洛谷P4191 [CTSC2010]性能优化(再探FFT)

洛谷P4191 [CTSC2010]性能优化(再探FFT)

题目大意

求循环卷积快速幂，保证 \(n\) 的所有质因子小于 10，且 \(n+1\) 是质数

解题思路

做这题要更深入的理解 FFT

其实正常我们做的 FFT 就是循环卷积，只不过是因为 \(\bmod 2^n\)，我们把模数故意比项数略高一些导致没有循环的效果

一般来讲我们有

\[f * g = \sum_{r=0,i,j}[i + j \bmod k = r]f_i *g_j \]

其中 k 是单位根的底数

那么在这题里我们就直接把项数设成 n 做循环卷积即可

具体来说，我们要先将每个 \(w_n^i\) 带进去求点值，然后直接快速幂最后再代入负单位根即可

代入的过程中要进行奇怪的分治，比如分成 3 项等

\[A(w_n^i) = \sum_{j=0}^{k-1}w_n^{ij}A(w_{\frac nk}^{i}) \]

代码

#include <queue>
#include <vector>
#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#define MP make_pair
#define ll long long
#define fi first
#define se second
using namespace std;

template <typename T>
void read(T &x) {
    x = 0; bool f = 0;
    char c = getchar();
    for (;!isdigit(c);c=getchar()) if (c=='-') f=1;
    for (;isdigit(c);c=getchar()) x=x*10+(c^48);
    if (f) x=-x;
}

template<typename F>
inline void write(F x, char ed = '\n')
{
	static short st[30];short tp=0;
	if(x<0) putchar('-'),x=-x;
	do st[++tp]=x%10,x/=10; while(x);
	while(tp) putchar('0'|st[tp--]);
	putchar(ed);
}

template <typename T>
inline void Mx(T &x, T y) { x < y && (x = y); }

template <typename T>
inline void Mn(T &x, T y) { x > y && (x = y); }

int P;
ll fpw(ll x, ll mi) {
	ll res = 1;
	for (; mi; mi >>= 1, x = x * x % P)
		if (mi & 1) res = res * x % P;
	return res;
}

const int N = 600500;
int prime[] = {0, 2, 3, 5, 7};
int st[N], tot, G, n, C, cnt;
int getG(int n) {
	for (int i = 2;i <= n; i++) {
		bool fl = 1;
		if ((n % 2 == 0) && (fpw(i, n / 2) == 1)) fl = 0;
		if ((n % 3 == 0) && (fpw(i, n / 3) == 1)) fl = 0;
		if ((n % 5 == 0) && (fpw(i, n / 5) == 1)) fl = 0;
		if ((n % 7 == 0) && (fpw(i, n / 7) == 1)) fl = 0;
		if (fl) return i;
	}
	return 0;
}

void NTT(ll *f, int deg, int d) {
	if (deg == 1) return; int t = deg / st[d];
	ll g[st[d]][t];
	for (int i = 0;i < deg; i++) g[i % st[d]][i / st[d]] = f[i];
	for (int i = 0;i < st[d]; i++) NTT(g[i], t, d + 1);
	ll wn = fpw(G, n / deg), ww = 1;
	for (int i = 0;i < deg; ww = ww * wn % P, i++) {
		f[i] = 0; int k = i % t; ll w = 1; 
		for (int j = 0;j < st[d]; j++, w = w * ww % P) 
			f[i] = (f[i] + w * g[j][k]) % P;
	}
}

ll a[N], b[N];
int main() {
	read(n), read(C); int x = n; P = n + 1;
	for (int i = 0;i < n; i++) read(a[i]);
	for (int j = 0;j < n; j++) read(b[j]);
	for (int i = 1;i <= 4; i++)
		while (x % prime[i] == 0) x /= prime[i], st[++cnt] = prime[i];
	G = getG(n); NTT(a, n, 1), NTT(b, n, 1);
	while (C) {
		if (C & 1) {
			for (int i = 0;i < n; i++) a[i] = a[i]* b[i] % P;
		}
		for (int i = 0;i < n; i++) b[i] = b[i] * b[i] % P;
		C >>= 1;
	}
	NTT(a, n, 1), reverse(a + 1, a + n);
	for (int i = 0;i < n; i++) write(a[i] * n % P);
	return 0;
}