Computational complexity of matrix multiplication: Difference between revisions

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| date=Mar 1990

| doi=10.1016/S0747-7171(08)80013-2

| doi-access=free

}}</ref>

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| 2010 || 2.3737 || Stothers<ref>

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Using a naive lower bound and schoolbook matrix multiplication for the upper bound, one can straightforwardly conclude that {{math|2 ≤ ω ≤ 3}}. Whether {{math|1=ω = 2}} is a major open question in [[theoretical computer science]], and there is a line of research developing matrix multiplication algorithms to get improved bounds on {{math|ω}}.

The current best bound on {{math|ω}} is {{math|ω < 2.3728596}}, by Josh Alman and [[Virginia Vassilevska Williams]].<ref name="aw20"/> This algorithm, like all other recent algorithms in this line of research, uses the ''laser method'', a generalization of the Coppersmith–Winograd algorithm, which was given by [[Don Coppersmith]] and [[Shmuel Winograd]] in 1990 and was the best matrix multiplication algorithm until 2010.<ref name="coppersmith">{{Citation|doi=10.1016/S0747-7171(08)80013-2 |title=Matrix multiplication via arithmetic progressions |url=http://www.cs.umd.edu/~gasarch/TOPICS/ramsey/matrixmult.pdf |year=1990 |last1=Coppersmith |first1=Don |last2=Winograd |first2=Shmuel |journal=Journal of Symbolic Computation |volume=9|issue=3|pages=251|doi-access=free }}</ref> The conceptual idea of these algorithms are similar to Strassen's algorithm: a way is devised for multiplying two {{math|''k'' × ''k''}}-matrices with fewer than {{math|''k''³}} multiplications, and this technique is applied recursively. The laser method has limitations to its power, and cannot be used to show that {{math|ω < 2.3725}}.<ref>{{Cite journal|last=Ambainis|first=Andris|last2=Filmus|first2=Yuval|last3=Le Gall|first3=François|date=2015-06-14|title=Fast Matrix Multiplication: Limitations of the Coppersmith-Winograd Method|url=https://doi.org/10.1145/2746539.2746554|journal=Proceedings of the forty-seventh annual ACM symposium on Theory of Computing|series=STOC '15|___location=Portland, Oregon, USA|publisher=Association for Computing Machinery|pages=585–593|doi=10.1145/2746539.2746554|isbn=978-1-4503-3536-2}}</ref>

=== Group theory reformulation of matrix multiplication algorithms ===