Computational complexity of matrix multiplication: Difference between revisions

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== Simple algorithms ==

If ''A'', ''B'' are two {{math|''n'' × ''n''}} matrices over a field, then their product ''AB'' is also an {{math|''n'' × ''n''}} matrix over that field, defined entrywise as

:<math>

(AB)_{ij} = \sum_{k = 1}^n A_{ik} B_{kj}.

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'''output''' ''C'' (as A*B)

This [[algorithm]] requires, in the [[worst-case complexity|worst case]], {{tmath|n^3}} multiplications of scalars and {{tmath|n^3 - n^2}} additions of scalars for computing the product of two square {{math|''n''×''n''}} matrices. Its [[computational complexity]] is therefore {{tmath|O(n^3)}}, in a [[model of computation]] where field operations (addition and multiplication) take constant time (in practice, this is the case for [[floating point]] numbers, but not necessarily for integers).

=== Strassen's algorithm ===

{{Main|Strassen algorithm}}

Strassen's algorithm improves on naive matrix multiplication through a [[Divide-and-conquer algorithm|divide-and-conquer]] approach. The key observation is that multiplying two {{math|2 × 2}} matrices can be done with only 7seven multiplications, instead of the usual 8eight (at the expense of 11 additional addition and subtraction operations). This means that, treating the input {{math|''n''×''n''}} matrices as [[Block matrix|block]] {{math|2 × 2}} matrices, the task of multiplying two {{math|''n''×''n''}} matrices can be reduced to 7seven subproblems of multiplying two {{math|''n/2''×''n/2''}} matrices. Applying this recursively gives an algorithm needing <math>O( n^{\log_{2}7}) \approx O(n^{2.807})</math> field operations.

Unlike algorithms with faster asymptotic complexity, Strassen's algorithm is used in practice. The [[numerical stability]] is reduced compared to the naive algorithm,<ref>{{cite journal | last1=Miller | first1=Webb | title=Computational complexity and numerical stability | citeseerx = 10.1.1.148.9947 | year=1975 | journal=SIAM News | volume=4 | issue=2 | pages=97–107 | doi=10.1137/0204009}}</ref> but it is faster in cases where {{math|''n'' > 100}} or so<ref name="skiena">{{cite book |first=Steven |last=Skiena |date=2012 |author-link=Steven Skiena |title=The Algorithm Design Manual |url=https://archive.org/details/algorithmdesignm00skie_772 |url-access=limited |publisher=Springer |pages=[https://archive.org/details/algorithmdesignm00skie_772/page/n56 45]–46, 401–403 |doi=10.1007/978-1-84800-070-4_4|chapter=Sorting and Searching |isbn=978-1-84800-069-8 }}</ref> and appears in several libraries, such as [[Basic Linear Algebra Subprograms|BLAS]].<ref>{{cite book |last1=Press |first1=William H. |last2=Flannery |first2=Brian P. |last3=Teukolsky |first3=Saul A. |author3-link=Saul Teukolsky |last4=Vetterling |first4=William T. |title=Numerical Recipes: The Art of Scientific Computing |publisher=[[Cambridge University Press]] |edition=3rd |isbn=978-0-521-88068-8 |year=2007 |page=[https://archive.org/details/numericalrecipes00pres_033/page/n131 108]|title-link=Numerical Recipes }}</ref> Fast matrix multiplication algorithms cannot achieve ''component-wise stability'', but some can be shown to exhibit ''norm-wise stability''.<ref name="bdl16">{{cite journal | last1=Ballard | first1=Grey | last2=Benson | first2=Austin R. | last3=Druinsky | first3=Alex | last4=Lipshitz | first4=Benjamin | last5=Schwartz | first5=Oded | title=Improving the numerical stability of fast matrix multiplication | year=2016 | journal=SIAM Journal on Matrix Analysis and Applications | volume=37 | issue=4 | pages=1382–1418 | doi=10.1137/15M1032168 | arxiv=1507.00687| s2cid=2853388 }}</ref> It is very useful for large matrices over exact domains such as [[finite field]]s, where numerical stability is not an issue.

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| pages=234&ndash;235

| date=Jun 1979

| url-access=subscription

}}</ref>

|-

| 1981 || 2.522 || [[Arnold Schönhage|Schönhage]]<ref>

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

[[Henry Cohn]], [[Robert Kleinberg]], [[Balázs Szegedy]] and [[Chris Umans]] put methods such as the Strassen and Coppersmith–Winograd algorithms in an entirely different [[group theory|group-theoretic]] context, by utilising triples of subsets of finite groups which satisfy a disjointness property called the [[Triple product property|triple product property (TPP)]]. They also give conjectures that, if true, would imply that there are matrix multiplication algorithms with essentially quadratic complexity. This implies that the optimal exponent of matrix multiplication is 2, which most researchers believe is indeed the case.<ref name="robinson"/> One such conjecture is that families of [[wreath product]]s of [[Abelian group]]s with symmetric groups realise families of subset triples with a simultaneous version of the TPP.<ref>{{Cite book | last1 = Cohn | first1 = H. | last2 = Kleinberg | first2 = R. | last3 = Szegedy | first3 = B. | last4 = Umans | first4 = C. | chapter = Group-theoretic Algorithms for Matrix Multiplication | doi = 10.1109/SFCS.2005.39 | title = 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05) | pages = 379 | year = 2005 | arxiv = math/0511460 | isbn = 0-7695-2468-0 | s2cid = 41278294 | url = https://authors.library.caltech.edu/23966/ }}</ref><ref>{{cite book |first1=Henry |last1=Cohn |first2=Chris |last2=Umans |chapter=A Group-theoretic Approach to Fast Matrix Multiplication |arxiv=math.GR/0307321 |title=Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 11–14 October 2003 |year=2003 |publisher=IEEE Computer Society |pages=438–449 |doi=10.1109/SFCS.2003.1238217 |isbn=0-7695-2040-5 |s2cid=5890100 }}</ref> Several of their conjectures have since been disproven by Blasiak, Cohn, Church, Grochow, Naslund, Sawin, and Umans using the Slice Rank method.<ref name=":0">{{Cite book | last1 = Blasiak | first1 = J. | last2 = Cohn | first2 = H. | last3 = Church | first3 = T. | last4 = Grochow | first4 = J. | last5 = Naslund | first5= E. | last6 = Sawin | first6 = W. | last7=Umans | first7= C.| chapter= On cap sets and the group-theoretic approach to matrix multiplication | doi = 10.19086/da.1245 | title = Discrete Analysis | year = 2017 | page = 1245 | s2cid = 9687868 | url = http://discreteanalysisjournal.com/article/1245-on-cap-sets-and-the-group-theoretic-approach-to-matrix-multiplication}}</ref> Further, Alon, Shpilka and [[Chris Umans]] have recently shown that some of these conjectures implying fast matrix multiplication are incompatible with another plausible conjecture, the [[sunflower conjecture]],<ref>{{cite journal |journal=Electronic Colloquium on Computational Complexity |date=April 2011 |author1-link=Noga Alon |last1=Alon |first1=N. |last2=Shpilka |first2=A. |last3=Umans |first3=C. |url=http://eccc.hpi-web.de/report/2011/067/ |title=On Sunflowers and Matrix Multiplication |id=TR11-067 }}</ref> which in turn is related to the [[Cap set#Matrix multiplication algorithms|cap set problem.]]<ref name=":0" />

=== Lower bounds for ω ===

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| volume = 114

| year = 2023}}</ref><ref>{{cite journal |last1=Makarov |first1=O. M. |title=An algorithm for multiplying 3×3 matrices |journal=Zhurnal Vychislitel'noi Matematiki I Matematicheskoi Fiziki |volume=26 |issue=2 |year=1986 |pages=293–294 |url=https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=zvmmf&paperid=4056&option_lang=eng |access-date=5 October 2022}}

:Also in {{cite journal |doi=10.1016/0041-5553(86)90203-X |title=An algorithm for multiplying 3×3 matrices |year=1986 |last1=Makarov |first1=O. M. |journal=USSR Computational Mathematics and Mathematical Physics |volume=26 |pages=179–180 }}</ref> (23 if non-commutative<ref>{{Cite journal |last=Laderman |first=Julian D. |date=1976 |title=A noncommutative algorithm for multiplying 3×3 matrices using 23 multiplications |url=https://www.ams.org/bull/1976-82-01/S0002-9904-1976-13988-2/ |journal=Bulletin of the American Mathematical Society |language=en |volume=82 |issue=1 |pages=126–128 |doi=10.1090/S0002-9904-1976-13988-2 |issn=0002-9904|doi-access=free }}</ref>). The lower bound of multiplications needed is 2''mn''+2''n''−''m''−2 (multiplication of ''n''×''m''- matrices with ''m''×''n''- matrices using the substitution method, ''<math>m''⩾'' \ge n''⩾3 \ge 3</math>), which means n=3 case requires at least 19 multiplications and n=4 at least 34.<ref>{{Cite journal |last=Bläser |first=Markus |date=February 2003 |title=On the complexity of the multiplication of matrices of small formats |journal=Journal of Complexity |language=en |volume=19 |issue=1 |pages=43–60 |doi=10.1016/S0885-064X(02)00007-9|doi-access=free }}</ref> For n=2 optimal 7seven multiplications and 15 additions are minimal, compared to only 4four additions for 8eight multiplications.<ref>{{Cite journal |last=Winograd |first=S. |date=1971-10-01 |title=On multiplication of 2 × 2 matrices |journal=Linear Algebra and Its Applications |language=en |volume=4 |issue=4 |pages=381–388 |doi=10.1016/0024-3795(71)90009-7 |issn=0024-3795|doi-access=free }}</ref><ref>{{Cite book |last=L. |first=Probert, R. |url=http://worldcat.org/oclc/1124200063 |title=On the complexity of matrix multiplication |date=1973 |publisher=University of Waterloo |oclc=1124200063}}</ref>

==See also==