Content deleted Content added
Dominic3203 (talk | contribs) m →Computational complexity of multiplication: Fixed typo Tags: Mobile edit Mobile app edit Android app edit App section source |
Citation bot (talk | contribs) Altered title. | Use this bot. Report bugs. | Suggested by Dominic3203 | Category:Multiplication | #UCB_Category 33/42 |
||
Line 272:
{{unsolved|computer science|What is the fastest algorithm for multiplication of two <math>n</math>-digit numbers?}}
A line of research in [[theoretical computer science]] is about the number of single-bit arithmetic operations necessary to multiply two <math>n</math>-bit integers. This is known as the [[computational complexity]] of multiplication. Usual algorithms done by hand have asymptotic complexity of <math>O(n^2)</math>, but in 1960 [[Anatoly Karatsuba]] discovered that better complexity was possible (with the [[Karatsuba algorithm]]).<ref>{{cite web | url=https://youtube.com/watch?v=AMl6EJHfUWo | title= The Genius Way Computers Multiply Big Numbers| website=[[YouTube]]}}</ref>
Currently, the algorithm with the best computational complexity is a 2019 algorithm of [[David Harvey (mathematician)|David Harvey]] and [[Joris van der Hoeven]], which uses the strategies of using [[number-theoretic transform]]s introduced with the [[Schönhage–Strassen algorithm]] to multiply integers using only <math>O(n\log n)</math> operations.<ref>{{cite journal | last1 = Harvey | first1 = David | last2 = van der Hoeven | first2 = Joris | author2-link = Joris van der Hoeven | doi = 10.4007/annals.2021.193.2.4 | issue = 2 | journal = [[Annals of Mathematics]] | mr = 4224716 | pages = 563–617 | series = Second Series | title = Integer multiplication in time <math>O(n \log n)</math> | volume = 193 | year = 2021| s2cid = 109934776 | url = https://hal.archives-ouvertes.fr/hal-02070778v2/file/nlogn.pdf }}</ref> This is conjectured to be the best possible algorithm, but lower bounds of <math>\Omega(n\log n)</math> are not known.
| |||