Ant colony optimization for triple product property triples to fast matrix multiplication

Lai, Xinsheng; Zhou, Yuren; Xiang, Yi

doi:10.1007/s00500-016-2259-y

Ant colony optimization for triple product property triples to fast matrix multiplication

Methodologies and Application
Published: 11 July 2016

Volume 21, pages 7159–7171, (2017)
Cite this article

Soft Computing Aims and scope Submit manuscript

Xinsheng Lai^1,2,
Yuren Zhou³ &
Yi Xiang³

293 Accesses
4 Citations
Explore all metrics

Abstract

The group-theoretic approach to fast matrix multiplication has been introduced by Cohn and Umans (proceedings of 44th annual IEEE symposium on foundations of computer science, IEEE Computer Society, 2003). In this approach, there is a challenging problem. It is to find three subsets of a given group satisfying the so-called triple product property such that the product of their sizes is as large as possible. For this challenging problem, some brute-force algorithm has been proposed, which is exact but time-consuming. The ant colony optimization is a randomized heuristic algorithm and finds extensive applications in many fields. We use it to solve the problem of searching for three subsets of a given group such that they satisfy the triple product property and the product of their sizes reaches the maximum. Experimental results show that the ant colony optimization is efficient for this problem. Using this approach, we obtain three subsets for each nonabelian group of order not larger than 44 which satisfy the triple product property and the product of whose sizes reaches the best value found so far.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic

$34.99 /Month

Get 10 units per month
Download Article/Chapter or eBook
1 Unit = 1 Article or 1 Chapter
Cancel anytime

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Ant Colony Optimization: Overview and Recent Advances

Two-Dimensional Pheromone in Ant Colony Optimization

Ant Colony Optimization—Recent Variants, Application and Perspectives

Discover the latest articles, news and stories from top researchers in related subjects.

Artificial Intelligence

References

Ambainis A, Filmus Y, Le Gall F (2015) Fast matrix multiplication: limitations of the coppersmith-winograd method. In: Proceedings of the forty-seventh annual acm symposium on theory of computing, pp 585–593
Bini D, Capovani M, Romani F, Lotti G (1979) $\text{ O }(n^{2.7799})$ complexity for $n\times n$ approximate matrix multiplication. Inf Process Lett 8(5):234–235
Article MATH Google Scholar
Bshouty NH (1995) On the additive complexity of $2\times 2$ matrix multiplication. Inf Process Lett 56(6):329–335
Article MATH Google Scholar
Cohn H, Umans C (2003) A group-theoretic approach to fast matrix multiplication. In: Proceedings of 44th annual IEEE symposium on foundations of computer science, IEEE Computer Society, pp 438–449
Coppersmith D, Winograd S (1982) On the asymptotic complexity of matrix multiplication. SIAM J Comput 11(3):472–492
Article MATH MathSciNet Google Scholar
Coppersmith D, Winograd S (1990) Matrix multiplication via arithmetic progressions. J Symb Comput 9(3):251–280
Article MATH MathSciNet Google Scholar
Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern Part B Cybern 26(1):29–41
Article Google Scholar
Hart S, Hedtke I, Müller-Hannemann M, Murthy S (2015) A fast search algorithm for $<m, m, m>$ triple product property triples and an application for 5 $\times $ 5 matrix multiplication. Groups Complexity Cryptol 7(1):31–46
Article MATH MathSciNet Google Scholar
Hedtke I, Murthy S (2012) Search and test algorithms for triple product property triples. Groups Complex Cryptol 4(1):111–133
Article MATH MathSciNet Google Scholar
Hopcroft JE, Kerr LR (1969) Some techniques for proving certain simple programs optimal. In: IEEE conference record of 10th annual symposium on switching and automata theory, pp 36–45
Jansen T, Wegener I (2001) Evolutionary algorithms: how to cope with plateaus of constant fitness and when to reject strings of the same fitness. IEEE Trans Evol Comput 5(6):589–599
Article Google Scholar
Le Gall F (2014) Powers of tensors and fast matrix multiplication. Eprint Arxiv pp 296–303
Neumann F, Sudholt D, Witt C (2009) Analysis of different MMAS ACO algorithms on unimodal functions and plateaus. Swarm Intell 3(1):35–68
Article Google Scholar
Neumann PM (2011) A note on the triple product property for subsets of finite groups. LMS J Comput Math 14(14):232–237
Article MATH MathSciNet Google Scholar
Pan VY (1978) Strassen’s algorithm is not optimal, trilinear technique of aggregating, uniting and canceling for constructing fast algorithms for matrix operations. In: Proceedings of the 19th annual symposium on foundations of computer science, pp 166–176
Pan VY (1980) New fast algorithms for matrix operations. SIAM J Comput 9(2):321–342
Article MATH MathSciNet Google Scholar
Rai P (2014) Edge detection of medical images using modified ant colony optimization algorithm based on weighted heuristics. Int J Image Gr Signal Process 6(3):21–26
Article Google Scholar
Salama K, Abdelbar AM (2014) A novel ant colony algorithm for building neural network topologies. In: Proceedings of the 9th international conference on ant colony optimization and swarm intelligence, ANTS 2014, Brussels, Belgium, September 10–12, pp 1–12
Schönhage A (1981) Partial and total matrix multiplication. SIAM J Comput 10(3):434–455
Article MATH MathSciNet Google Scholar
Stothers A (2010) On the complexity of matrix multiplication. Ph.D. thesis, University of Edinburgh
Strassen V (1969) Gaussian elimination is not optimal. Numer Math 13(4):354–356
Article MATH MathSciNet Google Scholar
Strassen V (1987) Relative bilinear complexity and matrix multiplication. J Für Die Reine Und Angewandte Math 375–376:406–443
MATH MathSciNet Google Scholar
Stützle T, Hoos HH (2000) Max–Min ant system. Future Gener Comput Syst 16(8):889–914
Article MATH Google Scholar
Williams VV (2012) Multiplying matrices faster than coppersmith-winograd. In: Proceedings of the forty-fourth annual acm symposium on theory of computing, pp 887–898
Winograd S (1971) On multiplication of $2\times 2$ matrices. Linear Algebra Appl 4(4):381–388
Article MATH MathSciNet Google Scholar

Download references

Author information

Authors and Affiliations

School of Computer Science and Engineering, South China University of Technology, Guangzhou, 510006, China
Xinsheng Lai
School of Mathematics and Computer Science, Shangrao Normal University, Shangrao, 334001, China
Xinsheng Lai
School of Data and Computer Science, Sun Yat-sen University, Guangzhou, 510006, China
Yuren Zhou & Yi Xiang

Authors

Xinsheng Lai
View author publications
You can also search for this author in PubMed Google Scholar
Yuren Zhou
View author publications
You can also search for this author in PubMed Google Scholar
Yi Xiang
View author publications
You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuren Zhou.

Ethics declarations

Funding

This study was funded by the National Natural Science Foundation of China (Nos. 61472143, 61562071), the Scientific Research Special Plan of Guangzhou Science and Technology Programme (No. 201607010045), and the Natural Science Foundation of Jiangxi Province (No. 20151BAB207020).

Conflict of interest

The authors declare that they have no conflict of interest

Human and Participants

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Communicated by V. Loia.

Appendix

GAP Id	$\langle n,m,p \rangle $	Details subsets
[36, 1]	$\langle 6, 4, 2\rangle $	$G:=$SmallGroup(36,1), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_1f_2f_3,f_1f_2f_4,f_1f_2f_3^2$,
		$f_2f_3f_4^2,f_2f_3^2f_4^2]$;
		$T=[e,f_1,f_2,f_1*f_2]$;
		$U=[e,f_1f_2f_4^2]$;
[36, 3]	$\langle 9$, 3, $2\rangle $	$G:=$SmallGroup(36,3), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_2,f_1f_4,f_2^2,f_1^2f_3,f_1f_2f_4$,
		$f_1^2f_2f_3,f_1f_2^2f_4,f_1^2f_2^2f_3]$;
		$T=[e,f_1,f_1^2*f_2]$;
		$U=[e,f_3*f_4]$;

[36, 4]	$\langle $12, 2$, 2\rangle $	$G:=$SmallGroup(36,4), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_3,f_1f_3,f_2f_3,f_1f_3^2,f_1f_4^2$,
		$f_2f_4^2,f_3^2f_4,f_1f_2f_3*f_4$,
		$f_1f_2f_4^2,f_2f_3^2f_4^2$,
		$f_1f_2f_3^2*f_4^2]$;
		$T=[e, f_1f_3^2f_4]$;
		$U=[e, f_1f_3^2f_4^2]$;
[36, 6]	$\langle 12, 2, 2\rangle $	$G:=$SmallGroup(36,6), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_3,f_2^2,f_1f_2f_4,f_1f_4^2,f_2^2f_3$,
		$f_2f_4^2,f_1f_2f_3f_4,f_1f_3f_4^2$,
		$f_2f_3f_4^2,f_1f_2^2f_4^2$,
		$f_1f_2^2f_3*f_4^2]$;
		$T=[e,f_1*f_4]$;
		$U=[e,f_1]$;
[36, 7]	$\langle $12, 2$, 2\rangle $	$G:=$SmallGroup(36,7), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_2,f_4,f_1f_4,f_2f_4,f_1f_2f_4$,
		$f_1f_4^2,f_3^2f_4,f_1f_2f_4^2,f_2f_3^2f_4$,
		$f_1f_3^2f_4^2,f_1f_2f_3^2*f_4^2]$;
		$T=[e,f_1f_2f_3^2]$;
		$U=[e,f_1f_2f_3*f_4^2]$;
[36, 9]	$\langle $12, 2$, 2\rangle $	$G:=$SmallGroup(36,9), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_1f_3,f_2f_3,f_3*f_4,f_4^2$,
		$f_1f_3f_4,f_1f_2f_3^2,f_1f_3f_4^2$,
		$f_2f_3^2f_4,f_2f_3f_4^2$,
		$f_1f_2f_3^2f_4$,$f_1f_2f_3^2f_4^2]$;
		$T=[e,f_2]$;
		$U=[e,f_2*f_3^2]$;
[36, 10]	$\langle 12, 2,2\rangle $	$G:=$SmallGroup(36,10), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_1,f_3,f_1f_3,f_2f_4,f_1f_2f_4$,
		$f_1f_4^2,f_2f_3^2,f_3f_4^2,f_1f_2*f_4^2$,
		$f_2f_3^2f_4,f_1f_2f_3^2*f_4^2]$;
		$T=[e,f_1*f_2]$;
		$U=[e,f_1f_2f_3*f_4]$;
[36, 11]	$\langle $8, 2$, 3\rangle $	$G:=$SmallGroup(36,11), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_2,f_1f_4,f_1^2f_2,f_1^2f_3f_4$,
		$f_1f_2^2f_3,f_2^2f_3f_4$,
		$f_1^2f_2^2f_3*f_4]$;
		$T=[e,f_3*f_4]$;
		$U=[e,f_1f_2f_3f_4,f_1^2f_2^2*f_4]$;
[36, 12]	$\langle 12, 2,2\rangle $	$G:=$SmallGroup(36,12), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_1f_2,f_3^2,f_1f_3^2,f_1f_3f_4$,
		$f_1f_4^2,f_2f_3^2,f_2f_3f_4,f_2*f_4^2$,
		$f_3f_4^2,f_1f_2f_3^2,f_1f_2f_3f_4^2]$;
		$T=[e,f_1f_2f_4]$;
		$U=[e,f_1f_2f_4^2]$;
[36, 13]	$\langle 12, 2,2\rangle $	$G:=$SmallGroup(36,13), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_2,f_3,f_1f_2,f_1f_4,f_4^2$,
		$f_1f_2f_4,f_1f_3^2,f_2f_3*f_4$,
		$f_1f_2f_3f_4,f_1f_3*f_4^2$,
		$f_2f_3^2f_4^2]$;
		$T=[e,f_1f_2f_3]$;
		$U=[e,f_1f_2f_3^2*f_4^2]$;

[38, 1]	$\langle 12,2,2\rangle $	$G:=$SmallGroup(38,1), $e=$One(G), $f_1=G.1$
		$f_2=G.2$
		$S=[e,f_1f_2^2,f_2^3,f_1f_2^5,f_2^6,f_1*f_2^8$,
		$f_1f_2^9,f_1f_2^{10},f_2^{11},f_1*f_2^{13}$,
		$f_2^{14},f_1*f_2^{16}]$;
		$T=[e,f_1*f_2^{11}]$;
		$U=[e,f_1*f_2]$;
[39, 1]	$\langle 6,3,3\rangle $	$G:=$SmallGroup(39,1), $e=$One(G), $f_1=G.1$
		$f_2=G.2$
		$S=[e,f_2,f_1^2f_2^2,f_2^9,f_1f_2^{10},f_1^2*f_2^{11}]$;
		$T=[e,f_1^2f_2^5,f_1f_2^{11}]$;
		$U=[e,f_1f_2^5,f_1^2f_2^7]$;
[40, 1]	$\langle 12,2,2\rangle $	$G:=$SmallGroup(40,1), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_1,f_2,f_3,f_1f_2,f_1f_3,f_2*f_3$,
		$f_1f_2f_3,f_1f_4^4,f_1f_2*f_4^4$,
		$f_1f_3f_4^4,f_1f_2f_3*f_4^4]$;
		$T=[e,f_1f_2f_3*f_4^3]$;
		$U=[e,f_1*f_4]$;
[40, 3]	$\langle 12,2,2\rangle $	$G:=$SmallGroup(40,3), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_1,f_3,f_1f_3,f_1f_4,f_2*f_4$,
		$f_2f_3f_4,f_1f_2f_4^3,f_2*f_4^4$,
		$f_1f_3f_4,f_1f_2f_3*f_4^3$,
		$f_2f_3f_4^4]$;
		$T=[e,f_2*f_4^2]$;
		$U=[e,f_2f_3f_4^3]$;
[40, 4]	$\langle 12,2,2\rangle $	$G:=$SmallGroup(40,4), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_3,f_4^2,f_1f_4^2,f_2f_4^2,f_3*f_4^2$,
		$f_1f_2f_4^2,f_1f_3f_4^2,f_2f_3f_4^2$,
		$f_1f_2f_3f_4^2,f_1f_2*f_4^4$,
		$f_1f_2f_3*f_4^4]$;
		$T=[e,f_1f_3f_4^3]$;
		$U=[e,f_1*f_4^4]$;
[40, 5]	$\langle 12,2,2\rangle $	$G:=$SmallGroup(40,5), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_1f_3,f_2f_4^2,f_1f_2f_4^2,f_1*f_4^3$,
		$f_2f_3f_4^2,f_2f_4^3,f_3f_4^3$,
		$f_1f_2f_3f_4^2,f_2f_3f_4^3,f_3f_4^4$,
		$f_1f_3f_4^4 ]$;
		$T=[e,f_1f_3f_4^3]$;
		$U=[e,f_1f_3f_4]$;
[40, 6]	$\langle 13,2,2\rangle $	$G:=$SmallGroup(40,6), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_4,f_1f_3,f_1f_2f_4,f_1f_4^2$,
		$f_2f_3f_4,f_1f_2f_4^2,f_1*f_4^3$,
		$f_2f_3f_4^2,f_3f_4^3,f_2f_4^4$,
		$f_1f_2f_3f_4^3,f_1f_2f_3f_4^4]$;
		$T=[e,f_1f_2f_3]$;
		$U=[e,f_1f_3f_4^3]$;
[40, 7]	$\langle 12,2,2\rangle $	$G:=$SmallGroup(40,7), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_1,f_2,f_3,f_1f_3,f_2f_3,f_2*f_4^2$,
		$f_2f_3f_4^2,f_2f_4^3,f_1f_2*f_4^3$,
		$f_2f_3f_4^3,f_1f_2f_3*f_4^3]$;
		$T=[e,f_1f_2f_3*f_4]$;
		$U=[e,f_1*f_4^2]$;

[40, 8]	$\langle 12,2,2\rangle $	$G:=$SmallGroup(40,8), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_1f_3,f_1f_2f_3,f_3f_4^2,f_1*f_4^3$,
		$f_3f_4^3,f_4^4,f_2f_3f_4^3,f_2f_4^4$,
		$f_1f_2f_4^4,f_2f_3f_4^4$,
		$f_1f_2f_3*f_4^4]$;
		$T=[e,f_1f_3f_4^4]$;
		$U=[e,f_1f_3f_4^2]$;
[40, 10]	$\langle 10,2,2\rangle $	$G:=$SmallGroup(40,10), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_3,f_1f_2f_4,f_1f_3^2,f_1f_3*f_4$,
		$f_2f_3^2f_4,f_3^3f_4,f_1f_3^3*f_4$,
		$f_2f_3^4,f_1f_3^4*f_4]$;
		$T=[e,f_4]$;
		$U=[e,f_2]$;
[40, 11]	$\langle 10,2,2\rangle $	$G:=$SmallGroup(40,11), $e=$One(G), $f_1=G.1$
		$S=[e,f_4,f_1f_2f_3,f_2*f_3^2,f_3^3$,
		$f_1f_2f_3f_4,f_2f_3^2*f_4,f_3^4$,
		$f_3^3f_4,f_3^4f_4]$;
		$T=[e,f_1]$;
		$U=[e,f_2]$;
		$f_2=G.2,f_3=G.3,f_4=G.4$
[40, 12]	$\langle 12,2,2\rangle $	$G:=$SmallGroup(40,12), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_2,f_3,f_1f_4,f_2f_4,f_1f_3f_4$,
		$f_2f_3f_4,f_1f_2f_3f_4,f_4^4,f_1f_4^4$,
		$f_1f_2f_4^4,f_1f_2f_3*f_4^4]$;
		$T=[e,f_3*f_4^4]$;
		$U=[e,f_3*f_4^2]$;
[40, 13]	$\langle 12,2,2\rangle $	$G:=$SmallGroup(40,13), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3,f_4=G.4$
		$S=[e,f_2,f_1f_3,f_3f_4,f_1f_2f_3$,
		$f_1f_4^2,f_2f_3f_4$,$f_4^3,f_1f_2*f_4^2$,
		$f_2f_4^3,f_3f_4^3,f_1f_2f_3*f_4^3]$;
		$T=[e,f_1f_2f_4]$;
		$U=[e,f_1]$;
[42, 1]	$\langle 12,2,2\rangle $	$G:=$SmallGroup(42,1), $e=$One(G), $f_1=G.1$
		$f_2=G.2,f_3=G.3$
		$S=[e,f_3,f_2^2,f_3^2,f_1f_3^2,f_2f_3^2$,
		$f_1f_2f_3^2,f_2^2f_3^3,f_2^2f_3^4,f_2*f_3^5$,
		$f_1f_2^2f_3^4,f_1f_2f_3^6]$;
		$T=[e,f_1*f_3^6]$;
		$U=[e,f_1*f_3^3]$;
[42, 2]	$\langle 6,3,3\rangle $	$G:=$SmallGroup(42,2), $e=$One(G),
		$f_1=G.1,f_2=G.2,f_3=G.3$
		$S=[e,f_2f_3,f_2^2f_3^3,f_1f_2f_3^4$,
		$f_1f_2f_3^5$, $f_1f_2^2f_3^6]$;
		$T=[e,f_2^2f_3,f_2f_3^5]$;
		$U=[e,f_2^2f_3^2,f_2f_3^3]$;
[42, 3]	$\langle 14,2,2\rangle $	$G:=$SmallGroup(42,3), $e=$One(G),
		$f_1=G.1,f_2=G.2,f_3=G.3$
		$S=[e,f_1f_2,f_1f_3,f_2^2,f_2*f_3^2$,
		$f_1f_2^3,f_1f_2^2f_3,f_2^4f_3,f_2^3*f_3^2$,
		$f_2^6,f_2^5f_3,f_1f_2^4*f_3^2$,
		$f_1f_2^6f_3,f_1f_2^5f_3^2]$;
		$T=[e,f_1]$;
		$U=[e,f_1*f_3^2]$;

[42, 4]	$\langle 12,2,2\rangle $	$G:=$SmallGroup(42,4), $e=$One(G),
		$f_1=G.1,f_2=G.2,f_3=G.3$
		$S=[e,f_1,f_1f_2,f_1f_3,f_2^2f_3^2,f_2f_3^3$,
		$f_1f_2f_3^3,f_1f_3^4,f_1f_2^2*f_3^3$,
		$f_1f_2f_3^4,f_2^2f_3^6,f_1f_2^2*f_3^6]$;
		$T=[e,f_1*f_3^5]$;
		$U=[e,f_1*f_3^3]$;
[42, 5]	$\langle 14,2,2\rangle $	$G:=$SmallGroup(42,5), $e=$One(G),
		$f_1=G.1,f_2=G.2,f_3=G.3$
		$S=[e,f_1f_2,f_2f_3,f_3^2,f_1f_2^2f_3$,
		$f_1f_2f_3^2,f_1f_3^3,f_2^2f_3^3,f_2*f_3^4$,
		$f_1f_2f_3^4,f_1f_3^5,f_2^2f_3^5,f_2*f_3^6$,
		$f_1f_2^2f_3^6]$;
		$T=[e,f_1f_2f_3^3]$;
		$U=[e,f_1f_2^2f_3^3]$;
[44, 1]	$\langle 11,2,2\rangle $	$G:=$SmallGroup(44,1), $e=$One(G),
		$f_1=G.1,f_2=G.2,f_3=G.3$
		$S=[e,f_3,f_2f_3^3,f_1f_2f_3^3,f_3^5,f_1f_3^5$,
		$f_1f_2f_3^7,f_1f_3^8,f_1f_2*f_3^9$,
		$f_2f_3^{10},f_1f_2*f_3^{10}]$;
		$T=[e,f_2]$;
		$U=[e,f_1f_2f_3]$;
[44, 3]	$\langle 14,2,2\rangle $	$G:=$SmallGroup(44,3), $e=$One(G),
		$f_1=G.1,f_2=G.2,f_3=G.3$
		$S=[e,f_1f_3,f_3^2,f_2f_3^2,f_1*f_3^4$,
		$f_3^5,f_1f_2f_3^4,f_1f_3^6,f_1f_3^7$,
		$f_2f_3^7,f_2f_3^8,f_3^{10},f_1f_2f_3^9$,
		$f_1f_2f_3^{10}]$;
		$T=[e,f_1]$;
		$U=[e,f_1f_2f_3]$;

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lai, X., Zhou, Y. & Xiang, Y. Ant colony optimization for triple product property triples to fast matrix multiplication. Soft Comput 21, 7159–7171 (2017). https://doi.org/10.1007/s00500-016-2259-y

Download citation

Published: 11 July 2016
Issue Date: December 2017
DOI: https://doi.org/10.1007/s00500-016-2259-y

Keywords

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic

$34.99 /Month

Get 10 units per month
Download Article/Chapter or eBook
1 Unit = 1 Article or 1 Chapter
Cancel anytime

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

GAP Id	\(\langle n,m,p \rangle \)	Details subsets
[36, 1]	\(\langle 6, 4, 2\rangle \)	\(G:=\)SmallGroup(36,1), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_1f_2f_3,f_1f_2f_4,f_1f_2f_3^2\),
		\(f_2f_3f_4^2,f_2f_3^2f_4^2]\);
		\(T=[e,f_1,f_2,f_1*f_2]\);
		\(U=[e,f_1f_2f_4^2]\);
[36, 3]	\(\langle 9\), 3, \(2\rangle \)	\(G:=\)SmallGroup(36,3), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_2,f_1f_4,f_2^2,f_1^2f_3,f_1f_2f_4\),
		\(f_1^2f_2f_3,f_1f_2^2f_4,f_1^2f_2^2f_3]\);
		\(T=[e,f_1,f_1^2*f_2]\);
		\(U=[e,f_3*f_4]\);

[36, 4]	\(\langle \)12, 2\(, 2\rangle \)	\(G:=\)SmallGroup(36,4), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_3,f_1f_3,f_2f_3,f_1f_3^2,f_1f_4^2\),
		\(f_2f_4^2,f_3^2f_4,f_1f_2f_3*f_4\),
		\(f_1f_2f_4^2,f_2f_3^2f_4^2\),
		\(f_1f_2f_3^2*f_4^2]\);
		\(T=[e, f_1f_3^2f_4]\);
		\(U=[e, f_1f_3^2f_4^2]\);
[36, 6]	\(\langle 12, 2, 2\rangle \)	\(G:=\)SmallGroup(36,6), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_3,f_2^2,f_1f_2f_4,f_1f_4^2,f_2^2f_3\),
		\(f_2f_4^2,f_1f_2f_3f_4,f_1f_3f_4^2\),
		\(f_2f_3f_4^2,f_1f_2^2f_4^2\),
		\(f_1f_2^2f_3*f_4^2]\);
		\(T=[e,f_1*f_4]\);
		\(U=[e,f_1]\);
[36, 7]	\(\langle \)12, 2\(, 2\rangle \)	\(G:=\)SmallGroup(36,7), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_2,f_4,f_1f_4,f_2f_4,f_1f_2f_4\),
		\(f_1f_4^2,f_3^2f_4,f_1f_2f_4^2,f_2f_3^2f_4\),
		\(f_1f_3^2f_4^2,f_1f_2f_3^2*f_4^2]\);
		\(T=[e,f_1f_2f_3^2]\);
		\(U=[e,f_1f_2f_3*f_4^2]\);
[36, 9]	\(\langle \)12, 2\(, 2\rangle \)	\(G:=\)SmallGroup(36,9), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_1f_3,f_2f_3,f_3*f_4,f_4^2\),
		\(f_1f_3f_4,f_1f_2f_3^2,f_1f_3f_4^2\),
		\(f_2f_3^2f_4,f_2f_3f_4^2\),
		\(f_1f_2f_3^2f_4\),\(f_1f_2f_3^2f_4^2]\);
		\(T=[e,f_2]\);
		\(U=[e,f_2*f_3^2]\);
[36, 10]	\(\langle 12, 2,2\rangle \)	\(G:=\)SmallGroup(36,10), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_1,f_3,f_1f_3,f_2f_4,f_1f_2f_4\),
		\(f_1f_4^2,f_2f_3^2,f_3f_4^2,f_1f_2*f_4^2\),
		\(f_2f_3^2f_4,f_1f_2f_3^2*f_4^2]\);
		\(T=[e,f_1*f_2]\);
		\(U=[e,f_1f_2f_3*f_4]\);
[36, 11]	\(\langle \)8, 2\(, 3\rangle \)	\(G:=\)SmallGroup(36,11), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_2,f_1f_4,f_1^2f_2,f_1^2f_3f_4\),
		\(f_1f_2^2f_3,f_2^2f_3f_4\),
		\(f_1^2f_2^2f_3*f_4]\);
		\(T=[e,f_3*f_4]\);
		\(U=[e,f_1f_2f_3f_4,f_1^2f_2^2*f_4]\);
[36, 12]	\(\langle 12, 2,2\rangle \)	\(G:=\)SmallGroup(36,12), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_1f_2,f_3^2,f_1f_3^2,f_1f_3f_4\),
		\(f_1f_4^2,f_2f_3^2,f_2f_3f_4,f_2*f_4^2\),
		\(f_3f_4^2,f_1f_2f_3^2,f_1f_2f_3f_4^2]\);
		\(T=[e,f_1f_2f_4]\);
		\(U=[e,f_1f_2f_4^2]\);
[36, 13]	\(\langle 12, 2,2\rangle \)	\(G:=\)SmallGroup(36,13), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_2,f_3,f_1f_2,f_1f_4,f_4^2\),
		\(f_1f_2f_4,f_1f_3^2,f_2f_3*f_4\),
		\(f_1f_2f_3f_4,f_1f_3*f_4^2\),
		\(f_2f_3^2f_4^2]\);
		\(T=[e,f_1f_2f_3]\);
		\(U=[e,f_1f_2f_3^2*f_4^2]\);

[38, 1]	\(\langle 12,2,2\rangle \)	\(G:=\)SmallGroup(38,1), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2\)
		\(S=[e,f_1f_2^2,f_2^3,f_1f_2^5,f_2^6,f_1*f_2^8\),
		\(f_1f_2^9,f_1f_2^{10},f_2^{11},f_1*f_2^{13}\),
		\(f_2^{14},f_1*f_2^{16}]\);
		\(T=[e,f_1*f_2^{11}]\);
		\(U=[e,f_1*f_2]\);
[39, 1]	\(\langle 6,3,3\rangle \)	\(G:=\)SmallGroup(39,1), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2\)
		\(S=[e,f_2,f_1^2f_2^2,f_2^9,f_1f_2^{10},f_1^2*f_2^{11}]\);
		\(T=[e,f_1^2f_2^5,f_1f_2^{11}]\);
		\(U=[e,f_1f_2^5,f_1^2f_2^7]\);
[40, 1]	\(\langle 12,2,2\rangle \)	\(G:=\)SmallGroup(40,1), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_1,f_2,f_3,f_1f_2,f_1f_3,f_2*f_3\),
		\(f_1f_2f_3,f_1f_4^4,f_1f_2*f_4^4\),
		\(f_1f_3f_4^4,f_1f_2f_3*f_4^4]\);
		\(T=[e,f_1f_2f_3*f_4^3]\);
		\(U=[e,f_1*f_4]\);
[40, 3]	\(\langle 12,2,2\rangle \)	\(G:=\)SmallGroup(40,3), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_1,f_3,f_1f_3,f_1f_4,f_2*f_4\),
		\(f_2f_3f_4,f_1f_2f_4^3,f_2*f_4^4\),
		\(f_1f_3f_4,f_1f_2f_3*f_4^3\),
		\(f_2f_3f_4^4]\);
		\(T=[e,f_2*f_4^2]\);
		\(U=[e,f_2f_3f_4^3]\);
[40, 4]	\(\langle 12,2,2\rangle \)	\(G:=\)SmallGroup(40,4), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_3,f_4^2,f_1f_4^2,f_2f_4^2,f_3*f_4^2\),
		\(f_1f_2f_4^2,f_1f_3f_4^2,f_2f_3f_4^2\),
		\(f_1f_2f_3f_4^2,f_1f_2*f_4^4\),
		\(f_1f_2f_3*f_4^4]\);
		\(T=[e,f_1f_3f_4^3]\);
		\(U=[e,f_1*f_4^4]\);
[40, 5]	\(\langle 12,2,2\rangle \)	\(G:=\)SmallGroup(40,5), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_1f_3,f_2f_4^2,f_1f_2f_4^2,f_1*f_4^3\),
		\(f_2f_3f_4^2,f_2f_4^3,f_3f_4^3\),
		\(f_1f_2f_3f_4^2,f_2f_3f_4^3,f_3f_4^4\),
		\(f_1f_3f_4^4 ]\);
		\(T=[e,f_1f_3f_4^3]\);
		\(U=[e,f_1f_3f_4]\);
[40, 6]	\(\langle 13,2,2\rangle \)	\(G:=\)SmallGroup(40,6), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_4,f_1f_3,f_1f_2f_4,f_1f_4^2\),
		\(f_2f_3f_4,f_1f_2f_4^2,f_1*f_4^3\),
		\(f_2f_3f_4^2,f_3f_4^3,f_2f_4^4\),
		\(f_1f_2f_3f_4^3,f_1f_2f_3f_4^4]\);
		\(T=[e,f_1f_2f_3]\);
		\(U=[e,f_1f_3f_4^3]\);
[40, 7]	\(\langle 12,2,2\rangle \)	\(G:=\)SmallGroup(40,7), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_1,f_2,f_3,f_1f_3,f_2f_3,f_2*f_4^2\),
		\(f_2f_3f_4^2,f_2f_4^3,f_1f_2*f_4^3\),
		\(f_2f_3f_4^3,f_1f_2f_3*f_4^3]\);
		\(T=[e,f_1f_2f_3*f_4]\);
		\(U=[e,f_1*f_4^2]\);

[40, 8]	\(\langle 12,2,2\rangle \)	\(G:=\)SmallGroup(40,8), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_1f_3,f_1f_2f_3,f_3f_4^2,f_1*f_4^3\),
		\(f_3f_4^3,f_4^4,f_2f_3f_4^3,f_2f_4^4\),
		\(f_1f_2f_4^4,f_2f_3f_4^4\),
		\(f_1f_2f_3*f_4^4]\);
		\(T=[e,f_1f_3f_4^4]\);
		\(U=[e,f_1f_3f_4^2]\);
[40, 10]	\(\langle 10,2,2\rangle \)	\(G:=\)SmallGroup(40,10), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_3,f_1f_2f_4,f_1f_3^2,f_1f_3*f_4\),
		\(f_2f_3^2f_4,f_3^3f_4,f_1f_3^3*f_4\),
		\(f_2f_3^4,f_1f_3^4*f_4]\);
		\(T=[e,f_4]\);
		\(U=[e,f_2]\);
[40, 11]	\(\langle 10,2,2\rangle \)	\(G:=\)SmallGroup(40,11), \(e=\)One(G), \(f_1=G.1\)
		\(S=[e,f_4,f_1f_2f_3,f_2*f_3^2,f_3^3\),
		\(f_1f_2f_3f_4,f_2f_3^2*f_4,f_3^4\),
		\(f_3^3f_4,f_3^4f_4]\);
		\(T=[e,f_1]\);
		\(U=[e,f_2]\);
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
[40, 12]	\(\langle 12,2,2\rangle \)	\(G:=\)SmallGroup(40,12), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_2,f_3,f_1f_4,f_2f_4,f_1f_3f_4\),
		\(f_2f_3f_4,f_1f_2f_3f_4,f_4^4,f_1f_4^4\),
		\(f_1f_2f_4^4,f_1f_2f_3*f_4^4]\);
		\(T=[e,f_3*f_4^4]\);
		\(U=[e,f_3*f_4^2]\);
[40, 13]	\(\langle 12,2,2\rangle \)	\(G:=\)SmallGroup(40,13), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3,f_4=G.4\)
		\(S=[e,f_2,f_1f_3,f_3f_4,f_1f_2f_3\),
		\(f_1f_4^2,f_2f_3f_4\),\(f_4^3,f_1f_2*f_4^2\),
		\(f_2f_4^3,f_3f_4^3,f_1f_2f_3*f_4^3]\);
		\(T=[e,f_1f_2f_4]\);
		\(U=[e,f_1]\);
[42, 1]	\(\langle 12,2,2\rangle \)	\(G:=\)SmallGroup(42,1), \(e=\)One(G), \(f_1=G.1\)
		\(f_2=G.2,f_3=G.3\)
		\(S=[e,f_3,f_2^2,f_3^2,f_1f_3^2,f_2f_3^2\),
		\(f_1f_2f_3^2,f_2^2f_3^3,f_2^2f_3^4,f_2*f_3^5\),
		\(f_1f_2^2f_3^4,f_1f_2f_3^6]\);
		\(T=[e,f_1*f_3^6]\);
		\(U=[e,f_1*f_3^3]\);
[42, 2]	\(\langle 6,3,3\rangle \)	\(G:=\)SmallGroup(42,2), \(e=\)One(G),
		\(f_1=G.1,f_2=G.2,f_3=G.3\)
		\(S=[e,f_2f_3,f_2^2f_3^3,f_1f_2f_3^4\),
		\(f_1f_2f_3^5\), \(f_1f_2^2f_3^6]\);
		\(T=[e,f_2^2f_3,f_2f_3^5]\);
		\(U=[e,f_2^2f_3^2,f_2f_3^3]\);
[42, 3]	\(\langle 14,2,2\rangle \)	\(G:=\)SmallGroup(42,3), \(e=\)One(G),
		\(f_1=G.1,f_2=G.2,f_3=G.3\)
		\(S=[e,f_1f_2,f_1f_3,f_2^2,f_2*f_3^2\),
		\(f_1f_2^3,f_1f_2^2f_3,f_2^4f_3,f_2^3*f_3^2\),
		\(f_2^6,f_2^5f_3,f_1f_2^4*f_3^2\),
		\(f_1f_2^6f_3,f_1f_2^5f_3^2]\);
		\(T=[e,f_1]\);
		\(U=[e,f_1*f_3^2]\);

[42, 4]	\(\langle 12,2,2\rangle \)	\(G:=\)SmallGroup(42,4), \(e=\)One(G),
		\(f_1=G.1,f_2=G.2,f_3=G.3\)
		\(S=[e,f_1,f_1f_2,f_1f_3,f_2^2f_3^2,f_2f_3^3\),
		\(f_1f_2f_3^3,f_1f_3^4,f_1f_2^2*f_3^3\),
		\(f_1f_2f_3^4,f_2^2f_3^6,f_1f_2^2*f_3^6]\);
		\(T=[e,f_1*f_3^5]\);
		\(U=[e,f_1*f_3^3]\);
[42, 5]	\(\langle 14,2,2\rangle \)	\(G:=\)SmallGroup(42,5), \(e=\)One(G),
		\(f_1=G.1,f_2=G.2,f_3=G.3\)
		\(S=[e,f_1f_2,f_2f_3,f_3^2,f_1f_2^2f_3\),
		\(f_1f_2f_3^2,f_1f_3^3,f_2^2f_3^3,f_2*f_3^4\),
		\(f_1f_2f_3^4,f_1f_3^5,f_2^2f_3^5,f_2*f_3^6\),
		\(f_1f_2^2f_3^6]\);
		\(T=[e,f_1f_2f_3^3]\);
		\(U=[e,f_1f_2^2f_3^3]\);
[44, 1]	\(\langle 11,2,2\rangle \)	\(G:=\)SmallGroup(44,1), \(e=\)One(G),
		\(f_1=G.1,f_2=G.2,f_3=G.3\)
		\(S=[e,f_3,f_2f_3^3,f_1f_2f_3^3,f_3^5,f_1f_3^5\),
		\(f_1f_2f_3^7,f_1f_3^8,f_1f_2*f_3^9\),
		\(f_2f_3^{10},f_1f_2*f_3^{10}]\);
		\(T=[e,f_2]\);
		\(U=[e,f_1f_2f_3]\);
[44, 3]	\(\langle 14,2,2\rangle \)	\(G:=\)SmallGroup(44,3), \(e=\)One(G),
		\(f_1=G.1,f_2=G.2,f_3=G.3\)
		\(S=[e,f_1f_3,f_3^2,f_2f_3^2,f_1*f_3^4\),
		\(f_3^5,f_1f_2f_3^4,f_1f_3^6,f_1f_3^7\),
		\(f_2f_3^7,f_2f_3^8,f_3^{10},f_1f_2f_3^9\),
		\(f_1f_2f_3^{10}]\);
		\(T=[e,f_1]\);
		\(U=[e,f_1f_2f_3]\);

Ant colony optimization for triple product property triples to fast matrix multiplication

Abstract

Access this article

Subscribe and save

Buy Now

Similar content being viewed by others

Ant Colony Optimization: Overview and Recent Advances

Two-Dimensional Pheromone in Ant Colony Optimization

Ant Colony Optimization—Recent Variants, Application and Perspectives

References

Author information

Authors and Affiliations

Corresponding author

Ethics declarations

Funding

Conflict of interest

Human and Participants

Additional information

Appendix

Rights and permissions

About this article

Cite this article

Keywords

Subscribe and save

Buy Now

Navigation

Ant colony optimization for triple product property triples to fast matrix multiplication

Abstract

Access this article

Subscribe and save

Buy Now

Similar content being viewed by others

Ant Colony Optimization: Overview and Recent Advances

Two-Dimensional Pheromone in Ant Colony Optimization

Ant Colony Optimization—Recent Variants, Application and Perspectives

Explore related subjects

References

Author information

Authors and Affiliations

Corresponding author

Ethics declarations

Funding

Conflict of interest

Human and Participants

Additional information

Appendix

Appendix

Rights and permissions

About this article

Cite this article

Share this article

Keywords

Subscribe and save

Buy Now

Search

Navigation