{"id":954,"date":"2020-11-07T14:07:45","date_gmt":"2020-11-07T06:07:45","guid":{"rendered":"http:\/\/www.zyhcoding.club\/?p=954"},"modified":"2020-11-07T23:30:54","modified_gmt":"2020-11-07T15:30:54","slug":"%e5%bf%ab%e9%80%9f%e5%82%85%e9%87%8c%e5%8f%b6%e5%8f%98%e6%8d%a2dftfft","status":"publish","type":"post","link":"http:\/\/www.zyhcoding.club\/index.php\/2020\/11\/07\/%e5%bf%ab%e9%80%9f%e5%82%85%e9%87%8c%e5%8f%b6%e5%8f%98%e6%8d%a2dftfft\/","title":{"rendered":"\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362,DFT,FFT"},"content":{"rendered":"

\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362FFT<\/h2>\n

\u5927\u6574\u6570\u4e58\u6cd5\u8ba1\u7b97\u7684\u53d1\u5c55\u5386\u7a0b\u662f\u7f13\u6162\u7684...\u5728\u6734\u7d20\u7684\u5927\u6574\u6570\u4e58\u6cd5\u8ba1\u7b97\u548c\u591a\u9879\u5f0f\u4e58\u6cd5\u4e2d\uff0c\u4e24\u4e2a\u591a\u9879\u5f0ff(x), g(x)\u7684\u4efb\u610f\u4e24\u9879\u4e4b\u95f4\u90fd\u8981\u76f8\u4e58\u4e00\u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3aO(n^2)<\/code>\u3002\u540e\u6765\u51fa\u73b0\u4e86\u7528\u5206\u6cbb\u6cd5\u8ba1\u7b97\u5927\u6574\u6570\u4e58\u6cd5\uff0c\u628a\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230O(n^{1.585})<\/code>\uff0c\u7136\u540e\u7ecf\u5386\u4e86\u76f8\u5f53\u957f\u7684\u65f6\u95f4\u540e\u624d\u53d1\u660e\u51fa\u4e86\u66f4\u5feb\u7684\u8ba1\u7b97\u65b9\u6cd5--\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362(Fast Fourier Transform)\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3aO(nlogn)<\/code>\uff0c\u4e3a\u76ee\u524d\u8ba1\u7b97\u5927\u6570\u4e58\u6cd5\u6700\u5feb\u7684\u7b97\u6cd5\u3002<\/p>\n

\u591a\u9879\u5f0f\u7684\u8868\u793a<\/h5>\n

\u4e00\u4e2a\u591a\u9879\u5f0f\u6709\u4e24\u79cd\u8868\u793a\u65b9\u6cd5\uff0c\u7cfb\u6570\u8868\u793a\u6cd5\u548c\u70b9\u503c\u8868\u793a\u6cd5\u3002\u7cfb\u6570\u8868\u793a\u6cd5\u5c31\u662f\u6211\u4eec\u6700\u5e38\u7528\u7684\u7528\u6765\u8868\u793a\u4e00\u4e2a\u591a\u9879\u5f0f\u7684\u65b9\u6cd5\uff0c\u5373\u591a\u9879\u5f0f\u6bcf\u4e2a\u9879\u7684\u7cfb\u6570\u7ec4\u6210\u7684\u96c6\u5408\u3002\u4e00\u4e2a\u6700\u9ad8n\u6b21\u7684\u591a\u9879\u5f0f\u53ef\u4ee5\u7528n+1\u4e2a\u7cfb\u6570\u6765\u8868\u793a\uff0c\u4e5f\u53ef\u4ee5\u7528n+1\u4e2a\u6a2a\u5750\u6807\u4e0d\u540c\u7684\u70b9\u6765\u8868\u793a\u3002n+1\u4e2a\u4e0d\u540c\u7684\u70b9\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e00\u4e2a\u6700\u9ad8n\u6b21\u591a\u9879\u5f0f\u3002<\/p>\n

\u7ed9\u5b9a\u4e24\u4e2a\u591a\u9879\u5f0ff(x), g(x)\uff0c\u73b0\u8ba1\u7b97\u5176\u4e58\u6cd5h(x) = f(x)*g(x)<\/code>\uff0c\u82e5\u8fd9\u4e24\u4e2a\u591a\u9879\u5f0f\u662f\u7528\u7cfb\u6570\u8868\u793a\u6cd5\u8868\u793a\u7684\uff0c\u90a3\u4e48\u9700\u8981O(n^2)<\/code>\u7684\u65f6\u95f4\u53bb\u8ba1\u7b97\uff1b\u82e5\u662f\u7528\u70b9\u503c\u8868\u793a\u6cd5\u8868\u793a\u7684\uff0c\u5047\u8bbe\u53d6\u5230\u7684\u70b9\u7684\u6a2a\u5750\u6807\u7684\u90fd\u662f\u76f8\u540c\u7684\uff0c\u5373(x1, f(x1)), (x2, f(x2)), (x3, f(x3))....(xn, f(xn))\u548c(x1, g(x1)), (x2, g(x2)), (x3, g(x3))....(xn, g(xn))\uff0c\u90a3\u4e48\u4ec5\u9700\u8981O(n)\u7684\u65f6\u95f4\u5373\u53ef\u8ba1\u7b97\u51fah(x)\u7684\u70b9\u503c\u8868\u793a\u6cd5(x1, f(x1)*g(x1)), (x2, f(x2)*g(x2)), (x3, f(x3)*g(x3))....(xn, f(xn)*g(xn))\u3002\u53ef\u4ee5\u5229\u7528\u8fd9\u4e00\u8fc7\u7a0b\u6765\u52a0\u901f\u8ba1\u7b97\u591a\u9879\u5f0f\u4e58\u6cd5\u3002<\/p>\n

\u96be\u70b9\u662f\u5982\u4f55\u5c06\u591a\u9879\u5f0f\u4ece\u7cfb\u6570\u8868\u793a\u6cd5\u8f6c\u5316\u4e3a\u70b9\u503c\u8868\u793a\u6cd5\u3002\u6211\u4eec\u77e5\u9053\u8ba1\u7b97\u4e00\u4e2an\u6b21\u591a\u9879\u5f0f\u7684\u4e00\u4e2af(x)\u9700\u8981O(n)\u7684\u65f6\u95f4\uff0c\u90a3\u8f6c\u6362\u6210\u70b9\u503c\u8868\u793a\u6cd5\u5c82\u4e0d\u662f\u4e5f\u9700\u8981O(n^2)<\/code>\u7684\u65f6\u95f4\u3002\u3002\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\u505a\u7684\u5de5\u4f5c\u5c31\u662f\u5feb\u901f\u8fdb\u884c\u8f6c\u6362\u8fd9\u4e2a\u5de5\u4f5c\u3002<\/p>\n

\u8fd9\u91cc\u9700\u8981\u5f15\u5165\u590d\u6570\u3002<\/p>\n

\u590d\u6570<\/h5>\n

\u5982\u4f55\u5feb\u901f\u4ece\u7cfb\u6570\u8868\u793a\u6cd5\u8f6c\u5316\u4e3a\u70b9\u503c\u8868\u793a\u6cd5\u5462\uff1f\u9996\u5148\u9700\u8981\u9009\u62e9n+1\u4e2a\u6a2a\u5750\u6807\uff0c\u8fd9\u91cc\u9009\u62e9x^n=1<\/code>\u5728\u590d\u6570\u610f\u4e49\u4e0a\u7684n\u4e2a\u5355\u4f4d\u6839\u6765\u8868\u793a\uff0c\u5229\u7528\u5355\u4f4d\u6839\u7684\u4e00\u4e9b\u6027\u8d28\u6765\u5feb\u901f\u5b8c\u6210\u8f6c\u5316\uff01
\n\u4e00\u4e2a\u590d\u6570a+bi<\/code>\u53ef\u4ee5\u50cf\u5411\u91cf\u4e00\u6837\u5728\u4e8c\u7ef4\u5750\u6807\u91cc\u8868\u793a\u4e3a\u5750\u6807(a, b)\uff0c\u4e5f\u53ef\u4ee5\u50cf\u6781\u5750\u6807\u4e00\u6837\u7528\u4e00\u5bf9(\u6a21\u89d2\uff0c\u5e45\u89d2)\u6765\u8868\u793a\uff0c\u6a21\u89d2\u662f\u590d\u6570\u7684\u957f\u5ea6\uff0c\u5e45\u89d2\u662f\u8be5\u590d\u6570\u548cx\u6b63\u534a\u8f74\u7684\u5939\u89d2\u3002\u4e24\u590d\u6570\u76f8\u4e58\u65f6\u6ee1\u8db3\uff1a\u5e45\u89d2\u76f8\u4e58\uff0c\u6a21\u89d2\u76f8\u52a0\u3002\u6240\u4ee5x^n=1<\/code>\u7684n\u4e2a\u6839\u5176\u5b9e\u5c31\u662f\u628a\u4e00\u4e2a\u5706\u5fc3\u5728\u539f\u70b9\u7684\u5355\u4f4d\u5143n\u7b49\u5206\uff0c\u5f97\u5230\u7684n\u4e2a\u70b9\u3002
\n\u8bbew_{n}^{k}<\/code>\u8868\u793an\u6b21\u590d\u6839\u7684\u7b2ck\u4e2a\u5355\u4f4d\u6839\u3002w_{n}^{k}<\/code>\u5bf9\u5e94\u7684\u5750\u6807\u4e3a(cos(\\frac{k}{n}2\\pi), sin(\\frac{k}{n}2\\pi))<\/code>\uff0c\u4e5f\u5c31\u662f\u590d\u6570cos(\\frac{k}{n}2\\pi)+i*sin(\\frac{k}{n}2\\pi)<\/code>\uff0c\u6839\u636e\u6b27\u62c9\u516c\u5f0f\u4e5f\u53ef\u8868\u793a\u6210e^{i\u00b7\\frac{k}{n}2\\pi}<\/code>\u3002\u7528\u6b27\u62c9\u516c\u5f0f\u53ef\u4ee5\u5f88\u5bb9\u6613\u7684\u89e3\u91ca\u4e3a\u4ec0\u4e48\u4e24\u4e2a\u590d\u6570\u76f8\u4e58\u7ed3\u679c\u662f\u6a21\u89d2\u76f8\u4e58\uff0c\u5e45\u89d2\u76f8\u52a0\u3002
\n\u4e09\u4e2a\u91cd\u8981\u6027\u8d28\uff1a<\/p>\n

\n

w_{n}^{n} = w_{n}^{0} = 1<\/code>
\nw_{n}^{k} = w_{2n}^{2k}<\/code>
\nw_{n}^{k+\\frac{n}{2}} = -w_{n}^{k}<\/code><\/p>\n<\/blockquote>\n

\u8fd9\u4e09\u4e2a\u6027\u8d28\u90fd\u5f88\u5bb9\u6613\u63a8\u51fa\u6765\u3002
\n\u7b2c\u4e09\u4e2a\u6027\u8d28\u4e2d\uff0c\u56e0\u4e3aw_{n}^{\\frac{n}{2}}<\/code>\u521a\u597d\u662f-1\uff0c\u6240\u4ee5\u53bb\u6389\u4e4b\u540e\u8fd8\u9700\u8981\u518d\u4e58\u4ee5-1.<\/p>\n

\u8774\u8776\u53d8\u6362<\/h5>\n

\u5982\u4f55\u628a\u590d\u6570\u5e94\u7528\u8fdb\u53bb\u5e94\u8be5\u662fFFT\u6700\u96be\u7406\u89e3\u7684\u90e8\u5206\u4e86\u3002<\/p>\n

\u6211\u4eec\u73b0\u5728\u8981\u505a\u7684\u662f\uff1a\u5bf9\u4e8e\u4e00\u4e2an-1\u6b21\u591a\u9879\u5f0ff(x)\uff0c\u628an\u4e2a\u5355\u4f4d\u590d\u6839\u5e26\u5165\uff0c\u6c42\u5f97n\u4e2a\u590d\u503c\uff0c\u5982\u4f55\u5feb\u901f\u7684\u6c42\u51fa\u3002<\/p>\n

\u6211\u4eec\u628af(x)\u7684\u5947\u6b21\u9879\u7684\u7cfb\u6570\u548c\u5076\u6b64\u9879\u7684\u7cfb\u6570\u5206\u522b\u8868\u793a\u6210\u4e00\u4e2a\u591a\u9879\u5f0f\uff0c\u5373
\nA_0(x) = a_0 + a_2x + a_4x^2 + ... + a_{n-2}x^{\\frac{n}{2}-1}<\/code>
\nA_1(x) = a_1 + a_3x + a_5x^2 + ... + a_{n-1}x^{\\frac{n}{2}-1}<\/code>
\n\u90a3\u4e48\u5c31\u6709f(x) = A_0(x^2) + x * A_1(x^2)<\/code>.<\/p>\n

\u6211\u4eec\u518d\u5c06n\u6b21\u5355\u4f4d\u590d\u6839\u5e26\u5165\uff0c\u8bbek<=\\frac{n}{2}<\/code>\uff0c\u5f97
\nf(w_{n}^{k}) = A_0(w_{n}^{2k}) + w_{n}^{k}*A_1(w_{n}^{2k}) = A_0(w_{\\frac{n}{2}}^{k}) + w_{n}^{k}*A_1(w_{\\frac{n}{2}}^{k})<\/code><\/p>\n

f(w_{n}^{k+\\frac{n}{2}}) = A_0(w_{n}^{2k+n}) + w_{n}^{k+\\frac{n}{2}}*A_1(w_{n}^{2k+n}) = A_0(w_{\\frac{n}{2}}^{k}) - w_{n}^{k}*A_1(w_{\\frac{n}{2}}^{k})<\/code><\/p>\n

f(x)\u662fn-1\u6b21\u591a\u9879\u5f0f\uff0c\u9700\u8981\u8ba1\u7b97n\u6b21\u5355\u4f4d\u590d\u6839\u5e26\u5165\u5f97\u5230\u7684\u503c\uff1bA_0(x), A_1(x)<\/code>\u662f\\frac{2}{n}-1<\/code>\u6b21\u591a\u9879\u5f0f\uff0c\u5df2\u7ecf\u8ba1\u7b97\u51fa\u5c06\\frac{2}{n}<\/code>\u6b21\u5355\u4f4d\u590d\u6839\u5e26\u5165\u5f97\u5230\u7684\u503c\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5728A_0(x), A_1(x)<\/code>\u7684\u57fa\u7840\u4e0a\u7ebf\u6027\u6c42\u51faf(x)\u7684n\u4e2a\u590d\u6839\uff01A_0(x), A_1(x)<\/code>\u4e5f\u53ef\u4ee5\u7528\u7c7b\u4f3c\u7684\u6c42\u6cd5\u9012\u5f52\u6c42\u4e0b\u53bb\uff0c\u5171\u9700\u8981\u6c42logn<\/code>\u5c42\uff0c\u6240\u4ee5\u603b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662fO(nlogn)<\/code>.
\n\u6ce8\u610f\u5230\u6bcf\u5c42\u5faa\u73af\u4e2dn\u90fd\u5fc5\u987b\u53ef\u4ee5\u88ab2\u6574\u9664\uff0c\u6240\u4ee5\u7b2c\u4e00\u5c42\u7684n\u5fc5\u987b\u662f2\u7684\u5e42\uff0c\u82e5\u6240\u7ed9\u7684\u591a\u9879\u5f0f\u7684\u6700\u9ad8\u6b21\u4e0d\u662f2\u7684\u5e42\uff0c\u90a3\u4e48\u5c31\u88650.<\/p>\n

\u5feb\u901f\u5085\u91cc\u53f6\u9006\u53d8\u6362IFFT<\/h5>\n

\u5f97\u5230\u76ee\u6807\u591a\u9879\u5f0f\u7684\u70b9\u503c\u8868\u793a\u6cd5\u4e4b\u540e\uff0c\u6211\u4eec\u8fd8\u8981\u628a\u5b83\u8f6c\u5316\u6210\u7cfb\u6570\u8868\u793a\u6cd5\uff0c\u8fd9\u4e00\u8fc7\u7a0b\u53eb\u5feb\u901f\u5085\u91cc\u53f6\u9006\u53d8\u6362\u3002<\/p>\n

\u628af(x)\u8f6c\u5316\u6210\u70b9\u503c\u8868\u793a\u6cd5\u540e\uff0c\u518d\u628a\u8fd9\u4e9b\u70b9\u503c\u4f5c\u4e3a\u53e6\u4e00\u4e2a\u591a\u9879\u5f0f\u7684\u7cfb\u6570\uff0c\u5e76\u628a\u5355\u4f4d\u590d\u6839\u7684\u5012\u6570\u4f5c\u4e3a\u6839\u4ee3\u5165\uff0c\u5f97\u5230\u7684\u503c\u5c31\u662ff(x)\u7684\u7cfb\u6570\uff0c\u6240\u4ee5\u8fd9\u91cc\u76f8\u5f53\u4e8e\u662f\u518d\u505a\u4e00\u6b21FFT\u5373\u53ef\u3002<\/p>\n

\u4ee3\u7801\uff1a<\/p>\n

const double PI = acos(-1.0);\nvoid change(Complex y[], int len){\n    for (int i=1, j=len\/2, k; i<len-1; i++){\n        if (i<j)    swap(y[i], y[j]);\n        k = len\/2;\n        while (j>=k){\n            j -= k;\n            k \/= 2;\n        }\n        if (j<k)    j += k;\n    }\n}\nvoid FFT(Complex y[], int len, int on){       \/\/\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff0cy\u8868\u793a\u8981\u5e26\u5165\u7684len\u4e2a\u590d\u6570\uff0con\u8868\u793a\u662f\u6c42\u6b63\u53d8\u6362\u8fd8\u662f\u9006\u53d8\u6362\u3002\n    change(y, len);                      \/\/\u76f4\u63a5\u5316\u6210\u9012\u5f52\u7684\u6700\u5e95\u4e00\u5c42\uff0c\u5373\u6700\u4f4e\u4e00\u5c42\u4e2d\u5404\u9879\u7cfb\u6570\u7684\u4f4d\u7f6e\u3002\n    for (int h=2; h<=len; h<<=1){        \/\/h\u8868\u793a\u8be5\u5c42\u6bcf\u4e2a\u51fd\u6570\u6709h\u4e2a\u5143\u7d20\uff0c\u5f00\u59cb\u5411\u4e0a\u8fed\u4ee3\n        Complex wn(cos(PI*2\/h), sin(PI*2*on\/h));      \/\/wn\u4e3a\u7b2c1\u4e2a\u5355\u4f4d\u6839\/\u5355\u4f4d\u6839\u5012\u6570\n        for (int j=0; j<len; j+=h){                   \/\/\u6bcf\u5c42\u7684\u7b2c\u4e00\u4e2a\u5143\u7d20\n            Complex w(1, 0);                          \/\/w\u8868\u793a\u7b2c\u51e0\u5355\u4f4d\u6839\uff0c\u6c42\u5b8c\u4e00\u6b21\u540e\uff0c\u90fd\u53d8\u6210\u4e0b\u4e00\u4e2a\u5355\u4f4d\u6839\u3002\n            for (int k=j; k<j+h\/2; k++){\n                Complex u = y[k], t = w*y[k+h\/2];\n                y[k] = u+t, y[k+h\/2] = u-t;\n                w = w*wn;\n            }\n        }\n    }\n    if (on==-1){                        \/\/\u82e5\u4e3a\u9006\u53d8\u6362\uff0c\u8fd8\u9700\u8981\u518d\u9664\u4ee5n\u624d\u80fd\u5f97\u5230\u539f\u591a\u9879\u5f0f\u7684\u7cfb\u6570\u3002\n        for (int i=0; i<len; i++)   y[i].x \/= len;\n    }\n}<\/code><\/pre>\n
\u590d\u6570\u7684\u4f7f\u7528\uff1a<\/p>\n
struct Complex{\n    double x, y;                \/\/\u5b9e\u90e8\u548c\u865a\u90e8\n    Complex(double a = 0.0, double b = 0.0){\n        x = a, y = b;\n    }\n    Complex operator + (const Complex &a) const {\n        return Complex(x+a.x, y+a.y);\n    }\n    Complex operator - (const Complex &a) const {\n        return Complex(x-a.x, y-a.y);\n    }\n    Complex operator * (const Complex &a) const {      \/\/\u7b80\u5355\u63a8\u51fa\n        return Complex(x*a.x-y*a.y, x*a.y+y*a.x);\n    }\n};<\/code><\/pre>\n
\u9898\u76ee
\nP1919 \u3010\u6a21\u677f\u3011A*B Problem\u5347\u7ea7\u7248<\/a>
\n\u5927\u6570\u76f8\u4e58\u4e0d\u540c\u4e8e\u591a\u9879\u5f0f\u4e58\u6cd5\u7684\u5730\u65b9\u662f\u5728\u8ba1\u7b97\u5b8c\u6210\u540e\u9700\u8981\u8fdb\u4f4d\u3002<\/p>\n
#include <bits\/stdc++.h>\n\nusing namespace std;\n\nconst int maxn = 1e6+500000;\nconst double PI = acos(-1.0);\nstruct Complex{\n    double x, y;\n    Complex(double a = 0.0, double b = 0.0){\n        x = a, y = b;\n    }\n    Complex operator + (const Complex &a) const {\n        return Complex(x+a.x, y+a.y);\n    }\n    Complex operator - (const Complex &a) const {\n        return Complex(x-a.x, y-a.y);\n    }\n    Complex operator * (const Complex &a) const {\n        return Complex(x*a.x-y*a.y, x*a.y+y*a.x);\n    }\n};\n\nchar s1[maxn], s2[maxn];\nstruct Complex x1[maxn*2], x2[maxn*2];\nint len;\nint ans[maxn*2];\n\nvoid change(Complex y[], int len){\n    for (int i=1, j=len\/2, k; i<len-1; i++){\n        if (i<j)    swap(y[i], y[j]);\n        k = len\/2;\n        while (j>=k){\n            j -= k;\n            k \/= 2;\n        }\n        if (j<k)    j += k;\n    }\n}\nvoid FFT(Complex y[], int len, int on){\n    change(y, len);                      \/\/\u76f4\u63a5\u5316\u6210\u9012\u5f52\u7684\u6700\u5e95\u4e00\u5c42\n    for (int h=2; h<=len; h<<=1){        \/\/h\u8868\u793a\u8be5\u5c42\u6bcf\u4e2a\u51fd\u6570\u6709h\u4e2a\u5143\u7d20\n        Complex wn(cos(PI*2\/h), sin(PI*2*on\/h));                    \/\/\n        for (int j=0; j<len; j+=h){        \/\/\u6bcf\u5c42\u7684\u7b2c\u4e00\u4e2a\u5143\u7d20\n            Complex w(1, 0);                                   \/\/w\u8868\u793a\u7b2c\u51e0\u5355\u4f4d\u6839\n            for (int k=j; k<j+h\/2; k++){\n                Complex u = y[k], t = w*y[k+h\/2];\n                y[k] = u+t, y[k+h\/2] = u-t;\n                w = w*wn;\n            }\n        }\n    }\n    if (on==-1){\n        for (int i=0; i<len; i++)   y[i].x \/= len;\n    }\n}\nint main(){\n\n    while (~scanf ("%s %s", s1, s2)){\n        int len1 = strlen(s1), len2 = strlen(s2);\n        len = 1;\n        while (len < len1+len2) len <<= 1;\n        for (int i=0; i<len1; i++)  x1[i] = Complex(s1[len1-i-1]-'0', 0.0);\n        for (int i=0; i<len2; i++)  x2[i] = Complex(s2[len2-i-1]-'0', 0.0);\n        for (int i=len1; i<len; i++)    x1[i] = Complex(0, 0);\n        for (int i=len2; i<len; i++)    x2[i] = Complex(0, 0);\n        FFT(x1, len, 1);\n        FFT(x2, len, 1);\n        for (int i=0; i<len; i++)  x1[i] = x1[i] * x2[i];\n        FFT(x1, len, -1);\n        for (int i=0; i<len; i++)  ans[i] = int(x1[i].x + 0.5);\n        for (int i=0; i<len; i++){\n            ans[i+1] += ans[i] \/ 10;\n            ans[i] %= 10;\n        }\n        while (ans[len-1]==0 && len>1) len--;\n        for (int i=len-1; i>=0; i--)   putchar(ans[i]+'0');\n        putchar('\\n');\n    }\n    return 0;\n}\n<\/code><\/pre>\n
P3803 \u3010\u6a21\u677f\u3011\u591a\u9879\u5f0f\u4e58\u6cd5\uff08FFT\uff09<\/a><\/p>\n
#include <bits\/stdc++.h>\n\nusing namespace std;\ntypedef long long ll;\nconst int maxn = 1e6+500000;\nconst double PI = acos(-1.0);\nstruct Complex{\n    double x, y;\n    Complex(double a = 0.0, double b = 0.0){\n        x = a, y = b;\n    }\n    Complex operator + (const Complex &a) const {\n        return Complex(x+a.x, y+a.y);\n    }\n    Complex operator - (const Complex &a) const {\n        return Complex(x-a.x, y-a.y);\n    }\n    Complex operator * (const Complex &a) const {\n        return Complex(x*a.x-y*a.y, x*a.y+y*a.x);\n    }\n};\n\nstruct Complex x1[maxn*2], x2[maxn*2];\nint len;\nll ans[maxn*2];\n\nvoid change(Complex y[], int len);\nvoid FFT(Complex y[], int len, int on);\nint main(){\n    int len1, len2, tmp;\n    while (~scanf ("%d %d", &len1, &len2)){\n        len = 1, len1++, len2++;\n        while (len < len1+len2) len <<= 1;\n        for (int i=0; i<len1; i++){\n            scanf ("%d", &tmp);\n            x1[i] = Complex(tmp, 0.0);\n        }\n        for (int i=0; i<len2; i++){\n            scanf ("%d", &tmp);\n            x2[i] = Complex(tmp, 0.0);\n        }\n        for (int i=len1; i<len; i++)    x1[i] = Complex(0, 0);\n        for (int i=len2; i<len; i++)    x2[i] = Complex(0, 0);\n        FFT(x1, len, 1);\n        FFT(x2, len, 1);\n        for (int i=0; i<len; i++)  x1[i] = x1[i] * x2[i];\n        FFT(x1, len, -1);\n        for (int i=0; i<len; i++)  ans[i] = ll(x1[i].x + 0.5);\n\n        while (ans[len-1]==0 && len>1) len--;\n        printf ("%lld", ans[0]);\n        len1--, len2--;\n        for (int i=1; i<=len1+len2; i++)   printf (" %lld", ans[i]);\n    }\n    return 0;\n}\n\nvoid change(Complex y[], int len){\n    for (int i=1, j=len\/2, k; i<len-1; i++){\n        if (i<j)    swap(y[i], y[j]);\n        k = len\/2;\n        while (j>=k){\n            j -= k;\n            k \/= 2;\n        }\n        if (j<k)    j += k;\n    }\n}\nvoid FFT(Complex y[], int len, int on){\n    change(y, len);                      \/\/\u76f4\u63a5\u5316\u6210\u9012\u5f52\u7684\u6700\u5e95\u4e00\u5c42\n    for (int h=2; h<=len; h<<=1){        \/\/h\u8868\u793a\u8be5\u5c42\u6bcf\u4e2a\u51fd\u6570\u6709h\u4e2a\u5143\u7d20\n        Complex wn(cos(PI*2\/h), sin(PI*2*on\/h));                    \/\/\n        for (int j=0; j<len; j+=h){        \/\/\u6bcf\u5c42\u7684\u7b2c\u4e00\u4e2a\u5143\u7d20\n            Complex w(1, 0);                                   \/\/w\u8868\u793a\u7b2c\u51e0\u5355\u4f4d\u6839\n            for (int k=j; k<j+h\/2; k++){\n                Complex u = y[k], t = w*y[k+h\/2];\n                y[k] = u+t, y[k+h\/2] = u-t;\n                w = w*wn;\n            }\n        }\n    }\n    if (on==-1){\n        for (int i=0; i<len; i++)   y[i].x \/= len;\n    }\n}\n<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362FFT \u5927\u6574\u6570\u4e58\u6cd5\u8ba1\u7b97\u7684\u53d1\u5c55\u5386\u7a0b\u662f\u7f13\u6162\u7684…\u5728\u6734\u7d20\u7684\u5927\u6574\u6570\u4e58\u6cd5\u8ba1\u7b97\u548c\u591a\u9879\u5f0f\u4e58\u6cd5\u4e2d\uff0c\u4e24 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[61],"tags":[],"_links":{"self":[{"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/posts\/954"}],"collection":[{"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/comments?post=954"}],"version-history":[{"count":38,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/posts\/954\/revisions"}],"predecessor-version":[{"id":957,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/posts\/954\/revisions\/957"}],"wp:attachment":[{"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/media?parent=954"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/categories?post=954"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.zyhcoding.club\/index.php\/wp-json\/wp\/v2\/tags?post=954"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}