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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fürer's algorithm ： ウィキペディア英語版 Fürer's algorithm Fürer's algorithm is an integer multiplication algorithm for very large numbers possessing a very low asymptotic complexity. It was created in 2007 by Swiss mathematician Martin Fürer of Pennsylvania State University〔Fürer, M. (2007). "(Faster Integer Multiplication )" in Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, June 11–13, 2007, San Diego, California, USA〕 as an asymptotically faster (when analysed on a multitape Turing machine) algorithm than its predecessor, the Schönhage–Strassen algorithm published in 1971.〔A. Schönhage and V. Strassen, "Schnelle Multiplikation großer Zahlen", Computing 7 (1971), pp. 281–292.〕 The predecessor to the Fürer algorithm, the Schönhage–Strassen algorithm, used fast Fourier transforms to compute integer products in time $O(n\; \backslash log\; n\; \backslash log\; \backslash log\; n)$ (in big O notation) and its authors, Arnold Schönhage and Volker Strassen, also conjectured a lower bound for the problem of Here, $n$ denotes the total number of bits in the two input numbers. Fürer's algorithm reduces the gap between these two bounds: it can be used to multiply binary integers of length $n$ in time $n\; \backslash log\; n\backslash ,2^$ (or by a circuit with that many logic gates), where represents the iterated logarithm operation. However, the difference between the $\backslash log\; \backslash log\; n$ and $2^$ factors in the time bounds of the Schönhage–Strassen algorithm and the Fürer algorithm for realistic values of $n$ is very small.〔 In 2008, Anindya De, Chandan Saha, Piyush Kurur and Ramprasad Saptharishi〔Anindya De, Piyush P Kurur, Chandan Saha, Ramprasad Saptharishi. Fast Integer Multiplication Using Modular Arithmetic. Symposium on Theory of Computation (STOC) 2008. 〕 gave a similar algorithm that relies on modular arithmetic instead of complex arithmetic to achieve the same running time. In 2014, David Harvey, Joris van der Hoeven, and Grégoire Lecerf〔David Harvey, Joris van der Hoeven, and Grégoire Lecerf, "Even faster integer multiplication", 2014, 〕 showed that an optimized version of Fürer's algorithm achieves a running time of $O(n\backslash log\; n2^)$, making the implied constant in the $O(\backslash log^$ * n) exponent explicit. They also gave a modification to Fürer's algorithm that achieves $O(n\backslash log\; n2^)$ In 2015 Svyatoslav Covanov and Emmanuel Thomé provided another modifications that achieves same running time.〔Svyatoslav Covanov and Emmanuel Thomé, "Fast arithmetic for faster integer multiplication", 2015 〕 Yet, as all the other implementation, it's still not practical. == See also == * Number-theoretic transform 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fürer's algorithm」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.