Solution -「Gym 102956A」Belarusian State University

$\mathcal{Description}$

  Link.

  给定两个不超过 $2^n-1$ 次的多项式 $A,B$，对于第 $i\in[0,n)$ 个二进制位，定义任意一个二元加法 $\oplus_i:\{0,1\}\times\{0,1\}\rightarrow\{0,1\}$，而对于两个整数 $u,v\in[0,2^n)$，定义 $u\oplus v=\sum_{i=0}^{n-1}(u_i\oplus_i v_i)2^i$。求 $A,B$ 的 $\oplus$ 卷积，保证答案任意系数不超过 $2^{63}-1$。

$\mathcal{Solution}$

  可以看出这个问题不弱于 FWT，所以我们大概需要根据 FWT 的思路，来 DIY 一个变换。

  在 FWT 的框架下，枚举每个二进制位，对 $16$ 种不同的加法分别构造各自的变换方式，构造时只需要考虑 $(a_0+a_1x)$ 与 $(b_0+b_1x)$ 的卷积，使得这个卷积合法即可。

  复杂度 $\mathcal O(2^nn)$。

$\mathcal{Code}$

  不要用 switch！不要用 switch！不要用 switch！这个语法真的离谱。

  啊……极少地在代码里爆了粗口，以记录我分类讨论加上被 switch 弄傻的愉悦！（

/*~Rainybunny~*/

#include <cstdio>
#include <cassert>

#define rep( i, l, r ) for ( int i = l, rep##i = r; i <= rep##i; ++i )
#define per( i, r, l ) for ( int i = r, per##i = l; i >= per##i; --i )

typedef long long LL;

const int MAXN = 18, MAXL = 1 << MAXN;
int n;
LL a[MAXL + 5], b[MAXL + 5];
char op[MAXN + 5][4];

inline int type( const char* o ) {
    int ret = 0;
    per ( i, 3, 0 ) ret = ret << 1 | ( o[i] ^ '0' );
    return ret;
}

inline void fuckin_wa_tle( const int len, LL* u, const int tp ) {
    // { -1: first item, 0: result, 1: second item }.
#define swp( a, b ) void( a ^= b ^= a ^= b )
    for ( int i = 0, stp = 1; stp < len; ++i, stp <<= 1 ) {
        for ( int j = 0; j < len; j += stp << 1 ) {
            rep ( k, j, j + stp - 1 ) {
                LL &p = u[k], &q = u[k + stp];
                switch ( type( op[i] ) ) {
                    // what fuckin stupid grammar??? I'll never `switch` again.
                    case 0:
                        if ( !~tp ) p += q, q = 0;
                        else if ( tp ) p += q, q = 0;
                        break;
                    case 1:
                        if ( !~tp ) swp( p, q ), p += q;
                        else if ( tp ) swp( p, q ), p += q;
                        else p -= q;
                        break;
                    case 2:
                        if ( !~tp ) swp( p, q ), p += q;
                        else if ( tp ) p += q;
                        else p -= q;
                        break;
                    case 3:
                        if ( !~tp ) swp( p, q );
                        else if ( tp ) p = q = p + q;
                        break;
                    case 4:
                        if ( !~tp ) p += q;
                        else if ( tp ) swp( p, q ), p += q;
                        else p -= q;
                        break;
                    case 5:
                        if ( !~tp ) p = q = p + q;
                        else if ( tp == 1 ) swp( p, q );
                        break;
                    case 6:
                        p += q, q = p - 2 * q;
                        if ( !tp ) {
                            assert( !( p % 2 ) && !( q % 2 ) );
                            p /= 2, q /= 2;
                        }
                        break;
                    case 7:
                        if ( !~tp ) swp( p, q ), q += p;
                        else if ( tp ) swp( p, q ), q += p;
                        else q -= p;
                        break;
                    case 8:
                        if ( !~tp ) p += q;
                        else if ( tp ) p += q;
                        else p -= q;
                        break;
                    case 9:
                        q += p, p = q - 2 * p;
                        if ( !tp ) {
                            assert( !( p % 2 ) && !( q % 2 ) );
                            p /= 2, q /= 2;
                        }
                        break;
                    case 10:
                        if ( !~tp ) p = q = p + q;
                        break;
                    case 11:
                        if ( !~tp ) swp( p, q ), q += p;
                        else if ( tp ) q += p;
                        else q -= p;
                        break;
                    case 12:
                        if ( tp == 1 ) p = q = p + q;
                        break;
                    case 13:
                        if ( !~tp ) q += p;
                        else if ( tp ) swp( p, q ), q += p;
                        else q -= p;
                        break;
                    case 14:
                        if ( !~tp ) q += p;
                        else if ( tp ) q += p;
                        else q -= p;
                        break;
                    case 15:
                        if ( !~tp ) q += p, p = 0;
                        else if ( tp ) q += p, p = 0;
                        break;
                    default: assert( false );
                }
            }
        }
    }
#undef swp
}

int main() {
    scanf( "%d", &n );
    rep ( i, 0, n - 1 ) scanf( "%s", op[i] );
    rep ( i, 0, ( 1 << n ) - 1 ) scanf( "%lld", &a[i] );
    rep ( i, 0, ( 1 << n ) - 1 ) scanf( "%lld", &b[i] );

    fuckin_wa_tle( 1 << n, a, -1 ), fuckin_wa_tle( 1 << n, b, 1 );
    rep ( i, 0, ( 1 << n ) - 1 ) a[i] *= b[i];
    fuckin_wa_tle( 1 << n, a, 0 );

    rep ( i, 0, ( 1 << n ) - 1 ) {
        printf( "%lld%c", a[i], i < repi ? ' ' : '\n' );
    }
    return 0;
}