Solution -「CF 623E」Transforming Sequence

题目

题意简述

  link.

  有一个 \(n\) 个元素的集合，你需要进行 \(m\) 次操作。每次操作选择集合的一个非空子集，要求该集合不是已选集合的并的子集。求操作的方案数，对 \(10^9+7\) 取模。

数据规模

  \(n\le3\times10^4\)。

\(\text{Solution}\)

  显然当 \(n<m\)，答案为 \(0\)，先特判掉。

  首先列一个 naive 的 DP 方程，令 \(f(i,j)\) 为前 \(i\) 次操作选出的集合并大小为 \(j\) 的方案数。易有：

\[f(i,j)=\sum_{k=1}^j2^{j-k}{n-(j-k)\choose k}f(i-1,j-k)
\]

  那么答案为 \(\sum_{i=1}^nf(m,i)\)。暴力 DP 是 \(\mathcal{O}(mn^2)\) 的。不过很显然这个式子是一个卷积的形式。把它变一下形：

\[f(i,j)=\frac{1}{(n-j)!}\sum_{k=1}^j\frac{1}{k!}\cdot2^{j-k}[n-(j-k)]!f(i-1,j-k)
\]

  忽略毒瘤的模数，每次转移 NTT 就可以了，复杂度 \(\mathcal{O}(mn\log n)\)。

  这个 \(f\) 不太好优化。我们换一种形式，令 \(g(i,j)\) 为前 \(i\) 次操作选出前 \(j\) 个元素的方案数。所以 \(g(i,j)=\frac{f(i,j)}{n\choose j}\)。那么对于 \(g(1)\)，有 \(g(1,i)=[0<i\le n]\)。来考虑两层状态的合并：

\[g(u+v,i)=\sum_{j=0}^ig(u,j)g(v,i-j){i\choose j}(2^j)^v
\]

  理解一下：把 \(u+v\) 次操作和 \(i\) 个选出的元素分拆成两步完成：第一步选 \(j\) 个元素，第二步选 \(i-j\) 个。由于要在 \(i\) 里拿出 \(j\) 个给第一步（或说拿出 \(i-j\) 给第二步），所以有系数 \(i\choose j\)。关键点在于 \((2^u)^j\)：在第一步中，我们已经选出了 \(j\) 个元素，所以在第二步的每次操作，都可以任意取 \(j\) 个元素的子集，故有系数 \((2^j)^v\)。

  向卷积形式靠拢：

\[g(u+v,i)=i!\sum_{j=0}^i\frac{2^{jv}g(u,j)}{j!}\cdot\frac{g(v,i-j)}{(i-j)!}
\]

  倍增处理 \(g\) 即可。复杂度 \(\mathcal{O}(n\log n\log m)\)。

代码

  这里用 MTT 实现任模 NTT。要特别留意每一步的取模嗷 qwq。

#include <cmath>
#include <cstdio>
#include <iostream>

typedef long long LL;

const int MAXN = 1 << 16, P = 1e9 + 7;
const double PI = acos ( -1 );
int n, m, len, lg, rev[MAXN + 5], fac[MAXN + 5], ifac[MAXN + 5], pwr[MAXN + 5];
int F[MAXN + 5], ans[MAXN + 5];

struct Complex {
	double x, y;
	Complex () {} Complex ( const double tx, const double ty ): x ( tx ), y ( ty ) {}
	inline Complex operator + ( const Complex t ) const { return Complex ( x + t.x, y + t.y ); }
	inline Complex operator - ( const Complex t ) const { return Complex ( x - t.x, y - t.y ); }
	inline Complex operator * ( const Complex t ) const { return Complex ( x * t.x - y * t.y, x * t.y + y * t.x ); }
	inline Complex operator / ( const double t ) const { return Complex ( x / t, y / t ); }
} omega[MAXN + 5];

inline void FFT ( const int n, Complex* A, const int tp ) {
	for ( int i = 0; i < n; ++ i ) if ( i < rev[i] ) std::swap ( A[i], A[rev[i]] );
	for ( int i = 2, stp = 1; i <= n; i <<= 1, stp <<= 1 ) {
		for ( int j = 0; j < n; j += i ) {
			for ( int k = 0; k < stp; ++ k ) {
				Complex w ( omega[n / stp * k].x, tp * omega[n / stp * k].y );
				Complex ev ( A[j + k] ), ov ( w * A[j + k + stp] );
				A[j + k] = ev + ov, A[j + k + stp] = ev - ov;
			}
		}
	}
	if ( ! ~ tp ) for ( int i = 0; i < n; ++ i ) A[i] = A[i] / n;
}

inline void initFFT ( const int lg ) {
	int n = 1 << lg;
	for ( int i = 0; i < n; ++ i ) rev[i] = ( rev[i >> 1] >> 1 ) | ( ( i & 1 ) << lg >> 1 );
	for ( int i = 1; i < n; i <<= 1 ) {
		for ( int k = 0; k < i; ++ k ) {
			omega[n / i * k] = Complex ( cos ( PI * k / i ), sin ( PI * k / i ) );
		}
	}
}

inline void polyConv ( const int n, const int* A, const int* B, int* C ) {
	static Complex highA[MAXN + 5], highB[MAXN + 5], lowA[MAXN + 5], lowB[MAXN + 5];
	for ( int i = 0; i < n; ++ i ) {
		lowA[i].x = A[i] & 0x7fff, highA[i].x = A[i] >> 15;
		lowB[i].x = B[i] & 0x7fff, highB[i].x = B[i] >> 15;
		lowA[i].y = highA[i].y = lowB[i].y = highB[i].y = 0;
	}
	FFT ( n, lowA, 1 ), FFT ( n, highA, 1 ), FFT ( n, lowB, 1 ), FFT ( n, highB, 1 );
	for ( int i = 0; i < n; ++ i ) {
		Complex la ( lowA[i] ), ha ( highA[i] ), lb ( lowB[i] ), hb ( highB[i] );
		lowA[i] = ha * hb, highA[i] = la * hb, lowB[i] = ha * lb, highB[i] = la * lb;
	}
	FFT ( n, lowA, -1 ), FFT ( n, highA, -1 ), FFT ( n, lowB, -1 ), FFT ( n, highB, -1 );
	for ( int i = 0; i < n; ++ i ) {
		C[i] = ( ( 1ll << 30 ) % P * ( LL ( round ( lowA[i].x ) ) % P ) % P
			 + ( LL ( round ( highA[i].x ) ) % P << 15 ) % P
			 + ( LL ( round ( lowB[i].x ) ) % P << 15 ) % P
			 + ( LL ( round ( highB[i].x ) ) % P ) ) % P;
	}
}

inline int qkpow ( int a, int b, const int p = P ) {
	int ret = 1;
	for ( ; b; a = 1ll * a * a % p, b >>= 1 ) ret = 1ll * ret * ( b & 1 ? a : 1 ) % p;
	return ret;
}

inline void init ( const int n ) {
	fac[0] = ifac[0] = pwr[0] = 1;
	for ( int i = 1; i <= n; ++ i ) {
		fac[i] = 1ll * i * fac[i - 1] % P;
		pwr[i] = ( pwr[i - 1] << 1 ) % P;
	}
	ifac[n] = qkpow ( fac[n], P - 2 );
	for ( int i = n - 1; i; -- i ) ifac[i] = ( i + 1ll ) * ifac[i + 1] % P;
}

inline void add ( const int fn, const int fm ) {
	if ( ! fn ) {
		for ( int i = 0; i <= n; ++ i ) ans[i] = F[i];
		return ;
	}
	static int A[MAXN + 5], B[MAXN + 5];
	for ( int i = 0; i <= n; ++ i ) {
		A[i] = 1ll * ans[i] * ifac[i] % P * qkpow ( 2, 1ll * i * fm % ( P - 1 ) ) % P;
		B[i] = 1ll * F[i] * ifac[i] % P;
	}
	for ( int i = n + 1; i < len; ++ i ) A[i] = B[i] = 0;
	polyConv ( len, A, B, ans );
	for ( int i = 0; i <= n; ++ i ) ans[i] = 1ll * ans[i] * fac[i] % P;
	for ( int i = n + 1; i < len; ++ i ) ans[i] = 0;
}

inline void shift ( const int cur ) {
	static int A[MAXN + 5], B[MAXN + 5];
	for ( int i = 0; i <= n; ++ i ) {
		A[i] = 1ll * F[i] * ifac[i] % P * qkpow ( 2, 1ll * i * cur % ( P - 1 ) ) % P;
		B[i] = 1ll * F[i] * ifac[i] % P;
	}
	for ( int i = n + 1; i < len; ++ i ) A[i] = B[i] = 0;
	polyConv ( len, A, B, F );
	for ( int i = 0; i <= n; ++ i ) F[i] = 1ll * F[i] * fac[i] % P;
	for ( int i = n + 1; i < len; ++ i ) F[i] = 0;
}

int main () {
	scanf ( "%d %d", &m, &n );
	if ( m > n ) return puts ( "0" ), 0;
	len = 1, lg = 0;
	for ( ; len <= n << 1; len <<= 1, ++ lg );
	initFFT ( lg ), init ( n );
	for ( int i = 1; i <= n; ++ i ) F[i] = 1;
	for ( int i = 0, cur = 0; 1 << i <= m; ++ i ) {
		if ( ( m >> i ) & 1 ) add ( cur, 1 << i ), cur |= 1 << i;
		shift ( 1 << i );
	}
	int sum = 0;
	for ( int i = 0; i <= n; ++ i ) {
		sum = ( sum + 1ll * ans[i] * fac[n] % P * ifac[i] % P * ifac[n - i] ) % P;
	}
	printf ( "%d\n", sum );
	return 0;
}