#define PROBLEM "https://yukicoder.me/problems/no/1938" #include #include #include using mint = atcoder::modint998244353; std::istream& operator>>(std::istream& in, mint &a) { long long e; in >> e; a = e; return in; } #include #include #include #include #include namespace suisen { template class inv_mods { public: inv_mods() {} inv_mods(int n) { ensure(n); } const mint& operator[](int i) const { ensure(i); return invs[i]; } static void ensure(int n) { int sz = invs.size(); if (sz < 2) invs = {0, 1}, sz = 2; if (sz < n + 1) { invs.resize(n + 1); for (int i = sz; i <= n; ++i) invs[i] = mint(mod - mod / i) * invs[mod % i]; } } private: static std::vector invs; static constexpr int mod = mint::mod(); }; template std::vector inv_mods::invs{}; } namespace suisen { template using convolution_t = std::vector (*)(const std::vector &, const std::vector &); template class FPS : public std::vector { public: using std::vector::vector; FPS(const std::initializer_list l) : std::vector::vector(l) {} FPS(const std::vector &v) : std::vector::vector(v) {} FPS(std::vector &&v) : std::vector::vector(std::move(v)) {} static void set_multiplication(convolution_t multiplication) { FPS::mult = multiplication; } inline const mint operator[](int n) const noexcept { return n <= deg() ? unsafe_get(n) : 0; } inline mint& operator[](int n) noexcept { ensure_deg(n); return unsafe_get(n); } inline int size() const noexcept { return std::vector::size(); } inline int deg() const noexcept { return size() - 1; } inline int normalize() { while (this->size() and this->back() == 0) this->pop_back(); return deg(); } inline FPS& pre_inplace(int max_deg) noexcept { if (deg() > max_deg) this->resize(std::max(0, max_deg + 1)); return *this; } inline FPS pre(int max_deg) const noexcept { return FPS(*this).pre_inplace(max_deg); } inline FPS operator+() const { return FPS(*this); } FPS operator-() const { FPS f(*this); for (auto &e : f) e = mint::mod() - e; return f; } inline FPS& operator++() { ++(*this)[0]; return *this; } inline FPS& operator--() { --(*this)[0]; return *this; } inline FPS& operator+=(const mint x) { (*this)[0] += x; return *this; } inline FPS& operator-=(const mint x) { (*this)[0] -= x; return *this; } FPS& operator+=(const FPS &g) { ensure_deg(g.deg()); for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) += g.unsafe_get(i); return *this; } FPS& operator-=(const FPS &g) { ensure_deg(g.deg()); for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) -= g.unsafe_get(i); return *this; } inline FPS& operator*=(const FPS &g) { return *this = FPS::mult(*this, g); } inline FPS& operator*=( FPS &&g) { return *this = FPS::mult(*this, g); } inline FPS& operator*=(const mint x) { for (auto &e : *this) e *= x; return *this; } FPS& operator/=(FPS &&g) { const int fd = normalize(), gd = g.normalize(); assert(gd >= 0); if (fd < gd) { this->clear(); return *this; } if (gd == 0) return *this *= g.unsafe_get(0).inv(); static constexpr int THRESHOLD_NAIVE_POLY_QUOTIENT = 256; if (gd <= THRESHOLD_NAIVE_POLY_QUOTIENT) { *this = std::move(naive_div_inplace(std::move(g), gd).first); return *this; } std::reverse(this->begin(), this->end()), std::reverse(g.begin(), g.end()); const int k = fd - gd; *this *= g.inv_inplace(k), this->resize(k + 1); std::reverse(this->begin(), this->end()); return *this; } FPS& operator%=(FPS &&g) { int fd = normalize(), gd = g.normalize(); assert(gd >= 0); if (fd < gd) return *this; if (gd == 0) { this->clear(); return *this; } static constexpr int THRESHOLD_NAIVE_REMAINDER = 256; if (gd <= THRESHOLD_NAIVE_REMAINDER) return naive_div_inplace(std::move(g), gd).second; *this -= g * (*this / g); return pre_inplace(gd - 1); } inline FPS& operator/=(const FPS &g) { return *this /= FPS(g); } inline FPS& operator%=(const FPS &g) { return *this %= FPS(g); } FPS& operator<<=(const int shamt) { this->insert(this->begin(), shamt, 0); return *this; } FPS& operator>>=(const int shamt) { if (shamt > size()) this->clear(); else this->erase(this->begin(), this->begin() + shamt); return *this; } inline FPS operator+(FPS &&g) const { return FPS(*this) += std::move(g); } inline FPS operator-(FPS &&g) const { return FPS(*this) -= std::move(g); } inline FPS operator*(FPS &&g) const { return FPS(*this) *= std::move(g); } inline FPS operator/(FPS &&g) const { return FPS(*this) /= std::move(g); } inline FPS operator%(FPS &&g) const { return FPS(*this) %= std::move(g); } inline FPS operator+(const FPS &g) const { return FPS(*this) += g; } inline FPS operator+(const mint x) const { return FPS(*this) += x; } inline FPS operator-(const FPS &g) const { return FPS(*this) -= g; } inline FPS operator-(const mint x) const { return FPS(*this) -= x; } inline FPS operator*(const FPS &g) const { return FPS(*this) *= g; } inline FPS operator*(const mint x) const { return FPS(*this) *= x; } inline FPS operator/(const FPS &g) const { return FPS(*this) /= g; } inline FPS operator%(const FPS &g) const { return FPS(*this) %= g; } inline friend FPS operator*(const mint x, const FPS &f) { return f * x; } inline friend FPS operator*(const mint x, FPS &&f) { return f *= x; } inline FPS operator<<(const int shamt) { return FPS(*this) <<= shamt; } inline FPS operator>>(const int shamt) { return FPS(*this) >>= shamt; } friend bool operator==(const FPS &f, const FPS &g) { int n = f.size(), m = g.size(); if (n < m) return g == f; for (int i = 0; i < m; ++i) if (f.unsafe_get(i) != g.unsafe_get(i)) return false; for (int i = m; i < n; ++i) if (f.unsafe_get(i) != 0) return false; return true; } FPS& diff_inplace() { if (this->size() == 0) return *this; for (int i = 1; i <= deg(); ++i) unsafe_get(i - 1) = unsafe_get(i) * i; this->pop_back(); return *this; } FPS& intg_inplace() { int d = deg(); ensure_deg(d + 1); for (int i = d; i >= 0; --i) unsafe_get(i + 1) = unsafe_get(i) * invs[i + 1]; unsafe_get(0) = 0; return *this; } FPS& inv_inplace(const int max_deg) { FPS res { unsafe_get(0).inv() }; for (int k = 1; k <= max_deg; k *= 2) { FPS tmp(this->pre(k * 2) * (res * res)); res *= 2, res -= tmp.pre_inplace(2 * k); } return *this = std::move(res), pre_inplace(max_deg); } FPS& log_inplace(const int max_deg) { FPS f_inv = inv(max_deg); diff_inplace(), *this *= f_inv, pre_inplace(max_deg - 1), intg_inplace(); return *this; } FPS& exp_inplace(const int max_deg) { FPS res {1}; for (int k = 1; k <= max_deg; k *= 2) res *= ++(pre(k * 2) - res.log(k * 2)), res.pre_inplace(k * 2); return *this = std::move(res), pre_inplace(max_deg); } FPS& pow_inplace(const long long k, const int max_deg) { int tlz = 0; while (tlz <= deg() and unsafe_get(tlz) == 0) ++tlz; if (tlz * k > max_deg) { this->clear(); return *this; } *this >>= tlz; mint base = (*this)[0]; *this *= base.inv(), log_inplace(max_deg), *this *= k, exp_inplace(max_deg), *this *= base.pow(k); return *this <<= tlz * k, pre_inplace(max_deg); } inline FPS diff() const { return FPS(*this).diff_inplace(); } inline FPS intg() const { return FPS(*this).intg_inplace(); } inline FPS inv(const int max_deg) const { return FPS(*this).inv_inplace(max_deg); } inline FPS log(const int max_deg) const { return FPS(*this).log_inplace(max_deg); } inline FPS exp(const int max_deg) const { return FPS(*this).exp_inplace(max_deg); } inline FPS pow(const long long k, const int max_deg) const { return FPS(*this).pow_inplace(k, max_deg); } private: static inline inv_mods invs; static convolution_t mult; inline void ensure_deg(int d) { if (deg() < d) this->resize(d + 1, 0); } inline const mint& unsafe_get(int i) const { return std::vector::operator[](i); } inline mint& unsafe_get(int i) { return std::vector::operator[](i); } std::pair naive_div_inplace(FPS &&g, const int gd) { const int k = deg() - gd; mint head_inv = g.unsafe_get(gd).inv(); FPS q(k + 1); for (int i = k; i >= 0; --i) { mint div = this->unsafe_get(i + gd) * head_inv; q.unsafe_get(i) = div; for (int j = 0; j <= gd; ++j) this->unsafe_get(i + j) -= div * g.unsafe_get(j); } return {q, pre_inplace(gd - 1)}; } }; template convolution_t FPS::mult = [](const auto &, const auto &) { std::cerr << "convolution function is not available." << std::endl; assert(false); return std::vector{}; }; } // namespace suisen template auto sqrt(suisen::FPS a) -> decltype(mint::mod(), suisen::FPS{}) { assert(false); } template auto log(suisen::FPS a) -> decltype(mint::mod(), suisen::FPS{}) { return a.log(a.deg()); } template auto exp(suisen::FPS a) -> decltype(mint::mod(), mint()) { return a.exp(a.deg()); } template auto pow(suisen::FPS a, T b) -> decltype(mint::mod(), mint()) { return a.pow(b, a.deg()); } template auto inv(suisen::FPS a) -> decltype(mint::mod(), suisen::FPS{}) { return a.inv(a.deg()); } namespace suisen { template std::vector multi_point_eval(const FPS &f, const std::vector &xs) { int m = xs.size(); int k = 1; while (k < m) k <<= 1; std::vector> seg(2 * k); for (int i = 0; i < m; ++i) seg[k + i] = FPS {-xs[i], 1}; for (int i = m; i < k; ++i) seg[k + i] = FPS {1}; for (int i = k - 1; i> 0; --i) seg[i] = seg[i * 2] * seg[i * 2 + 1]; seg[1] = f % seg[1]; for (int i = 2; i < k + m; ++i) seg[i] = seg[i / 2] % seg[i]; std::vector ys(m); for (int i = 0; i < m; ++i) ys[i] = seg[k + i][0]; return ys; } } // namespace suisen namespace suisen { /** * O(N(logN)^2) * return the vector p of length xs.size() s.t. p[i]=Π[j!=i](x[i]-x[j]) */ template std::vector product_of_differences(const std::vector& xs) { // f(x):=Π_i(x-x[i]) // => f'(x)=Σ_i Π[j!=i](x-x[j]) // => f'(x[i])=Π[j!=i](x[i]-x[j]) const int n = xs.size(); std::deque> dq; for (int i = 0; i < n; ++i) dq.push_back(FPS{ -xs[i], mint{ 1 } }); while (dq.size() >= 2) { auto f = std::move(dq.front()); dq.pop_front(); auto g = std::move(dq.front()); dq.pop_front(); dq.push_back(f * g); } auto f = std::move(dq.front()); f.diff_inplace(); return multi_point_eval(f, xs); } } // namespace suisen int main() { suisen::FPS::set_multiplication([](const auto &a, const auto &b) { return atcoder::convolution(a, b); }); int n; mint x; std::cin >> n >> x; std::vector xs(n), ys(n); for (int i = 0; i < n; ++i) { std::cin >> xs[i] >> ys[i]; } std::vector w = suisen::product_of_differences(xs); mint s = 0; for (int i = 0; i < n; ++i) { s += ys[i] / w[i]; } mint p = 1; for (int i = 0; i < n; ++i) { p *= x - xs[i]; } mint ans = 0; for (int i = 0; i < n; ++i) { if (x == xs[i]) { ans += n * ys[i] - s * w[i]; } else { ans += n * ys[i] * p / (w[i] * (x - xs[i])) - s * p / (x - xs[i]); } } std::cout << ans.val() << std::endl; return 0; }