Nowcoder Circulant Matrix ( FWT )

题目链接

题意 :

给你一个a数组和b数组，构造出A[i][j]矩阵（A[i][j] = a[i xor j]）

给出等式 A * x = b ( mod p )

n等于4的时候有：

A[0][0]*x[0] + A[0][1]*x[1] + A[0][2]*x[2] + A[0][3]*x[3] = b[0] (mod p)
A[1][0]*x[0] + A[1][1]*x[1] + A[1][2]*x[2] + A[1][3]*x[3] = b[1] (mod p)
A[2][0]*x[0] + A[2][1]*x[1] + A[2][2]*x[2] + A[2][3]*x[3] = b[2] (mod p)
A[3][0]*x[0] + A[3][1]*x[1] + A[3][2]*x[2] + A[3][3]*x[3] = b[3] (mod p)

分析 :

对于给出的矩阵乘法式子、你随便取出第一行会发现一个规律

例如 A[0][0]*x[0] + A[0][1]*x[1] + A[0][2]*x[2] + A[0][3]*x[3] = b[0] (mod p)

A[0][0]*x[0] => a[0^0] * x[0] => a[0] * x[0] = b[0]

A[0][1]*x[1] => a[0^1] * x[1] => a[1] * x[1] = b[0]

......

然后你会发现当把所有的 A[i][j] 变成 a[i^j] 后

每一条恒等式都变成了异或卷积的形式

当然这个规律也可以这么看，你会发现 A 的第二维下标永远和 x 的下标一样

所以变成卷积形式的话，那么的出来的异或值永远为 i ==> A[i][j] * x[j] + A[i][j+1]*x[j+1] + .... = b[i] ==> i^j^j = i

所以可以用 FWT 思考

对于普通的 FWT 优化的是 

for(int i=0; i<n; i++) for(int j=0; j<n; j++) b[i^j] += a[i] * x[j]

这里我们已知 a 和 b 要求 x、可以将下标变化一下有

for(int i=0; i<n; i++) for(int j=0; j<n; j++) b[i] += a[j] * x[j^i]

for(int i=0; i<n; i++) for(int j=0; j<n; j++) x[i^j] += b[i] / a[j]

这样就变成了熟悉的异或卷积的形式

只不过乘法运算变成了除法运算、这里还好是求模意义下的

故可以用乘法逆元来将除法变成乘法

for(int i=0; i<n; i++) for(int j=0; j<n; j++) x[i^j] = b[i] * inv_a[j]

这题就做完了

#include<bits/stdc++.h>
#define LL long long
#define ULL unsigned long long

#define scl(i) scanf("%lld", &i)
#define scll(i, j) scanf("%lld %lld", &i, &j)
#define sclll(i, j, k) scanf("%lld %lld %lld", &i, &j, &k)
#define scllll(i, j, k, l) scanf("%lld %lld %lld %lld", &i, &j, &k, &l)

#define scs(i) scanf("%s", i)
#define sci(i) scanf("%d", &i)
#define scd(i) scanf("%lf", &i)
#define scIl(i) scanf("%I64d", &i)
#define scii(i, j) scanf("%d %d", &i, &j)
#define scdd(i, j) scanf("%lf %lf", &i, &j)
#define scIll(i, j) scanf("%I64d %I64d", &i, &j)
#define sciii(i, j, k) scanf("%d %d %d", &i, &j, &k)
#define scddd(i, j, k) scanf("%lf %lf %lf", &i, &j, &k)
#define scIlll(i, j, k) scanf("%I64d %I64d %I64d", &i, &j, &k)
#define sciiii(i, j, k, l) scanf("%d %d %d %d", &i, &j, &k, &l)
#define scdddd(i, j, k, l) scanf("%lf %lf %lf %lf", &i, &j, &k, &l)
#define scIllll(i, j, k, l) scanf("%I64d %I64d %I64d %I64d", &i, &j, &k, &l)

#define lson l, m, rt<<1
#define rson m+1, r, rt<<1|1
#define lowbit(i) (i & (-i))
#define mem(i, j) memset(i, j, sizeof(i))

#define fir first
#define sec second
#define VI vector<int>
#define ins(i) insert(i)
#define pb(i) push_back(i)
#define pii pair<int, int>
#define VL vector<long long>
#define mk(i, j) make_pair(i, j)
#define all(i) i.begin(), i.end()
#define pll pair<long long, long long>

#define _TIME 0
#define _INPUT 0
#define _OUTPUT 0
clock_t START, END;
void __stTIME();
void __enTIME();
void __IOPUT();
using namespace std;
const LL mod = 1e9 + 7;
LL inv2 = (mod + 1)>>1;


void FWT(LL f[], int n, int op) {
    int mx = 0;
    while((1LL<<mx) < n) mx++;
    for (int i = 1; i <= mx; ++i) {
        int m = (1 << i), len = m >> 1;
        for (int r = 0; r < n; r += m) {
            int t1 = r, t2 = r + len;
            for (int j = 0; j < len; ++j, ++t1, ++t2) {
                LL x1 = f[t1], x2 = f[t2];
                if (op == 1) {   //xor
                    f[t1] = x1 + x2;
                    f[t2] = (x1 - x2 + mod) % mod;
                    if(f[t1] >= mod) f[t1] -= mod;
                    if(f[t2] < 0) f[t2] += mod;
                }
                if (op == 2) {   //and
                    f[t1] = x1 + x2;
                    f[t2] = x2;
                    if(f[t1] >= mod) f[t1] -= mod;
                }
                if (op == 3) {   //or
                    f[t1] = x1;
                    f[t2] = x2 + x1;
                    if(f[t2] >= mod) f[t2] -= mod;
                }
            }
        }
    }
}
void IFWT(LL f[], int n, int op) {
    int mx = 0;
    while((1LL<<mx) < n) mx++;
    for (int i = mx; i >= 1; --i) {
        int m = (1 << i), len = m >> 1;
        for (int r = 0; r < n; r += m) {
            int t1 = r, t2 = r + len;
            for (int j = 0; j < len; ++j, ++t1, ++t2) {
                LL x1 = f[t1], x2 = f[t2];
                if (op == 1) {   //xor
//                    f[t1] = (x1 + x2) / 2;
//                    f[t2] = (x1 - x2) / 2;
                    f[t1] = (x1 + x2) * inv2;
                    f[t2] = (x1 - x2) * inv2;
                    if(f[t1] >= mod) f[t1] %= mod;
                    if(f[t2] >= mod) f[t2] %= mod;
                    if(f[t2] < 0) f[t2] = f[t2] % mod + mod;
                }
                if (op == 2) {   //and
                    f[t1] = x1 - x2;
                    f[t2] = x2;
                    if(f[t1] < 0) f[t1] += mod;
                }
                if (op == 3) {   //or
                    f[t1] = x1;
                    f[t2] = x2 - x1;
                    if(f[t2] < 0) f[t2] += mod;
                }
            }
        }
    }
}

LL pow_mod(LL a, LL b){
    a %= mod;
    LL ret = 1;
    while(b){
        if(b & 1) ret = (ret * a)%mod;
        a = (a * a)%mod;
        b >>= 1;
    }return ret;
}

const int maxn = 262144 + 10;
LL a[maxn], b[maxn], x[maxn];

int main(void){__stTIME();__IOPUT();


    int n;
    sci(n);

    for(int i=0; i<n; i++) scl(a[i]);
    for(int i=0; i<n; i++) scl(b[i]);

    FWT(a, n, 1);
    FWT(b, n, 1);

    for(int i=0; i<n; i++) x[i] = (b[i] * pow_mod(a[i], mod-2))%mod;

    IFWT(x, n, 1);

    for(int i=0; i<n; i++) printf("%lld
", x[i]);








__enTIME();return 0;}


void __stTIME()
{
    #if _TIME
        START = clock();
    #endif
}

void __enTIME()
{
    #if _TIME
        END = clock();
        cerr<<"execute time = "<<(double)(END-START)/CLOCKS_PER_SEC<<endl;
    #endif
}

void __IOPUT()
{
    #if _INPUT
        freopen("in.txt", "r", stdin);
    #endif
    #if _OUTPUT
        freopen("out.txt", "w", stdout);
    #endif
}