Polynomial Taylor Shift が使えた!
問題概要
頂点数 の木が与えられる。各 に対して、次の問いに答えよ。
- 頂点から 個の頂点を選ぶ 通りの方法それぞれについて
- その 個の頂点をすべて含む連結な部分グラフのサイズとして考えられる最小値を求め
- それらの総和をとった値を 924844033 で割った余りを求めよ
制約
考えたこと
頂点集合の部分集合 をすべて含む最小の連結成分を と書くことにする。
まず、 を固定したときの解法を考える。主客転倒して、さらに各頂点ごとに着目することにする。頂点 に対して、 が を含まない方法は、
- を除去してできる各部分木から 1 つ選び
- その部分木から 個を選ぶ方法
となる。これを から引いた値について、各 についての和をとればよい。
具体的には、この木の 個ある各全方位木 を考えて
これで "Easy Problem" が解けた。ここまでは ABC-F くらいの内容だと思われる。
各 に対して
ここから多項式で考える。各全方位木 ( 個ある) に対して、次の多項式 を考える。
これを各 に対して総和をとったもの
について、 次の係数を求めればよいこととなる。ここで、二項定理により と変形できることから、
で定まる多項式 を考えると、求める多項式 は
と表せることが分かった。よって、Polynomial Taylor Shift で解ける。
コード
#include<bits/stdc++.h> using namespace std; // modint template<int MOD> struct Fp { // inner value long long val; // constructor constexpr Fp() noexcept : val(0) { } constexpr Fp(long long v) noexcept : val(v % MOD) { if (val < 0) val += MOD; } constexpr long long get() const noexcept { return val; } constexpr int get_mod() const noexcept { return MOD; } // arithmetic operators constexpr Fp operator - () const noexcept { return val ? MOD - val : 0; } constexpr Fp operator + (const Fp &r) const noexcept { return Fp(*this) += r; } constexpr Fp operator - (const Fp &r) const noexcept { return Fp(*this) -= r; } constexpr Fp operator * (const Fp &r) const noexcept { return Fp(*this) *= r; } constexpr Fp operator / (const Fp &r) const noexcept { return Fp(*this) /= r; } constexpr Fp& operator ++ () noexcept { ++val; if (val >= MOD) val -= MOD; return *this; } constexpr Fp& operator -- () noexcept { if (val == 0) val += MOD; --val; return *this; } constexpr Fp& operator += (const Fp &r) noexcept { val += r.val; if (val >= MOD) val -= MOD; return *this; } constexpr Fp& operator -= (const Fp &r) noexcept { val -= r.val; if (val < 0) val += MOD; return *this; } constexpr Fp& operator *= (const Fp &r) noexcept { val = val * r.val % MOD; return *this; } constexpr Fp& operator /= (const Fp &r) noexcept { long long a = r.val, b = MOD, u = 1, v = 0; while (b) { long long t = a / b; a -= t * b, swap(a, b); u -= t * v, swap(u, v); } val = val * u % MOD; if (val < 0) val += MOD; return *this; } constexpr Fp pow(long long n) const noexcept { Fp res(1), mul(*this); while (n > 0) { if (n & 1) res *= mul; mul *= mul; n >>= 1; } return res; } constexpr Fp inv() const noexcept { Fp res(1), div(*this); return res / div; } // other operators constexpr bool operator == (const Fp &r) const noexcept { return this->val == r.val; } constexpr bool operator != (const Fp &r) const noexcept { return this->val != r.val; } friend constexpr istream& operator >> (istream &is, Fp<MOD> &x) noexcept { is >> x.val; x.val %= MOD; if (x.val < 0) x.val += MOD; return is; } friend constexpr ostream& operator << (ostream &os, const Fp<MOD> &x) noexcept { return os << x.val; } friend constexpr Fp<MOD> modpow(const Fp<MOD> &r, long long n) noexcept { return r.pow(n); } friend constexpr Fp<MOD> modinv(const Fp<MOD> &r) noexcept { return r.inv(); } }; namespace NTT { long long modpow(long long a, long long n, int mod) { long long res = 1; while (n > 0) { if (n & 1) res = res * a % mod; a = a * a % mod; n >>= 1; } return res; } long long modinv(long long a, int mod) { long long b = mod, u = 1, v = 0; while (b) { long long t = a / b; a -= t * b, swap(a, b); u -= t * v, swap(u, v); } u %= mod; if (u < 0) u += mod; return u; } int calc_primitive_root(int mod) { if (mod == 2) return 1; if (mod == 167772161) return 3; if (mod == 469762049) return 3; if (mod == 754974721) return 11; if (mod == 998244353) return 3; int divs[20] = {}; divs[0] = 2; int cnt = 1; long long x = (mod - 1) / 2; while (x % 2 == 0) x /= 2; for (long long i = 3; i * i <= x; i += 2) { if (x % i == 0) { divs[cnt++] = i; while (x % i == 0) x /= i; } } if (x > 1) divs[cnt++] = x; for (int g = 2;; g++) { bool ok = true; for (int i = 0; i < cnt; i++) { if (modpow(g, (mod - 1) / divs[i], mod) == 1) { ok = false; break; } } if (ok) return g; } } int get_fft_size(int N, int M) { int size_a = 1, size_b = 1; while (size_a < N) size_a <<= 1; while (size_b < M) size_b <<= 1; return max(size_a, size_b) << 1; } // number-theoretic transform template<class mint> void trans(vector<mint> &v, bool inv = false) { if (v.empty()) return; int N = (int)v.size(); int MOD = v[0].get_mod(); int PR = calc_primitive_root(MOD); static bool first = true; static vector<long long> vbw(30), vibw(30); if (first) { first = false; for (int k = 0; k < 30; ++k) { vbw[k] = modpow(PR, (MOD - 1) >> (k + 1), MOD); vibw[k] = modinv(vbw[k], MOD); } } for (int i = 0, j = 1; j < N - 1; j++) { for (int k = N >> 1; k > (i ^= k); k >>= 1); if (i > j) swap(v[i], v[j]); } for (int k = 0, t = 2; t <= N; ++k, t <<= 1) { long long bw = vbw[k]; if (inv) bw = vibw[k]; for (int i = 0; i < N; i += t) { mint w = 1; for (int j = 0; j < t/2; ++j) { int j1 = i + j, j2 = i + j + t/2; mint c1 = v[j1], c2 = v[j2] * w; v[j1] = c1 + c2; v[j2] = c1 - c2; w *= bw; } } } if (inv) { long long invN = modinv(N, MOD); for (int i = 0; i < N; ++i) v[i] = v[i] * invN; } } // for garner static constexpr int MOD0 = 754974721; static constexpr int MOD1 = 167772161; static constexpr int MOD2 = 469762049; using mint0 = Fp<MOD0>; using mint1 = Fp<MOD1>; using mint2 = Fp<MOD2>; static const mint1 imod0 = 95869806; // modinv(MOD0, MOD1); static const mint2 imod1 = 104391568; // modinv(MOD1, MOD2); static const mint2 imod01 = 187290749; // imod1 / MOD0; // small case (T = mint, long long) template<class T> vector<T> naive_mul(const vector<T> &A, const vector<T> &B) { if (A.empty() || B.empty()) return {}; int N = (int)A.size(), M = (int)B.size(); vector<T> res(N + M - 1); for (int i = 0; i < N; ++i) for (int j = 0; j < M; ++j) res[i + j] += A[i] * B[j]; return res; } // mint template<class mint> vector<mint> mul(const vector<mint> &A, const vector<mint> &B) { if (A.empty() || B.empty()) return {}; int N = (int)A.size(), M = (int)B.size(); if (min(N, M) < 30) return naive_mul(A, B); int MOD = A[0].get_mod(); int size_fft = get_fft_size(N, M); if (MOD == 998244353) { vector<mint> a(size_fft), b(size_fft), c(size_fft); for (int i = 0; i < N; ++i) a[i] = A[i]; for (int i = 0; i < M; ++i) b[i] = B[i]; trans(a), trans(b); vector<mint> res(size_fft); for (int i = 0; i < size_fft; ++i) res[i] = a[i] * b[i]; trans(res, true); res.resize(N + M - 1); return res; } vector<mint0> a0(size_fft, 0), b0(size_fft, 0), c0(size_fft, 0); vector<mint1> a1(size_fft, 0), b1(size_fft, 0), c1(size_fft, 0); vector<mint2> a2(size_fft, 0), b2(size_fft, 0), c2(size_fft, 0); for (int i = 0; i < N; ++i) a0[i] = A[i].val, a1[i] = A[i].val, a2[i] = A[i].val; for (int i = 0; i < M; ++i) b0[i] = B[i].val, b1[i] = B[i].val, b2[i] = B[i].val; trans(a0), trans(a1), trans(a2), trans(b0), trans(b1), trans(b2); for (int i = 0; i < size_fft; ++i) { c0[i] = a0[i] * b0[i]; c1[i] = a1[i] * b1[i]; c2[i] = a2[i] * b2[i]; } trans(c0, true), trans(c1, true), trans(c2, true); static const mint mod0 = MOD0, mod01 = mod0 * MOD1; vector<mint> res(N + M - 1); for (int i = 0; i < N + M - 1; ++i) { int y0 = c0[i].val; int y1 = (imod0 * (c1[i] - y0)).val; int y2 = (imod01 * (c2[i] - y0) - imod1 * y1).val; res[i] = mod01 * y2 + mod0 * y1 + y0; } return res; } // long long vector<long long> mul_ll(const vector<long long> &A, const vector<long long> &B) { if (A.empty() || B.empty()) return {}; int N = (int)A.size(), M = (int)B.size(); if (min(N, M) < 30) return naive_mul(A, B); int size_fft = get_fft_size(N, M); vector<mint0> a0(size_fft, 0), b0(size_fft, 0), c0(size_fft, 0); vector<mint1> a1(size_fft, 0), b1(size_fft, 0), c1(size_fft, 0); vector<mint2> a2(size_fft, 0), b2(size_fft, 0), c2(size_fft, 0); for (int i = 0; i < N; ++i) a0[i] = A[i], a1[i] = A[i], a2[i] = A[i]; for (int i = 0; i < M; ++i) b0[i] = B[i], b1[i] = B[i], b2[i] = B[i]; trans(a0), trans(a1), trans(a2), trans(b0), trans(b1), trans(b2); for (int i = 0; i < size_fft; ++i) { c0[i] = a0[i] * b0[i]; c1[i] = a1[i] * b1[i]; c2[i] = a2[i] * b2[i]; } trans(c0, true), trans(c1, true), trans(c2, true); static const long long mod0 = MOD0, mod01 = mod0 * MOD1; vector<long long> res(N + M - 1); for (int i = 0; i < N + M - 1; ++i) { int y0 = c0[i].val; int y1 = (imod0 * (c1[i] - y0)).val; int y2 = (imod01 * (c2[i] - y0) - imod1 * y1).val; res[i] = mod01 * y2 + mod0 * y1 + y0; } return res; } }; // Polynomial template<typename mint> struct Poly : vector<mint> { using vector<mint>::vector; // constructor constexpr Poly(const vector<mint> &r) : vector<mint>(r) {} // core operator constexpr mint eval(const mint &v) { mint res = 0; for (int i = (int)this->size()-1; i >= 0; --i) { res *= v; res += (*this)[i]; } return res; } constexpr Poly& normalize() { while (!this->empty() && this->back() == 0) this->pop_back(); return *this; } // basic operator constexpr Poly operator - () const noexcept { Poly res = (*this); for (int i = 0; i < (int)res.size(); ++i) res[i] = -res[i]; return res; } constexpr Poly operator + (const mint &v) const { return Poly(*this) += v; } constexpr Poly operator + (const Poly &r) const { return Poly(*this) += r; } constexpr Poly operator - (const mint &v) const { return Poly(*this) -= v; } constexpr Poly operator - (const Poly &r) const { return Poly(*this) -= r; } constexpr Poly operator * (const mint &v) const { return Poly(*this) *= v; } constexpr Poly operator * (const Poly &r) const { return Poly(*this) *= r; } constexpr Poly operator / (const mint &v) const { return Poly(*this) /= v; } constexpr Poly operator / (const Poly &r) const { return Poly(*this) /= r; } constexpr Poly operator % (const Poly &r) const { return Poly(*this) %= r; } constexpr Poly operator << (int x) const { return Poly(*this) <<= x; } constexpr Poly operator >> (int x) const { return Poly(*this) >>= x; } constexpr Poly& operator += (const mint &v) { if (this->empty()) this->resize(1); (*this)[0] += v; return *this; } constexpr Poly& operator += (const Poly &r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < (int)r.size(); ++i) (*this)[i] += r[i]; return this->normalize(); } constexpr Poly& operator -= (const mint &v) { if (this->empty()) this->resize(1); (*this)[0] -= v; return *this; } constexpr Poly& operator -= (const Poly &r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < (int)r.size(); ++i) (*this)[i] -= r[i]; return this->normalize(); } constexpr Poly& operator *= (const mint &v) { for (int i = 0; i < (int)this->size(); ++i) (*this)[i] *= v; return *this; } constexpr Poly& operator *= (const Poly &r) { return *this = NTT::mul((*this), r); } constexpr Poly& operator <<= (int x) { Poly res(x, 0); res.insert(res.end(), begin(*this), end(*this)); return *this = res; } constexpr Poly& operator >>= (int x) { Poly res; res.insert(res.end(), begin(*this) + x, end(*this)); return *this = res; } // division constexpr Poly& operator /= (const mint &v) { assert(v != 0); mint iv = modinv(v); for (int i = 0; i < (int)this->size(); ++i) (*this)[i] *= iv; return *this; } constexpr Poly pre(int siz) const { return Poly(begin(*this), begin(*this) + min((int)this->size(), siz)); } constexpr Poly rev() const { Poly res = *this; reverse(begin(res), end(res)); return res; } constexpr Poly inv(int deg) const { assert((*this)[0] != 0); if (deg < 0) deg = (int)this->size(); Poly res({mint(1) / (*this)[0]}); for (int i = 1; i < deg; i <<= 1) { res = (res + res - res * res * pre(i << 1)).pre(i << 1); } res.resize(deg); return res; } constexpr Poly inv() const { return inv((int)this->size()); } constexpr Poly& operator /= (const Poly &r) { assert(!r.empty()); assert(r.back() != 0); this->normalize(); if (this->size() < r.size()) { this->clear(); return *this; } int need = (int)this->size() - (int)r.size() + 1; *this = (rev().pre(need) * r.rev().inv(need)).pre(need).rev(); return *this; } constexpr Poly& operator %= (const Poly &r) { assert(!r.empty()); assert(r.back() != 0); this->normalize(); Poly q = (*this) / r; return *this -= q * r; } }; // Binomial coefficient template<class T> struct BiCoef { vector<T> fact_, inv_, finv_; constexpr BiCoef() {} constexpr BiCoef(int n) noexcept : fact_(n, 1), inv_(n, 1), finv_(n, 1) { init(n); } constexpr void init(int n) noexcept { fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1); int MOD = fact_[0].get_mod(); for(int i = 2; i < n; i++){ fact_[i] = fact_[i-1] * i; inv_[i] = -inv_[MOD%i] * (MOD/i); finv_[i] = finv_[i-1] * inv_[i]; } } constexpr T com(int n, int k) const noexcept { if (n < k || n < 0 || k < 0) return 0; return fact_[n] * finv_[k] * finv_[n-k]; } constexpr T fact(int n) const noexcept { if (n < 0) return 0; return fact_[n]; } constexpr T inv(int n) const noexcept { if (n < 0) return 0; return inv_[n]; } constexpr T finv(int n) const noexcept { if (n < 0) return 0; return finv_[n]; } }; // Polynomial Taylor Shift // given: f(x), c // find: coefficients of f(x + c) template<class mint> Poly<mint> PolynomialTaylorShift(const Poly<mint> &f, long long c) { int N = (int)f.size() - 1; BiCoef<mint> bc(N + 1); // convolution Poly<mint> p(N + 1), q(N + 1); for (int i = 0; i <= N; ++i) { p[i] = f[i] * bc.fact(i); q[N - i] = mint(c).pow(i) * bc.finv(i); } Poly<mint> pq = p * q; // result Poly<mint> res(N + 1); for (int i = 0; i <= N; ++i) res[i] = pq[i + N] * bc.finv(i); return res; } void AGC_005_F() { const int MOD = 924844033; using mint = Fp<MOD>; int N; cin >> N; BiCoef<mint> bc(N + 1); vector<vector<int>> G(N); for (int i = 0; i < N-1; ++i) { int a, b; cin >> a >> b; --a, --b; G[a].push_back(b); G[b].push_back(a); } Poly<mint> f(N+1, 0); vector<int> si(N, 1); auto rec = [&](auto self, int v, int p = -1) -> void { for (auto ch : G[v]) { if (ch == p) continue; self(self, ch, v); ++f[si[ch]]; si[v] += si[ch]; } ++f[N - si[v]]; }; rec(rec, 0); Poly<mint> g = PolynomialTaylorShift(f, 1); for (int k = 1; k <= N; ++k) cout << bc.com(N, k) * N - g[k] << endl; } int main() { AGC_005_F(); }