bzoj 3527: [Zjoi2014]力 FFT

首先我们知道\(\displaystyle E_j=\sum_{ij}\frac{q_i}{(i-j)^2}\),

设\(\displaystyle g[i]=\frac{1}{i^2}\),因为\(g\)是偶函数，所以\(\displaystyle E_j=\sum_{i=0}^{j-1} q_i g[j-i]-\sum_{i=j+1}^n q_ig[j-i]\)。

前面这东西很明显就是卷积，处理后面就要发挥人类智慧了，将\(q_i\) 翻转成新数组\(q_i'\)，原来的式子就成了

\(\displaystyle E_j=\sum_{i=0}^{j-1} q_i g[j-i]-\sum_{i=0}^{j-1} q_i'g[j-i]\),

后面这东西也变成卷积啦，用FFT处理一下就ok啦。

时间复杂度O(n log n).

#include
#include
#include
#define DB double
using namespace std;
int n, lim = 1, tmp;
const DB PI = acos(-1);
const int N = 400010;
struct lmaginary 
{
    DB x, y;
    lmaginary(double X = 0, double Y = 0) {x = X, y = Y;}
    friend lmaginary operator +(const lmaginary &a, const lmaginary &b)
    {return (lmaginary) {a.x + b.x, a.y + b.y};}
    friend lmaginary operator -(const lmaginary &a, const lmaginary &b)
    {return (lmaginary) {a.x - b.x, a.y - b.y};}
    friend lmaginary operator *(const lmaginary &a, const lmaginary &b)
    {return (lmaginary) {a.x*b.x - a.y*b.y, a.x*b.y + a.y*b.x};}
} f1[N], f2[N], g[N];
DB q[N];
int r[N];
void FFT(lmaginary *A, int lim, int opt) 
{
    for (int i = 0; i < lim; ++i)
        if (i < r[i])swap(A[i], A[r[i]]);
    for (int mid = 1; mid < lim; mid <<= 1) 
    { //长度的一半
        lmaginary wn(cos(PI / mid), opt * sin(PI / mid));
        for (int len = mid << 1, j = 0; j < lim; j += len) 
        {
            lmaginary w((DB)1, (DB)0);
            for (int k = j; k < mid + j; ++k, w = w * wn) 
            {
                lmaginary a = A[k], b = w * A[k + mid];
                A[k] = a + b;
                A[k + mid] = a - b;
            }
        }
    }
}
void YYCH() 
{
    for (int i = 1; i <= n; ++i) 
    {
        g[i].x = (1.0 / i / i);
        f1[i].x = f2[n - i + 1].x = q[i];
    }
    while (lim < 2 * n)lim <<= 1, ++tmp;
    for (int i = 0; i < lim; ++i)
        r[i] = (r[i >> 1] >> 1) | ((i & 1) << (tmp - 1));
}
int main() 
{
    cin >> n;
    for (int i = 1; i <= n; ++i)scanf("%lf", &q[i]);
    YYCH();
    FFT(f1, lim, 1); FFT(f2, lim, 1); FFT(g, lim, 1);
    for (int i = 0; i < lim; ++i)
        f1[i] = f1[i] * g[i], f2[i] = f2[i] * g[i];
    FFT(f1, lim, -1); FFT(f2, lim, -1);
    for (int i = 1; i <= n; ++i)
        printf("%f\n", (f1[i].x - f2[n - i + 1].x) / lim);
    return 0;
}