[洛谷P5050]【模板】多项式多点求值

题目大意：给你一个$n$次多项式$f(x)$，以及$m$个$x_i$，对于$i\in[1,m]$，求$f(x_i)$

题解：多项式多点求值

令$g(x)=\prod\limits_{i=1}^m(x-x_i)$，求出$R(x)$使得$f(x)=Q(x)\times g(x)+R(x)$。因为当$x=x_i$时，$g(x)=0$，即$f(x)=R(x)$，$f(x)$是$n$次的，$R(x)$是$m-1$次的，似乎可以使得问题缩小了

考虑分治，现在区间为$[l,r]$，令$g_L(x)=\prod\limits_{i=l}^{mid}(x-x_i)$，$g_R(x)=\prod\limits_{i=mid}^r(x-x_i)$，所以$R_L(x)=f(x)\bmod g_L(x)$，$R_R(x)=f(x)\bmod g_R(x)$。最后当$l=r$时，第$i$个的值就是当前$R(x)$的常数项。

那$g(x)$怎么算呢，分治$FFT$，可以先把每个的$g(x)$求出来，用$vector$保存一下就行了

卡点：不知道为什么，用$C++$会$MLE$，$C++11$就过了，有可能是$vector$初始化部分出锅了

UPDATE（2019-8-1）：除法的时候没有初始化，用的是求逆的初始化，若要使用$\dfrac 32n$优化，需要修改，新的多点求值可看[SOJ #559]鲲

 

C++ Code：

#include <cstdio>
#include <algorithm>
#include <vector>
const int mod = 998244353, G = 3;

namespace Math {
	inline int pw(int base, int p) {
		static int res;
		for (res = 1; p; p >>= 1, base = static_cast<long long> (base) * base % mod) if (p & 1) res = static_cast<long long> (res) * base % mod;
		return res;
	}
	inline int inv(int x) { return pw(x, mod - 2); }
}
inline void reduce(int &x) { x += x >> 31 & mod; }

#define maxn 65536
int a[maxn], ans[maxn];
namespace Poly {
#define N maxn
	int rev[N], lim, s, ilim;
	int Wn[N + 1];
	inline void clear(register int *l, const int *r) {
		if (l >= r) return ;
		while (l != r) *l++ = 0;
	}

	inline void init(const int n) {
		s = -1, lim = 1; while (lim < n) lim <<= 1, ++s; ilim = Math::inv(lim);
		for (register int i = 0; i < lim; ++i) rev[i] = rev[i >> 1] >> 1 | (i & 1) << s;
		const int t = Math::pw(G, (mod - 1) / lim);
		*Wn = 1; for (register int *i = Wn; i != Wn + lim; ++i) *(i + 1) = static_cast<long long> (*i) * t % mod;
	}
	inline void NTT(int *A, const int op = 1) {
		static int Wt[N];
		for (register int i = 1; i < lim; ++i) if (i < rev[i]) std::swap(A[i], A[rev[i]]);
		for (register int mid = 1; mid < lim; mid <<= 1) {
			const int t = lim / mid >> 1;
			*Wt = Wn[op ? 0 : lim];
			for (register int *i = Wt, W = 0; i != Wt + mid; ++i, W += t) *i = Wn[op ? W : lim - W];
			for (register int i = 0; i < lim; i += mid << 1) {
				for (register int j = 0; j < mid; ++j) {
					const int X = A[i + j], Y = static_cast<long long> (Wt[j]) * A[i + j + mid] % mod;
					reduce(A[i + j] += Y - mod), reduce(A[i + j + mid] = X - Y);
				}
			}
		}
		if (!op) for (register int *i = A; i != A + lim; ++i) *i = static_cast<long long> (*i) * ilim % mod;
	}

	std::vector<int> P[N << 1], S[N << 1];
	int C[N], D[N];
	void DC_NTT(int rt, int l, int r) {
		if (l == r) { P[rt] = {a[l], 1}; return ; }
		int mid = l + r >> 1;
		DC_NTT(rt << 1, l, mid), DC_NTT(rt << 1 | 1, mid + 1, r);
		int L = rt << 1, R = rt << 1 | 1;
		int n = P[L].size(), m = P[R].size();
		init(n + m - 1);
		std::copy(P[L].begin(), P[L].end(), C); clear(C + n, C + lim);
		std::copy(P[R].begin(), P[R].end(), D); clear(D + m, D + lim);
		NTT(C), NTT(D);
		for (int i = 0; i < lim; ++i) C[i] = static_cast<long long> (C[i]) * D[i] % mod;
		NTT(C, 0);
		P[rt].assign(C, C + n + m - 1);
	}

	int E[N];
	void INV(int *A, int *B, int n) {
		if (n == 1) {
			*B = Math::inv(*A);
			return ;
		}
		INV(A, B, n + 1 >> 1);
		init(n + n - 1);
		std::copy(A, A + n, E); clear(E + n, E + lim);
		clear(B + (n + 1 >> 1), B + lim);
		NTT(B), NTT(E);
		for (int i = 0; i < lim; ++i) B[i] = (2 + mod - static_cast<long long> (B[i]) * E[i] % mod) * B[i] % mod;
		NTT(B, 0); clear(B + n, B + lim);
	}
	int F[N];
	void DIV(int A, int n, int B, int m) {
		const int len = n - m + 1;
		init(len << 1);
		std::reverse_copy(S[A].begin(), S[A].end(), C); clear(C + len, C + lim);
		std::reverse_copy(P[B].begin(), P[B].end(), D); clear(D + len, D + lim);
		clear(F, F + lim);
		INV(D, F, len);
		NTT(C), NTT(F);
		for (int i = 0; i < lim; ++i) F[i] = static_cast<long long> (F[i]) * C[i] % mod;
		NTT(F, 0);
		clear(F + len, F + lim);
	}
	void __DIVMOD(int res, int A, int n, int B, int m) {
		if (n < m) {
			S[res].assign(S[A].begin(), S[A].end());
			return ;
		}
		DIV(A, n, B, m);
		init(n);
		std::reverse_copy(F, F + n - m + 1, C); clear(C + n - m + 1, C + lim);
		std::copy(P[B].begin(), P[B].end(), D); clear(D + m, D + lim);
		NTT(C), NTT(D);
		for (int i = 0; i < lim; ++i) C[i] = static_cast<long long> (C[i]) * D[i] % mod;
		NTT(C, 0);
		for (int i = 0; i < m - 1; ++i) reduce(C[i] = S[A][i] - C[i]);
		S[res].assign(C, C + m - 1);
	}
	void DIVMOD(int res, int A) {
		int n = S[A].size(), m = P[res].size();
		__DIVMOD(res, A, n, res, m);
	}

	void solve(int rt, int l, int r) {
		if (l == r) {
			ans[l] = S[rt][0];
			return ;
		}
		int mid = l + r >> 1;
		DIVMOD(rt << 1, rt), DIVMOD(rt << 1 | 1, rt);
		solve(rt << 1, l, mid), solve(rt << 1 | 1, mid + 1, r);
	}

	void work(int *f, int n, int m) {
		DC_NTT(1, 1, m);
		S[0].assign(f, f + n);
		DIVMOD(1, 0);
		solve(1, 1, m);
	}
#undef N
}

int n, m;
int f[maxn];
int main() {
	scanf("%d%d", &n, &m); if (!m) return 0; ++n;
	for (int i = 0; i < n; ++i) scanf("%d", f + i);
	for (int i = 1; i <= m; ++i) scanf("%d", a + i), reduce(a[i] = -a[i]);
	Poly::work(f, n, m);
	for (int i = 1; i <= m; ++i) printf("%d\n", ans[i]);
	return 0;
}