{"id":607,"date":"2019-02-11T20:23:42","date_gmt":"2019-02-11T12:23:42","guid":{"rendered":"http:\/\/SmokeyDays.top\/wordpress\/?p=607"},"modified":"2019-03-04T20:57:00","modified_gmt":"2019-03-04T12:57:00","slug":"lp3803-%e3%80%90%e6%a8%a1%e6%9d%bf%e3%80%91%e5%a4%9a%e9%a1%b9%e5%bc%8f%e4%b9%98%e6%b3%95%ef%bc%88fft%ef%bc%89","status":"publish","type":"post","link":"http:\/\/SmokeyDays.top\/wordpress\/2019\/02\/11\/lp3803-%e3%80%90%e6%a8%a1%e6%9d%bf%e3%80%91%e5%a4%9a%e9%a1%b9%e5%bc%8f%e4%b9%98%e6%b3%95%ef%bc%88fft%ef%bc%89\/","title":{"rendered":"lp3803 \u3010\u6a21\u677f\u3011\u591a\u9879\u5f0f\u4e58\u6cd5\uff08FFT\uff09"},"content":{"rendered":"\n

*\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\u662f\u4e00\u79cd\u88ab\u5e7f\u6cdb\u8fd0\u7528\u4e8e\u5404\u4e2a\u9886\u57df\u7b97\u6cd5\uff0c\u5c24\u5176\u662f\u901a\u4fe1\u9886\u57df\u3002\u5b83\u7684\u4e3b\u8981\u7528\u9014\u662f\u8ba1\u7b97\u4e24\u4e2a\u51fd\u6570\u7684\u5377\u79ef\u3002\u5728\u8fd9\u91cc\u6211\u4eec\u8ba8\u8bba\u7684\u662f\u7528\u4e8e\u591a\u9879\u5f0f\u4e58\u6cd5\u7684\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\u3002<\/p><\/blockquote>\n\n\n\n

\u4e00\u4e2a\\(n\\)\u6b21\u591a\u9879\u5f0f\u51fd\u6570\\(f\\)\u6709\u4e24\u79cd\u8868\u8fbe\u65b9\u5f0f\uff1a\u70b9\u503c\u8868\u8fbe\u6cd5\u548c\u7cfb\u6570\u8868\u8fbe\u6cd5\u3002
\u7cfb\u6570\u8868\u8fbe\u6cd5\uff1a
\u7cfb\u6570\u8868\u8fbe\u6cd5\u662f\u4e00\u4e2a\u591a\u9879\u5f0f\u51fd\u6570\u6700\u76f4\u63a5\u7684\u8868\u8fbe\u6cd5\u3002
\u5bf9\u4e8e\u4e00\u4e2a\u591a\u9879\u5f0f\u51fd\u6570\uff0c\u5b83\u603b\u662f\u53ef\u4ee5\u8868\u8fbe\u4e3a\u8fd9\u6837\u4e00\u4e2a\u5f62\u5f0f\uff1a
$$f(x)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\u2026a_{n}x^{n}$$
\u70b9\u503c\u8868\u8fbe\u6cd5\uff1a
\u5bf9\u4e8e\u4e00\u4e2a\u591a\u9879\u5f0f\u51fd\u6570\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5b83\u7684\u4e00\u4e2a\u53d6\u503c\u96c6\u5408\uff1a
$$\\{(x_{0},f(x_{0})),(x_{1},f(x_{1})),\u2026(x_{n-1},f(x_{n-1}))\\}$$
\u8fd9\u4e2a\u53d6\u503c\u96c6\u5408\u672c\u8eab\u4e5f\u4ee3\u8868\u4e86\u8fd9\u4e2a\u591a\u9879\u5f0f\u51fd\u6570\u3002\u56e0\u4e3a\uff0c\u5f53\u6211\u4eec\u9009\u53d6\u597d\u4e00\u4e2a\u503c\u4ee5\u540e\uff0c\u603b\u662f\u53ef\u4ee5\u901a\u8fc7\u4e00\u7cfb\u5217\u8ba1\u7b97\u5229\u7528\u8fd9\u4e2a\u53d6\u503c\u96c6\u5408\u5f97\u5230\u8fd9\u4e2a\u503c\u5173\u4e8e\u8fd9\u4e2a\u591a\u9879\u5f0f\u51fd\u6570\u7684\u6620\u5c04\u3002<\/p>\n\n\n\n

\u66b4\u529b\u8ba1\u7b97\u4e24\u4e2a\u591a\u9879\u5f0f\u7684\\(f,g\\)\u7684\u5377\u79ef\u7684\u590d\u6742\u5ea6\u6839\u636e\u5176\u8868\u8fbe\u6cd5\u7684\u4e0d\u540c\u6709\u7740\u4e0d\u540c\u7684\u590d\u6742\u5ea6\u3002
\u66b4\u529b\u8ba1\u7b97\u4e24\u4e2a\u7cfb\u6570\u8868\u8fbe\u7684\u591a\u9879\u5f0f\u7684\u5377\u79ef\u7684\u590d\u6742\u5ea6\u662f\\(O(n^2)\\)\u7684\uff0c\u5177\u4f53\u6765\u8bf4\u5c31\u662f\u5c06\u5b83\u4eec\u50cf\u7ad6\u5f0f\u4e58\u6cd5\u4e00\u6837\u76f8\u4e58\u3002\u5f62\u5982\uff1a
$$fg=\\sum_{i=0}^{n}\\sum_{i=0}^{m}a_{i}b_{j}x^{i+j}$$
\u66b4\u529b\u8ba1\u7b97\u4e24\u4e2a\u53d6\u503c\u76f8\u540c\u7684\u70b9\u503c\u8868\u8fbe\u7684\u591a\u9879\u5f0f\u7684\u5377\u79ef\u7684\u590d\u6742\u5ea6\u662f\\(O(n)\\)\u7684\uff0c\u5177\u4f53\u6765\u8bf4\u5c31\u662f\u5c06\u5b83\u4eec\u7684\u6bcf\u4e2a\u70b9\u503c\u8868\u8fbe\u5f0f\u7684\u6bcf\u4e00\u4f4d\u7684\u7ed3\u679c\u76f8\u4e58\u3002
$$fg={(x_{0},f(x_{0})g(x_{0})),(x_{1},f(x_{1})g(x_{1})),\u2026(x_{n+m},f(x_{n+m})g(x_{n+m})))}$$<\/p>\n\n\n\n

\u5728\u4e00\u4e9b\u5e94\u7528\u60c5\u51b5\uff0c\u6211\u4eec\u9700\u8981\u7528\u6bd4\\(O(n^2)\\)\u66f4\u4f18\u7684\u590d\u6742\u5ea6\u8ba1\u7b97\u4e24\u4e2a\u591a\u9879\u5f0f\u51fd\u6570\u7684\u7cfb\u6570\u8868\u8fbe\u6cd5\u5377\u79ef\u3002
\n\u5f88\u5bb9\u6613\u53ef\u4ee5\u60f3\u5230\u7684\u662f\u5c06\u4e00\u4e2a\u591a\u9879\u5f0f\u51fd\u6570\u7684\u7cfb\u6570\u8868\u8fbe\u5f0f\u8f6c\u6362\u4e3a\u70b9\u503c\u8868\u8fbe\u5f0f\uff0c\u8ba1\u7b97\u5377\u79ef\u540e\u518d\u8f6c\u6362\u56de\u53bb\uff0c\u8fd9\u6837\u53ef\u80fd\u5c31\u80fd\u591f\u4ee5\u4e00\u4e2a\u53ef\u4ee5\u63a5\u53d7\u7684\u590d\u6742\u5ea6\u8ba1\u7b97\u4e24\u4e2a\u591a\u9879\u5f0f\u7684\u7cfb\u6570\u8868\u8fbe\u5f0f\u3002
\n\u590d\u6742\u5ea6\u7684\u74f6\u9888\u5728\u4e8e\u70b9\u503c\u8868\u8fbe\u5f0f\u548c\u7cfb\u6570\u8868\u8fbe\u5f0f\u7684\u76f8\u4e92\u8f6c\u6362\u3002
\n\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362(Fast Fourier Transform)\u6cdb\u6307\u662f\u4e00\u79cd\u901a\u8fc7\u9009\u53d6\u5de7\u5999\u7684\u53d6\u503c\uff0c\u4f7f\u5f97\u70b9\u503c\u8868\u8fbe\u5f0f\u53ef\u4ee5\u5feb\u901f\u5730\u548c\u7cfb\u6570\u8868\u8fbe\u5f0f\u76f8\u4e92\u8f6c\u6362\u7684\u7b97\u6cd5\u3002\u5176\u4e2d\uff0c\u5c06\u7cfb\u6570\u8868\u8fbe\u5f0f\u8f6c\u6362\u4e3a\u70b9\u503c\u8868\u8fbe\u5f0f\u7684\u8f6c\u6362\u88ab\u79f0\u4e3a\u5feb\u901f\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u5c06\u70b9\u503c\u8868\u8fbe\u5f0f\u8f6c\u6362\u4e3a\u7cfb\u6570\u8868\u8fbe\u5f0f\u7684\u8f6c\u6362\u88ab\u79f0\u4e3a\u5feb\u901f\u9006\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\u3002
\n\u4e3a\u4e86\u5b8c\u6210\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u6211\u4eec\u9700\u8981\u4e86\u89e3\u6211\u4eec\u5373\u5c06\u9009\u53d6\u7684\u4e00\u7ec4\u53d6\u503c\u2014\u2014\u5355\u4f4d\u590d\u6839\uff0c\u4ee5\u53ca\u5b83\u7684\u6027\u8d28\u3002<\/p>\n\n\n\n

\u65b9\u7a0b\\(x^n=1\\)\u7684\u6240\u6709\u89e3\u6784\u6210\u4e86\\(n\\)\u4e2a\u5355\u4f4d\u590d\u6839\uff0c\u8bb0\u4f5c\\(\\omega_{n}\\)\uff0c\u5176\u4e2d\u7684\u7b2c\\(i\\)\u9879\u8bb0\u4f5c\\(\\omega_{n}^{i}\\)
\n\u591a\u9879\u5f0f\u7684\u5355\u4f4d\u590d\u6839\u5177\u6709\u4e00\u4e9b\u5947\u5999\u7684\u6027\u8d28\uff1a
\n\u6d88\u53bb\u5f15\u7406\uff1a
\n$$\\omega_{dn}^{dk}=e^{\\frac{2\\pi dki}{dn}}=e^{\\frac{2\\pi ki}{n}}=\\omega_{n}^{k}$$
\n\u6298\u534a\u5f15\u7406\uff1a
\n$$(\\omega_{n}^{k+\\frac{n}{2}})^2=(e^{\\frac{2\\pi (k+\\frac{n}{2})i}{n}})^2=(e^{\\frac{2\\pi ki}{n}}*(e^{\\pi i}))^2=(-e^{\\frac{2\\pi ki}{n}})^2=(-\\omega_{n}^{k})^2=(\\omega_{n}^{k})^2$$
\n\u6c42\u548c\u5f15\u7406\uff1a
\n$$\\sum_{j=0}^{n-1}(\\omega_{n}^{k})^j=\\frac{(\\omega_{n}^{k})^n-1}{\\omega_{n}^{k}-1}=\\frac{(e^{\\frac{2\\pi ki}{n}})^n-1}{\\omega_{n}^{k}-1}=\\frac{0}{\\omega_{n}^{k}-1}=0\\ (k>0)$$
\n\u6298\u534a\u5f15\u7406\u7684\u4e00\u4e2a\u63a8\u8bba\uff1a
\n$$\\sum_{k=0}^{n-1}\\omega_{n}^{k}=0$$<\/p>\n\n\n\n

\u73b0\u5728\u6211\u4eec\u8003\u8651\u5982\u4f55\u901a\u8fc7\u9009\u53d6\u5355\u4f4d\u590d\u6839\u6765\u5b8c\u6210\u5feb\u901f\u5730\u5c06\u51fd\u6570\u8868\u8fbe\u5728\u7cfb\u6570\u8868\u8fbe\u6cd5\u548c\u53d6\u503c\u8868\u8fbe\u6cd5\u4e4b\u95f4\u8f6c\u6362\u3002
\u6211\u4eec\u4e0d\u7531\u81ea\u4e3b\u5730\u9009\u62e9\u5206\u6cbb\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u3002\u4e3a\u4e86\u5b9e\u73b0\u5947\u5076\u5206\u6cbb\uff0c\u6211\u4eec\u9700\u8981\u5c06\u4e00\u4e2a\u51fd\u6570\u7684\u7cfb\u6570\u8868\u8fbe\u5f0f\u6269\u5145\u5230\\(2^k-1\\)\u6b21\u3002
\u7136\u540e\u6211\u4eec\u5206\u522b\u8003\u8651FFT\u548c\u9006FFT
\u6211\u4eec\u4f5c\u51fa\u4e86\u5982\u4e0b\u5b9a\u4e49\uff1a
$$f(x)=\\sum_{i=0}^{(2^n)-1}a_{i}x^{i}$$
$$f_0(x)=\\sum_{i=0}^{2^{(n-1)}}a_{2i}x^{i}$$ $$f_1(x)=\\sum_{i=0}^{2^{(n-1)}}a_{2i+1}x^{i}$$
\u5219\uff1a
$$f(x)=f_0(x^2)+xf_1(x^2)$$
\u9996\u5148\u6211\u4eec\u8003\u8651\u5982\u4f55\u53d6\u503c\u2014\u2014\u4e0b\u9762\u6211\u4eec\u8bc1\u660e\uff0c\u5f53\u6211\u4eec\u5df2\u7ecf\u62e5\u6709\u4e86\u4e00\u4e2a\u51fd\u6570\u7684\u5947\u5076\u5206\u6cbb\u540e\u7684\u51fd\u6570\u7684\u70b9\u503c\u8868\u8fbe\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u7ebf\u6027\u65f6\u95f4\u5185\u5f97\u5230\u8fd9\u4e2a\u51fd\u6570\u7684\u70b9\u503c\u8868\u8fbe\u5f0f\u3002
\u5bf9\u4e8e\u67d0\u4e00\u4e2a\u53d6\u503c\\(f(\\omega_{n}^{k})\\ (k\\le\\frac{n}{2})\\)\uff0c\u6211\u4eec\u6709\uff1a
$$f(\\omega_{n}^{k})=f_0(\\omega_{n}^{2k})+\\omega_{n}^{k}f_1(\\omega_{n}^{2k})$$
$$f(\\omega_{n}^{k})=f_0(\\omega_{\\frac{n}{2}}^{k})+\\omega_{n}^{k}f_1(\\omega_{\\frac{n}{2}}^{k})$$
$$f(\\omega_{n}^{k+\\frac{n}{2}})=f_0(\\omega_{n}^{2k+n})+\\omega_{n}^{k+\\frac{n}{2}}f_1(\\omega_{n}^{2k+n})$$
$$f(\\omega_{n}^{k+\\frac{n}{2}})=f_0(\\omega_{\\frac{n}{2}}^{k})-\\omega_{n}^{k}f_1(\\omega_{\\frac{n}{2}}^{k})$$
\u901a\u8fc7\u8fd9\u6837\u5730\u9012\u5f52\u53d6\u503c\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\\(O(nlog_2n)\\)\u7684\u65f6\u95f4\u5185\u5c06\u4e00\u4e2a\u591a\u9879\u5f0f\u4ece\u5b83\u7684\u7cfb\u6570\u8868\u8fbe\u6cd5\u8f6c\u5316\u4e3a\u70b9\u503c\u8868\u8fbe\u6cd5\u3002<\/p>\n\n\n\n

\u7136\u540e\u8003\u8651\u5982\u4f55\u5feb\u901f\u5730\u5c06\u70b9\u503c\u8868\u8fbe\u6cd5\u8f6c\u5316\u4e3a\u7cfb\u6570\u8868\u8fbe\u6cd5\u3002
\n\u6211\u4eec\u5c06\u70b9\u503c\u8868\u8fbe\u6cd5\u83b7\u5f97\u7684\u53d6\u503c\u4f5c\u4e3a\u7cfb\u6570\uff0c\u8bbe\u51fa\u4e86\u4e00\u4e2a\u51fd\u6570\\(h\\)\uff0c\u5b83\u7684\u6bcf\u4e00\u9879\u7684\u7cfb\u6570\u90fd\u662f\u539f\u51fd\u6570\u7684\u70b9\u503c\u8868\u8fbe\u5f0f\u4e2d\u7684\u53d6\u503c\uff1a
\n$$f'(x)=\\sum_{i=0}^{2^n-1}(a_{i}\\sum_{j=0}^{2^n-1}\\omega_{n}^{ij}x^{j})$$
\n\u636e\u8d85\u56fe\u7075\u673a\u6240\u8bf4\uff0c\u6211\u4eec\u5c06\\(\\omega_{n}^{-k}\\)\u4ee3\u5165\u8fd9\u4e2a\u51fd\u6570\uff0c\u53ef\u4ee5\u5f97\u5230\u5982\u4e0b\u5f0f\u5b50\uff1a
\n$$f'(\\omega_{n}^{-k})=\\sum_{i=0}^{2^n-1}(a_{i}\\sum_{j=0}^{2^n-1}\\omega_{n}^{ij}\\omega_{n}^{-kj})$$
\n$$f'(\\omega_{n}^{-k})=\\sum_{i=0}^{2^n-1}(a_{i}\\sum_{j=0}^{2^n-1}\\omega_{n}^{j(i-k)})$$
\n\u7531\u6c42\u548c\u5f15\u7406\u53ef\u5f97\uff0c\u5f53\u4e14\u4ec5\u5f53\\(i-k=0\\)\u65f6\uff0c\\(\\omega_{n}^{j(i-k)}=1\\)\u3002\u5176\u4ed6\u65f6\u5019\u5747\u6709\\(\\omega_{n}^{j(i-k)}=0\\)
\n\u6545\u6709\uff1a
\n$$f'(\\omega_{n}^{-k})=a_{k}\\sum_{j=0}^{2^n-1}\\omega_{n}^{0}=na_{k}$$
\n\u6545\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5c06\u51fd\u6570\\(f’\\)\u901a\u8fc7\u53d6\u6bcf\u4e2a\u5355\u4f4d\u590d\u6839\u7684\u5012\u6570\u6765\u6c42\u5f97\u539f\u51fd\u6570\u7684\u7cfb\u6570\u3002
\n\u800c\u8fd9\u4e00\u8fc7\u7a0b\u4e5f\u53ef\u4ee5\u540c\u6837\u53ef\u4ee5\u901a\u8fc7\u4e0a\u8ff0\u7684\u5206\u6cbb\u601d\u60f3\u6765\u4f18\u5316\u3002<\/p>\n\n\n\n

\u901a\u8fc7\u5feb\u901f\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\u5c06\u51fd\u6570\u4ece\u7cfb\u6570\u8868\u8fbe\u5f0f\u8f6c\u5316\u6210\u70b9\u503c\u8868\u8fbe\u5f0f\uff0c\u7136\u540e\u5c06\u70b9\u503c\u8868\u8fbe\u5f0f\u5377\u79ef\u8d77\u6765\uff0c\u518d\u5c06\u5377\u79ef\u4ee5\u540e\u7684\u70b9\u503c\u8868\u8fbe\u5f0f\u901a\u8fc7\u5feb\u901f\u79bb\u6563\u5085\u91cc\u53f6\u9006\u53d8\u6362\u8f6c\u5316\u4e3a\u7cfb\u6570\u8868\u8fbe\u5f0f\uff0c\u8fd9\u6837\u5c31\u5728\\(O(nlog_2n)\\)\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5185\u5b8c\u6210\u4e86\u8ba1\u7b97\u591a\u9879\u5f0f\u51fd\u6570\u7684\u7cfb\u6570\u8868\u8fbe\u5f0f\u7684\u5377\u79ef\u3002<\/p>\n\n\n\n

#include<iostream>\n#include<cstdio>\n#include<cmath>\n#define MID (LEN>>1)\n#define PI 3.141592653589793238462\n\nstruct Complex{\n\tdouble r;\n\tdouble i;\n\tComplex(double rIN=0.0,double iIN=0.0):r(rIN),i(iIN){}\n\tinline Complex operator+(const Complex &B)const{\n\t\treturn (Complex){r+B.r,i+B.i};\n\t}\n\tinline Complex operator-(const Complex &B)const{\n\t\treturn (Complex){r-B.r,i-B.i};\n\t}\n\tinline Complex operator*(const Complex &B)const{\n\t\treturn (Complex){r*B.r-i*B.i,r*B.i+i*B.r};\n\t}\n}a[4000005],b[4000005],c[4000005];\n\ninline void FFT(int LEN,Complex *A,Complex *B,int typ){\n\tif(LEN==1){\n\t\tB[0]=A[0];\n\t\treturn;\n\t}\n\/\/\t\u5947\u5076\u9879\u5206\u6cbb \n\tfor(int i=0;i<MID;++i){\n\t\tB[i]=A[i<<1];\n\t\tB[i+MID]=A[i<<1|1];\n\t}\n\tfor(int i=0;i<LEN;++i){\n\t\tA[i]=B[i];\n\t}\n\tFFT(MID,A,B,typ),FFT(MID,A+MID,B+MID,typ);\n\tComplex bs(cos((PI*2.0)\/LEN),typ*sin((PI*2.0)\/LEN)),nw(1.0,0.0);\n\tfor(int i=0;i<MID;++i){\n\t\tA[i]=B[i]+nw*B[i+MID];\n\t\tA[i+MID]=B[i]-nw*B[i+MID];\n\t\tnw=nw*bs;\n\t}\n\tfor(int i=0;i<LEN;++i){\n\t\tB[i]=A[i];\n\t}\n}\n\n\nint n,m,cnt;\nvoid init(){\n\tscanf(\"%d%d\",&n,&m);\n\tfor(int i=0;i<=n;++i){\n\t\tscanf(\"%lf\",&a[i].r);\n\t}\n\tfor(int i=0;i<=m;++i){\n\t\tscanf(\"%lf\",&b[i].r);\n\t}\n\tcnt=1;\n\twhile(cnt<=n+m){\n\t\tcnt<<=1;\n\t}\n\tFFT(cnt,a,c,1);\n\tFFT(cnt,b,c,1);\n\tfor(int i=0;i<cnt;++i){\n\t\ta[i]=a[i]*b[i];\n\t}\n\tFFT(cnt,a,c,-1);\n\tfor(int i=0;i<=n+m;++i){\n\t\tprintf(\"%d \",(int)(a[i].r\/cnt+0.5));\n\t}\n}\n\nint main(){\n\tinit();\n\treturn 0;\n}<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
*\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\u662f\u4e00\u79cd\u88ab\u5e7f\u6cdb\u8fd0\u7528\u4e8e\u5404\u4e2a\u9886\u57df\u7b97\u6cd5\uff0c\u5c24\u5176\u662f\u901a\u4fe1\u9886\u57df\u3002\u5b83\u7684\u4e3b\u8981\u7528\u9014\u662f\u8ba1\u7b97\u4e24\u4e2a\u51fd\u6570\u7684\u5377\u79ef\u3002\u5728\u8fd9\u91cc\u6211\u4eec\u8ba8 … <\/p>\n
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