -= q * b, b);
      std::swap(u -= q * v, v);
    }
    return u;
  }

  static MInt fact(const int n) {
    static std::vector<MInt> factorial{1};
    if (const int prev = factorial.size(); n >= prev) {
      factorial.resize(n + 1);
      for (int i = prev; i <= n; ++i) {
        factorial[i] = factorial[i - 1] * i;
      }
    }
    return factorial[n];
  }

  static MInt fact_inv(const int n) {
    static std::vector<MInt> f_inv{1};
    if (const int prev = f_inv.size(); n >= prev) {
      f_inv.resize(n + 1);
      f_inv[n] = inv(fact(n).v);
      for (int i = n; i > prev; --i) {
        f_inv[i - 1] = f_inv[i] * i;
      }
    }
    return f_inv[n];
  }

  static MInt nCk(const int n, const int k) {
    if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
    return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) :
                                  fact_inv(n - k) * fact_inv(k));
  }
  static MInt nPk(const int n, const int k) {
    return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k);
  }
  static MInt nHk(const int n, const int k) {
    return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k));
  }

  static MInt large_nCk(long long n, const int k) {
    if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
    inv<true>(k);
    MInt res = 1;
    for (int i = 1; i <= k; ++i) {
      res *= inv(i) * n--;
    }
    return res;
  }

  MInt pow(long long exponent) const {
    MInt res = 1, tmp = *this;
    for (; exponent > 0; exponent >>= 1) {
      if (exponent & 1) res *= tmp;
      tmp *= tmp;
    }
    return res;
  }

  MInt& operator+=(const MInt& x) {
    if ((v += x.v) >= mod()) v -= mod();
    return *this;
  }
  MInt& operator-=(const MInt& x) {
    if ((v += mod() - x.v) >= mod()) v -= mod();
    return *this;
  }
  MInt& operator*=(const MInt& x) {
    v = (unsigned long long){v} * x.v % mod();
    return *this;
    }
  MInt& operator/=(const MInt& x) { return *this *= inv(x.v); }

  auto operator<=>(const MInt& x) const = default;

  MInt& operator++() {
    if (++v == mod()) [[unlikely]] v = 0;
    return *this;
  }
  MInt operator++(int) {
    const MInt res = *this;
    ++*this;
    return res;
  }
  MInt& operator--() {
    v = (v == 0 ? mod() - 1 : v - 1);
    return *this;
  }
  MInt operator--(int) {
    const MInt res = *this;
    --*this;
    return res;
  }

  MInt operator+() const { return *this; }
  MInt operator-() const { return raw(v ? mod() - v : 0); }

  MInt operator+(const MInt& x) const { return MInt(*this) += x; }
  MInt operator-(const MInt& x) const { return MInt(*this) -= x; }
  MInt operator*(const MInt& x) const { return MInt(*this) *= x; }
  MInt operator/(const MInt& x) const { return MInt(*this) /= x; }

  friend std::ostream& operator<<(std::ostream& os, const MInt& x) {
    return os << x.v;
  }
  friend std::istream& operator>>(std::istream& is, MInt& x) {
    long long v;
    is >> v;
    x = MInt(v);
    return is;
  }

 private:
  static unsigned int& mod() {
    static unsigned int divisor = 0;
    return divisor;
  }
};
#endif  // ARBITRARY_MODINT

}  // namespace emthrm


#line 18 "include/emthrm/math/convolution/number_theoretic_transform.hpp"

namespace emthrm {

#if __has_include(<atcoder/convolution>)

template <unsigned int T>
struct NumberTheoreticTransform {
  using ModInt = MInt<T>;

  NumberTheoreticTransform() = default;

  template <typename U>
  std::vector<ModInt> dft(const std::vector<U>& a);

  void idft(std::vector<ModInt>* a);

  template <typename U>
  std::vector<ModInt> convolution(
      const std::vector<U>& a, const std::vector<U>& b) const {
    const int a_size = a.size(), b_size = b.size();
    std::vector<atcoder::static_modint<T>> c(a_size), d(b_size);
    for (int i = 0; i < a_size; ++i) {
      c[i] = atcoder::static_modint<T>::raw(ModInt(a[i]).v);
    }
    for (int i = 0; i < b_size; ++i) {
      d[i] = atcoder::static_modint<T>::raw(ModInt(b[i]).v);
    }
    c = atcoder::convolution(c, d);
    const int c_size = c.size();
    std::vector<ModInt> res(c_size);
    for (int i = 0; i < c_size; ++i) {
      res[i] = ModInt::raw(c[i].val());
    }
    return res;
  }
};

#else  // __has_include(<atcoder/convolution>)

template <unsigned int T>
struct NumberTheoreticTransform {
  using ModInt = MInt<T>;

  NumberTheoreticTransform()
      : n_max(1 << init().first), root(ModInt::raw(init().second)) {}

  template <typename U>
  std::vector<ModInt> dft(const std::vector<U>& a) {
    std::vector<ModInt> b(std::bit_ceil(a.size()), 0);
    std::ranges::copy(a, b.begin());
    calc(&b);
    return b;
  }

  void idft(std::vector<ModInt>* a) {
    assert(std::has_single_bit(a->size()));
    calc(a);
    std::reverse(std::next(a->begin()), a->end());
    const int n = a->size();
    const ModInt inv_n = ModInt::inv(n);
    for (int i = 0; i < n; ++i) {
      (*a)[i] *= inv_n;
    }
  }

  template <typename U>
  std::vector<ModInt> convolution(
      const std::vector<U>& a, const std::vector<U>& b) {
    const int a_size = a.size(), b_size = b.size();
    const int c_size = a_size + b_size - 1;
    if (std::min(a_size, b_size) <= 60) {
      std::vector<ModInt> c(c_size, 0);
      if (a_size > b_size) {
        for (int i = 0; i < a_size; ++i) {
          for (int j = 0; j < b_size; ++j) {
            c[i + j] += ModInt(a[i]) * b[j];
          }
        }
      } else {
        for (int j = 0; j < b_size; ++j) {
          for (int i = 0; i < a_size; ++i) {
            c[i + j] += ModInt(b[j]) * a[i];
          }
        }
      }
      return c;
    }
    const int n = std::bit_ceil(static_cast<unsigned int>(c_size));
    std::vector<ModInt> c(n, 0), d(n, 0);
    std::ranges::copy(a, c.begin());
    calc(&c);
    std::ranges::copy(b, d.begin());
    calc(&d);
    for (int i = 0; i < n; ++i) {
      c[i] *= d[i];
    }
    idft(&c);
    c.resize(c_size);
    return c;
  }

 private:
  static std::pair<int, int> init() {
    static const std::map<int, std::pair<int, int>> primes{
        {16957441, {14, 102066830}},  // 329
        {17006593, {15, 608991743}},  // 26
        {19529729, {17, 927947839}},  // 770
        {167772161, {25, 243}},  // 3
        {469762049, {26, 2187}},  // 3
        {645922817, {23, 680782677}},  // 3
        {897581057, {23, 126991183}},  // 3
        {924844033, {21, 480100938}},  // 5
        {935329793, {22, 945616399}},  // 3
        {943718401, {22, 39032610}},  // 7
        {950009857, {21, 912960248}},  // 7
        {962592769, {21, 762567211}},  // 7
        {975175681, {21, 973754139}},  // 17
        {976224257, {20, 168477898}},  // 3
        {985661441, {22, 157780640}},  // 3
        {998244353, {23, 15311432}},  // 3
        {1004535809, {21, 840453100}},  // 3
        {1007681537, {20, 283888334}},  // 3
        {1012924417, {21, 428116421}},  // 5
        {1045430273, {20, 328125745}},  // 3
        {1051721729, {20, 234350985}},  // 6
        {1053818881, {20, 309635616}},  // 7
        {1224736769, {24, 304180829}}};  // 3
    return primes.at(T);
  }

  const int n_max;
  const ModInt root;

  std::vector<int> butterfly{0};
  std::vector<std::vector<ModInt>> omega{{1}};

  void calc(std::vector<ModInt>* a) {
    const int n = a->size(), prev_n = butterfly.size();
    if (n > prev_n) {
      assert(n <= n_max);
      butterfly.resize(n);
      const int prev_lg = omega.size(), lg = std::countr_zero(a->size());
      for (int i = 1; i < prev_n; ++i) {
        butterfly[i] <<= lg - prev_lg;
      }
      for (int i = prev_n; i < n; ++i) {
        butterfly[i] = (butterfly[i >> 1] >> 1) | ((i & 1) << (lg - 1));
      }
      omega.resize(lg);
      for (int i = prev_lg; i < lg; ++i) {
        omega[i].resize(1 << i);
        const ModInt tmp = root.pow((ModInt::get_mod() - 1) >> (i + 1));
        for (int j = 0; j < (1 << (i - 1)); ++j) {
          omega[i][j << 1] = omega[i - 1][j];
          omega[i][(j << 1) + 1] = omega[i - 1][j] * tmp;
        }
      }
    }
    const int shift =
        std::countr_zero(butterfly.size()) - std::countr_zero(a->size());
    for (int i = 0; i < n; ++i) {
      const int j = butterfly[i] >> shift;
      if (i < j) std::swap((*a)[i], (*a)[j]);
    }
    for (int block = 1, den = 0; block < n; block <<= 1, ++den) {
      for (int i = 0; i < n; i += (block << 1)) {
        for (int j = 0; j < block; ++j) {
          const ModInt tmp = (*a)[i + j + block] * omega[den][j];
          (*a)[i + j + block] = (*a)[i + j] - tmp;
          (*a)[i + j] += tmp;
        }
      }
    }
  }
};

#endif  // __has_include(<atcoder/convolution>)

}  // namespace emthrm


#line 17 "test/dynamic_programming/convert_online_dp_to_offline_dp.test.cpp"

int main() {
  constexpr int MOD = 998244353;
  using ModInt = emthrm::MInt<MOD>;
  int n, m, t;
  std::cin >> n >> m >> t;
  std::vector<int> a(m), b(m);
  std::vector p(m, std::vector(t + 1, ModInt(0)));
  for (int i = 0; i < m; ++i) {
    std::cin >> a[i] >> b[i];
    --a[i]; --b[i];
    for (int j = 1; j <= t; ++j) {
      std::cin >> p[i][j];
    }
  }
  std::vector dp(n, std::vector(t + 1, ModInt(0)));
  dp[0][0] = 1;
  const std::function<void(int, int, int)> induce =
      [m, &a, &b, &p, &dp](const int l, const int mid, const int r) -> void {
        static emthrm::NumberTheoreticTransform<MOD> ntt;
        for (int id = 0; id < m; ++id) {
          std::vector<ModInt> dp_id(mid - l), p_id(r - l);
          std::copy(std::next(dp[a[id]].begin(), l),
                    std::next(dp[a[id]].begin(), mid), dp_id.begin());
          std::copy(p[id].begin(), std::next(p[id].begin(), r - l),
                    p_id.begin());
          std::vector<ModInt> c = ntt.convolution(dp_id, p_id);
          for (int i = mid; i < r; ++i) {
            dp[b[id]][i] += c[i - l];
          }
          std::copy(std::next(dp[b[id]].begin(), l),
                    std::next(dp[b[id]].begin(), mid), dp_id.begin());
          c = ntt.convolution(dp_id, p_id);
          for (int i = mid; i < r; ++i) {
            dp[a[id]][i] += c[i - l];
          }
        }
      };
  emthrm::convert_online_dp_to_offline_dp(t + 1, induce);
  std::cout << dp[0][t] << '\n';
  return 0;
}
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