【51nod】1773 A国的贸易

题解

FWT板子题

可以发现

\(dp[i][u] = \sum_{i = 0}^{N - 1} dp[i - 1][u xor (2^i)] + dp[i - 1][u]\)

然后如果把异或提出来可以变成一个异或卷积

也就是另一个数组里只有\(0\),\(2^0\),\(2^1\)...\(2^{n - 1}\)有值

用FWT变换一下，然后快速幂，之后和原数组卷积起来就是答案了

代码

#include <iostream>
#include <algorithm>
#include <cstdio>
#include <cstring>
#define enter putchar('\n')
#define space putchar(' ')
#define MAXN 2000005
//#define ivorysi
using namespace std;
typedef long long int64;
template<class T>
void read(T &res) {
    res = 0;char c = getchar();T f = 1;
    while(c < '0' || c > '9') {
	if(c == '-') f = -1;
	c = getchar();
    }
    while(c >= '0' && c <= '9') {
	res = res * 10 - '0' + c;
	c = getchar();
    }
    res = res * f;
}
template<class T>
void out(T x) {
    if(x < 0) {x = -x;putchar('-');}
    if(x >= 10) out(x / 10);
    putchar('0' + x % 10);
}
const int MOD = 1000000007;
int N,T,g[MAXN],f[MAXN],Inv2 = (MOD + 1) / 2;

int inc(int a,int b) {
    return a + b >= MOD ? a + b - MOD : a + b;
}
int mul(int a,int b) {
    return 1LL * a * b % MOD;
}
void FWT(int *a) {
    for(int i = 1 ; i < (1 << N) ; i <<= 1) {
	for(int j = 0 ; j < (1 << N) ; j += (i << 1)) {
	    for(int k = 0 ; k < i ; ++k) {
		int t0 = a[j + k],t1 = a[j + k + i];
		a[j + k] = inc(t0,t1);
		a[j + k + i] = inc(t0,MOD - t1);
	    }
	}
    }
}
void IFWT(int *a) {
    for(int i = 1 ; i < (1 << N) ; i <<= 1) {
	for(int j = 0 ; j < (1 << N) ; j += (i << 1)) {
	    for(int k = 0 ; k < i ; ++k) {
		int t0 = a[j + k],t1 = a[j + k + i];
		a[j + k] = mul(inc(t0,t1),Inv2);
		a[j + k + i] = mul(inc(t0,MOD - t1),Inv2);
	    }
	}
    }
}
void conv(int *a,int *b) {
    for(int i = 0 ; i < (1 << N) ; ++i) a[i] = mul(a[i],b[i]);
}
void fpow(int *a,int *ans,int c) {
    static int t[MAXN];
    for(int i = 0 ; i < (1 << N) ; ++i) t[i] = a[i],ans[i] = a[i];
    --c;
    while(c) {
	if(c & 1) conv(ans,t);
	conv(t,t);
	c >>= 1;
    }
}
void Solve() {
    read(N);read(T);
    g[0] = 1;
    for(int i = 0 ; i < N ; ++i) g[1 << i] = 1;
    for(int i = 0 ; i < (1 << N) ; ++i) read(f[i]);
    FWT(g);FWT(f);
    fpow(g,g,T);
    conv(f,g);
    IFWT(f);
    for(int i = 0 ; i < (1 << N) ; ++i) {
	out(f[i]);if(i != (1 << N) - 1) space;
    }
    enter;
}
int main() {
#ifdef ivorysi
    freopen("f1.in","r",stdin);
#endif
    Solve();
}