test/verify/yosupo-shift-of-sampling-points-of-polynomial.test.cpp

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• Last update: 2026-08-10 14:34:25+00:00
• Problem: https://judge.yosupo.jp/problem/shift_of_sampling_points_of_polynomial

Depends on

• Lagrange Polynomial(多項式補間, 値) (math/combinatorics/lagrange-polynomial-3.hpp)
• Montgomery Mod Int (math/combinatorics/montgomery-mod-int.hpp)
• Number Theoretic Transform Friendly ModInt (math/fft/number-theoretic-transform-friendly-mod-int.hpp)

Code

// clang-format off
// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/shift_of_sampling_points_of_polynomial
// clang-format on

#include <iostream>
#include <vector>

#include "../../math/combinatorics/lagrange-polynomial-3.hpp"
#include "../../math/combinatorics/montgomery-mod-int.hpp"
#include "../../math/fft/number-theoretic-transform-friendly-mod-int.hpp"

using namespace std;

using mint = modint998244353;

template <typename Sequence>
void read_sequence(Sequence& values) {
  for (auto& value : values) cin >> value;
}

template <typename Sequence>
void print_sequence(const Sequence& values) {
  for (int i = 0; i < static_cast<int>(values.size()); i++) {
    if (i > 0) cout << " ";
    cout << values[i];
  }
  cout << "\n";
}

int main() {
  int N, T, M;
  cin >> N >> T >> M;
  vector<mint> ys(N);
  read_sequence(ys);
  NumberTheoreticTransformFriendlyModInt<mint> v;
  auto multiply = [&](const vector<mint>& a, const vector<mint>& b) {
    return v.multiply(a, b);
  };
  print_sequence(lagrange_polynomial(ys, M, T, multiply));
}

#line 1 "test/verify/yosupo-shift-of-sampling-points-of-polynomial.test.cpp"
// clang-format off
// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/shift_of_sampling_points_of_polynomial
// clang-format on

#include <iostream>
#include <vector>

#line 2 "math/combinatorics/lagrange-polynomial-3.hpp"

#include <algorithm>
#include <cstdint>
#include <iterator>
#line 7 "math/combinatorics/lagrange-polynomial-3.hpp"

/**
 * @brief Lagrange Polynomial(多項式補間, 値)
 */
template <typename Mint, typename F>
std::vector<Mint> lagrange_polynomial(const std::vector<Mint>& y,
                                      std::int64_t T, const int& m,
                                      const F& multiply) {
  int k = (int)y.size() - 1;
  T %= Mint::mod();
  if (T <= k) {
    std::vector<Mint> ret(m);
    int ptr = 0;
    for (std::int64_t i = T; i <= k and ptr < m; i++) {
      ret[ptr++] = y[i];
    }
    if (k + 1 < T + m) {
      auto suf = lagrange_polynomial(y, k + 1, m - ptr, multiply);
      for (int i = k + 1; i < T + m; i++) {
        ret[ptr++] = suf[i - (k + 1)];
      }
    }
    return ret;
  }
  if (T + m > Mint::mod()) {
    auto pref = lagrange_polynomial(y, T, Mint::mod() - T, multiply);
    auto suf = lagrange_polynomial(y, 0, m - pref.size(), multiply);
    std::copy(std::begin(suf), std::end(suf), std::back_inserter(pref));
    return pref;
  }

  std::vector<Mint> finv(k + 1, 1), d(k + 1);
  for (int i = 2; i <= k; i++) finv[k] *= i;
  finv[k] = Mint(1) / finv[k];
  for (int i = k; i >= 1; i--) finv[i - 1] = finv[i] * i;
  for (int i = 0; i <= k; i++) {
    d[i] = finv[i] * finv[k - i] * y[i];
    if ((k - i) & 1) d[i] = -d[i];
  }

  std::vector<Mint> h(m + k);
  for (int i = 0; i < m + k; i++) {
    h[i] = Mint(1) / (T - k + i);
  }

  auto dh = multiply(d, h);

  std::vector<Mint> ret(m);
  Mint cur = T;
  for (int i = 1; i <= k; i++) cur *= T - i;
  for (int i = 0; i < m; i++) {
    ret[i] = cur * dh[k + i];
    cur *= T + i + 1;
    cur *= h[i];
  }
  return ret;
}
#line 2 "math/combinatorics/montgomery-mod-int.hpp"

#line 5 "math/combinatorics/montgomery-mod-int.hpp"

template <std::uint32_t mod_, bool fast = false>
struct MontgomeryModInt {
 private:
  using mint = MontgomeryModInt;
  using i32 = std::int32_t;
  using i64 = std::int64_t;
  using u32 = std::uint32_t;
  using u64 = std::uint64_t;

  static constexpr u32 get_r() {
    u32 ret = mod_;
    for (i32 i = 0; i < 4; i++) ret *= 2 - mod_ * ret;
    return ret;
  }

  static constexpr u32 r = get_r();

  static constexpr u32 n2 = -u64(mod_) % mod_;

  static_assert(r * mod_ == 1, "invalid, r * mod != 1");
  static_assert(mod_ < (1 << 30), "invalid, mod >= 2 ^ 30");
  static_assert((mod_ & 1) == 1, "invalid, mod % 2 == 0");

  u32 x;

 public:
  MontgomeryModInt() : x{} {}

  MontgomeryModInt(const i64& a)
      : x(reduce(u64(fast ? a : (a % mod() + mod())) * n2)) {}

  static constexpr u32 reduce(const u64& b) {
    return u32(b >> 32) + mod() - u32((u64(u32(b) * r) * mod()) >> 32);
  }

  mint& operator+=(const mint& p) {
    if (i32(x += p.x - 2 * mod()) < 0) x += 2 * mod();
    return *this;
  }

  mint& operator-=(const mint& p) {
    if (i32(x -= p.x) < 0) x += 2 * mod();
    return *this;
  }

  mint& operator*=(const mint& p) {
    x = reduce(u64(x) * p.x);
    return *this;
  }

  mint& operator/=(const mint& p) {
    *this *= p.inv();
    return *this;
  }

  mint operator-() const { return mint() - *this; }

  mint operator+(const mint& p) const { return mint(*this) += p; }

  mint operator-(const mint& p) const { return mint(*this) -= p; }

  mint operator*(const mint& p) const { return mint(*this) *= p; }

  mint operator/(const mint& p) const { return mint(*this) /= p; }

  bool operator==(const mint& p) const {
    return (x >= mod() ? x - mod() : x) == (p.x >= mod() ? p.x - mod() : p.x);
  }

  bool operator!=(const mint& p) const {
    return (x >= mod() ? x - mod() : x) != (p.x >= mod() ? p.x - mod() : p.x);
  }

  u32 val() const {
    u32 ret = reduce(x);
    return ret >= mod() ? ret - mod() : ret;
  }

  mint pow(u64 n) const {
    mint ret(1), mul(*this);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }

  mint inv() const { return pow(mod() - 2); }

  friend std::ostream& operator<<(std::ostream& os, const mint& p) {
    return os << p.val();
  }

  friend std::istream& operator>>(std::istream& is, mint& a) {
    i64 t;
    is >> t;
    a = mint(t);
    return is;
  }

  static constexpr u32 mod() { return mod_; }
};

template <std::uint32_t mod>
using modint = MontgomeryModInt<mod>;
using modint998244353 = modint<998244353>;
using modint1000000007 = modint<1000000007>;
#line 2 "math/fft/number-theoretic-transform-friendly-mod-int.hpp"

#include <cassert>
#line 5 "math/fft/number-theoretic-transform-friendly-mod-int.hpp"

/**
 * @brief Number Theoretic Transform Friendly ModInt
 */
template <typename Mint>
struct NumberTheoreticTransformFriendlyModInt {
  static std::vector<Mint> roots, iroots, rate2, irate2, rate3, irate3;
  static int max_base;

  NumberTheoreticTransformFriendlyModInt() = default;

  static void init() {
    if (roots.empty()) {
      const unsigned mod = Mint::mod();
      assert(mod >= 3 && mod % 2 == 1);
      auto tmp = mod - 1;
      max_base = 0;
      while (tmp % 2 == 0) tmp >>= 1, max_base++;
      Mint root = 2;
      while (root.pow((mod - 1) >> 1) == 1) {
        root += 1;
      }
      assert(root.pow(mod - 1) == 1);

      roots.resize(max_base + 1);
      iroots.resize(max_base + 1);
      rate2.resize(max_base + 1);
      irate2.resize(max_base + 1);
      rate3.resize(max_base + 1);
      irate3.resize(max_base + 1);

      roots[max_base] = root.pow((mod - 1) >> max_base);
      iroots[max_base] = Mint(1) / roots[max_base];
      for (int i = max_base - 1; i >= 0; i--) {
        roots[i] = roots[i + 1] * roots[i + 1];
        iroots[i] = iroots[i + 1] * iroots[i + 1];
      }
      {
        Mint prod = 1, iprod = 1;
        for (int i = 0; i <= max_base - 2; i++) {
          rate2[i] = roots[i + 2] * prod;
          irate2[i] = iroots[i + 2] * iprod;
          prod *= iroots[i + 2];
          iprod *= roots[i + 2];
        }
      }
      {
        Mint prod = 1, iprod = 1;
        for (int i = 0; i <= max_base - 3; i++) {
          rate3[i] = roots[i + 3] * prod;
          irate3[i] = iroots[i + 3] * iprod;
          prod *= iroots[i + 3];
          iprod *= roots[i + 3];
        }
      }
    }
  }

  static void ntt(std::vector<Mint>& a) {
    init();
    const int n = (int)a.size();
    assert((n & (n - 1)) == 0);
    int h = __builtin_ctz(n);
    assert(h <= max_base);
    int len = 0;
    Mint imag = roots[2];
    if (h & 1) {
      int p = 1 << (h - 1);
      for (int i = 0; i < p; i++) {
        auto r = a[i + p];
        a[i + p] = a[i] - r;
        a[i] += r;
      }
      len++;
    }
    for (; len + 1 < h; len += 2) {
      int p = 1 << (h - len - 2);
      {  // s = 0
        for (int i = 0; i < p; i++) {
          auto a0 = a[i];
          auto a1 = a[i + p];
          auto a2 = a[i + 2 * p];
          auto a3 = a[i + 3 * p];
          auto a1na3imag = (a1 - a3) * imag;
          auto a0a2 = a0 + a2;
          auto a1a3 = a1 + a3;
          auto a0na2 = a0 - a2;
          a[i] = a0a2 + a1a3;
          a[i + 1 * p] = a0a2 - a1a3;
          a[i + 2 * p] = a0na2 + a1na3imag;
          a[i + 3 * p] = a0na2 - a1na3imag;
        }
      }
      Mint rot = rate3[0];
      for (int s = 1; s < (1 << len); s++) {
        int offset = s << (h - len);
        Mint rot2 = rot * rot;
        Mint rot3 = rot2 * rot;
        for (int i = 0; i < p; i++) {
          auto a0 = a[i + offset];
          auto a1 = a[i + offset + p] * rot;
          auto a2 = a[i + offset + 2 * p] * rot2;
          auto a3 = a[i + offset + 3 * p] * rot3;
          auto a1na3imag = (a1 - a3) * imag;
          auto a0a2 = a0 + a2;
          auto a1a3 = a1 + a3;
          auto a0na2 = a0 - a2;
          a[i + offset] = a0a2 + a1a3;
          a[i + offset + 1 * p] = a0a2 - a1a3;
          a[i + offset + 2 * p] = a0na2 + a1na3imag;
          a[i + offset + 3 * p] = a0na2 - a1na3imag;
        }
        rot *= rate3[__builtin_ctz(~s)];
      }
    }
  }

  static void intt(std::vector<Mint>& a, bool f = true) {
    init();
    const int n = (int)a.size();
    assert((n & (n - 1)) == 0);
    int h = __builtin_ctz(n);
    assert(h <= max_base);
    int len = h;
    Mint iimag = iroots[2];
    for (; len > 1; len -= 2) {
      int p = 1 << (h - len);
      {  // s = 0
        for (int i = 0; i < p; i++) {
          auto a0 = a[i];
          auto a1 = a[i + 1 * p];
          auto a2 = a[i + 2 * p];
          auto a3 = a[i + 3 * p];
          auto a2na3iimag = (a2 - a3) * iimag;
          auto a0na1 = a0 - a1;
          auto a0a1 = a0 + a1;
          auto a2a3 = a2 + a3;
          a[i] = a0a1 + a2a3;
          a[i + 1 * p] = (a0na1 + a2na3iimag);
          a[i + 2 * p] = (a0a1 - a2a3);
          a[i + 3 * p] = (a0na1 - a2na3iimag);
        }
      }
      Mint irot = irate3[0];
      for (int s = 1; s < (1 << (len - 2)); s++) {
        int offset = s << (h - len + 2);
        Mint irot2 = irot * irot;
        Mint irot3 = irot2 * irot;
        for (int i = 0; i < p; i++) {
          auto a0 = a[i + offset];
          auto a1 = a[i + offset + 1 * p];
          auto a2 = a[i + offset + 2 * p];
          auto a3 = a[i + offset + 3 * p];
          auto a2na3iimag = (a2 - a3) * iimag;
          auto a0na1 = a0 - a1;
          auto a0a1 = a0 + a1;
          auto a2a3 = a2 + a3;
          a[i + offset] = a0a1 + a2a3;
          a[i + offset + 1 * p] = (a0na1 + a2na3iimag) * irot;
          a[i + offset + 2 * p] = (a0a1 - a2a3) * irot2;
          a[i + offset + 3 * p] = (a0na1 - a2na3iimag) * irot3;
        }
        irot *= irate3[__builtin_ctz(~s)];
      }
    }
    if (len >= 1) {
      int p = 1 << (h - 1);
      for (int i = 0; i < p; i++) {
        auto ajp = a[i] - a[i + p];
        a[i] += a[i + p];
        a[i + p] = ajp;
      }
    }
    if (f) {
      Mint inv_sz = Mint(1) / n;
      for (int i = 0; i < n; i++) a[i] *= inv_sz;
    }
  }

  /**
   * @brief Transpose of ntt()
   */
  static void transposed_ntt(std::vector<Mint>& a) {
    init();
    const int n = (int)a.size();
    assert((n & (n - 1)) == 0);
    const int h = __builtin_ctz(n);
    assert(h <= max_base);

    int len = h;
    const Mint imag = roots[2];
    while (len > 0) {
      if (len == 1) {
        const int p = 1 << (h - len);
        Mint rot = 1;
        for (int s = 0; s < (1 << (len - 1)); s++) {
          const int offset = s << (h - len + 1);
          for (int i = 0; i < p; i++) {
            const auto lhs = a[i + offset];
            const auto rhs = a[i + offset + p];
            a[i + offset] = lhs + rhs;
            a[i + offset + p] = (lhs - rhs) * rot;
          }
          rot *= rate2[__builtin_ctz(~s)];
        }
        len--;
      } else {
        const int p = 1 << (h - len);
        Mint rot = 1;
        for (int s = 0; s < (1 << (len - 2)); s++) {
          const int offset = s << (h - len + 2);
          const Mint rot2 = rot * rot;
          const Mint rot3 = rot2 * rot;
          for (int i = 0; i < p; i++) {
            const auto a0 = a[i + offset];
            const auto a1 = a[i + offset + p];
            const auto a2 = a[i + offset + 2 * p];
            const auto a3 = a[i + offset + 3 * p];
            const auto x = (a2 - a3) * imag;
            a[i + offset] = a0 + a1 + a2 + a3;
            a[i + offset + p] = (a0 - a1 + x) * rot;
            a[i + offset + 2 * p] = (a0 + a1 - a2 - a3) * rot2;
            a[i + offset + 3 * p] = (a0 - a1 - x) * rot3;
          }
          rot *= rate3[__builtin_ctz(~s)];
        }
        len -= 2;
      }
    }
  }

  /**
   * @brief Transpose of intt()
   */
  static void transposed_intt(std::vector<Mint>& a, bool f = true) {
    init();
    const int n = (int)a.size();
    assert((n & (n - 1)) == 0);
    const int h = __builtin_ctz(n);
    assert(h <= max_base);

    if (f) {
      const Mint inv_sz = Mint(1) / n;
      for (auto& value : a) value *= inv_sz;
    }

    int len = 0;
    const Mint iimag = iroots[2];
    while (len < h) {
      if (len == h - 1) {
        const int p = 1 << (h - len - 1);
        Mint irot = 1;
        for (int s = 0; s < (1 << len); s++) {
          const int offset = s << (h - len);
          for (int i = 0; i < p; i++) {
            const auto lhs = a[i + offset];
            const auto rhs = a[i + offset + p] * irot;
            a[i + offset] = lhs + rhs;
            a[i + offset + p] = lhs - rhs;
          }
          irot *= irate2[__builtin_ctz(~s)];
        }
        len++;
      } else {
        const int p = 1 << (h - len - 2);
        Mint irot = 1;
        for (int s = 0; s < (1 << len); s++) {
          const Mint irot2 = irot * irot;
          const Mint irot3 = irot2 * irot;
          const int offset = s << (h - len);
          for (int i = 0; i < p; i++) {
            const auto a0 = a[i + offset];
            const auto a1 = a[i + offset + p] * irot;
            const auto a2 = a[i + offset + 2 * p] * irot2;
            const auto a3 = a[i + offset + 3 * p] * irot3;
            const auto x = (a1 - a3) * iimag;
            a[i + offset] = a0 + a2 + a1 + a3;
            a[i + offset + p] = a0 + a2 - a1 - a3;
            a[i + offset + 2 * p] = a0 - a2 + x;
            a[i + offset + 3 * p] = a0 - a2 - x;
          }
          irot *= irate3[__builtin_ctz(~s)];
        }
        len += 2;
      }
    }
  }

  static std::vector<Mint> multiply(std::vector<Mint> a, std::vector<Mint> b) {
    int need = a.size() + b.size() - 1;
    int nbase = 1;
    while ((1 << nbase) < need) nbase++;
    int sz = 1 << nbase;
    a.resize(sz, 0);
    b.resize(sz, 0);
    ntt(a);
    ntt(b);
    Mint inv_sz = Mint(1) / sz;
    for (int i = 0; i < sz; i++) a[i] *= b[i] * inv_sz;
    intt(a, false);
    a.resize(need);
    return a;
  }
};

template <typename Mint>
std::vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::roots =
    std::vector<Mint>();
template <typename Mint>
std::vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::iroots =
    std::vector<Mint>();
template <typename Mint>
std::vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::rate2 =
    std::vector<Mint>();
template <typename Mint>
std::vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::irate2 =
    std::vector<Mint>();
template <typename Mint>
std::vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::rate3 =
    std::vector<Mint>();
template <typename Mint>
std::vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::irate3 =
    std::vector<Mint>();
template <typename Mint>
int NumberTheoreticTransformFriendlyModInt<Mint>::max_base = 0;
#line 11 "test/verify/yosupo-shift-of-sampling-points-of-polynomial.test.cpp"

using namespace std;

using mint = modint998244353;

template <typename Sequence>
void read_sequence(Sequence& values) {
  for (auto& value : values) cin >> value;
}

template <typename Sequence>
void print_sequence(const Sequence& values) {
  for (int i = 0; i < static_cast<int>(values.size()); i++) {
    if (i > 0) cout << " ";
    cout << values[i];
  }
  cout << "\n";
}

int main() {
  int N, T, M;
  cin >> N >> T >> M;
  vector<mint> ys(N);
  read_sequence(ys);
  NumberTheoreticTransformFriendlyModInt<mint> v;
  auto multiply = [&](const vector<mint>& a, const vector<mint>& b) {
    return v.multiply(a, b);
  };
  print_sequence(lagrange_polynomial(ys, M, T, multiply));
}

Test cases

Env	Name	Status	Elapsed	Memory
g++	N_1_00	AC	122 ms	10 MB
g++	c_0_00	AC	267 ms	21 MB
g++	example_00	AC	2 ms	4 MB
g++	example_01	AC	2 ms	3 MB
g++	max_random_00	AC	469 ms	34 MB
g++	max_random_01	AC	468 ms	34 MB
g++	max_random_02	AC	475 ms	34 MB
g++	max_random_03	AC	469 ms	34 MB
g++	medium_random_00	AC	11 ms	4 MB
g++	medium_random_01	AC	10 ms	4 MB
g++	medium_random_02	AC	10 ms	4 MB
g++	medium_random_03	AC	10 ms	4 MB
g++	small_random_00	AC	2 ms	4 MB
g++	small_random_01	AC	2 ms	4 MB
g++	small_random_02	AC	2 ms	3 MB
g++	small_random_03	AC	2 ms	4 MB
g++	type0_random_00	AC	115 ms	5 MB
g++	type0_random_01	AC	129 ms	5 MB
g++	type0_random_02	AC	18 ms	4 MB
g++	type0_random_03	AC	131 ms	5 MB
g++	type1_random_00	AC	416 ms	32 MB
g++	type1_random_01	AC	385 ms	31 MB
g++	type1_random_02	AC	36 ms	6 MB
g++	type1_random_03	AC	295 ms	22 MB
g++	type2_random_00	AC	395 ms	31 MB
g++	type2_random_01	AC	437 ms	32 MB
g++	type2_random_02	AC	134 ms	12 MB
g++	type2_random_03	AC	257 ms	20 MB
g++	type3_random_00	AC	284 ms	20 MB
g++	type3_random_01	AC	407 ms	31 MB
g++	type3_random_02	AC	166 ms	11 MB
g++	type3_random_03	AC	257 ms	20 MB
clang++	N_1_00	AC	138 ms	10 MB
clang++	c_0_00	AC	294 ms	21 MB
clang++	example_00	AC	2 ms	4 MB
clang++	example_01	AC	2 ms	4 MB
clang++	max_random_00	AC	515 ms	34 MB
clang++	max_random_01	AC	507 ms	34 MB
clang++	max_random_02	AC	506 ms	34 MB
clang++	max_random_03	AC	523 ms	34 MB
clang++	medium_random_00	AC	11 ms	4 MB
clang++	medium_random_01	AC	11 ms	4 MB
clang++	medium_random_02	AC	10 ms	4 MB
clang++	medium_random_03	AC	11 ms	4 MB
clang++	small_random_00	AC	2 ms	4 MB
clang++	small_random_01	AC	2 ms	4 MB
clang++	small_random_02	AC	2 ms	4 MB
clang++	small_random_03	AC	2 ms	4 MB
clang++	type0_random_00	AC	120 ms	5 MB
clang++	type0_random_01	AC	131 ms	5 MB
clang++	type0_random_02	AC	18 ms	4 MB
clang++	type0_random_03	AC	126 ms	5 MB
clang++	type1_random_00	AC	428 ms	32 MB
clang++	type1_random_01	AC	409 ms	31 MB
clang++	type1_random_02	AC	38 ms	6 MB
clang++	type1_random_03	AC	310 ms	22 MB
clang++	type2_random_00	AC	427 ms	31 MB
clang++	type2_random_01	AC	459 ms	32 MB
clang++	type2_random_02	AC	157 ms	12 MB
clang++	type2_random_03	AC	273 ms	20 MB
clang++	type3_random_00	AC	305 ms	20 MB
clang++	type3_random_01	AC	444 ms	31 MB
clang++	type3_random_02	AC	165 ms	11 MB
clang++	type3_random_03	AC	270 ms	20 MB

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