Научная электронная библиотека ULRICH'S Periodicals Directory

Сибирский государственный университет
науки и технологий имени академика М.Ф. Решетнева

Сибирский аэрокосмический журнал
ISSN 2712-8970 (Print) ISSN 2782-5760 (on-line)

Сведения об образовательной организации

UDK 519.6
PERSPECTIVE TOPOLOGIES OF MULTIPROCESSOR COMPUTING SYSTEMS BASED ON THE CAYLEY GRAPHS OF GROUPS OF PERIOD 4
А. A. Kuznetsov1, A. S. Kuznetsova2
1Reshetnev Siberian State Aerospace University 31, Krasnoyarskiy Rabochiy Av., Krasnoyarsk, 660037, Russian Federation 2Krasnoyarsk State Agrarian University 90, Mira Av., Krasnoyarsk, 660049, Russian Federation
The definition of the Cayley graph was given to the famous English mathematician Arthur Cayley in the XIX century to represent algebraic group defined by a fixed set of generating elements. At present, Cayley graphs are widely used both in mathematics and in applications. In particular, these graphs are used to represent computer networks, including the modeling of topologies of multiprocessor computer systems (MСS) – supercomputers. Since this direction is actively developed. This is due to the fact that Cayley graphs have many attractive properties such as regularity, vertex transitive, small diameter and degree at a sufficiently large number of vertices in the graph. For example, such a basic network topology as the “ring”, “hypercube” and “torus” are the Cayley graphs. One of the commonly used topologies MCS is a k-dimensional hypercube. This graph is given by k-generated Burnside groups of period 2, which is denoted B(k,2). The group B(k,2) has a simple structure and is equal to the direct product of k copies of the cyclic group of order 2. A generalization of the hypercube is a n-dimensional torus, which is generated by the direct product of n copies of the cyclic groups whose orders may be different. In this work, We researched the structure of the Cayley graphs of group B(k,4) – Burnside k-generated groups of period 4 (and some their quotients). Then we compared these graphs with the corresponding toruses and hypercubes. The analysis is shown that the graphs B(k,4) have better properties in comparison with hypercubes and toruses. Therefore, they deserve attention in the design of advanced topologies of MCS.
Keywords: the Cayley graph, a multiprocessor computing system.
References

1. Akers S., Krishnamurthy B. A group theoretic model for symmetric interconnection networks. Proceedings of the International Conference on Parallel Processing, 1986, P. 216–223.

2. Schibell S., Stafford R. Processor interconnection networks and Cayley graphs. Discrete Applied Mathematics, 1992, Vol. 40, P. 337–357.

3. Cooperman G., Finkelstein L. New methods for using Cayley graphs in interconnection networks. Discrete Applied Mathematics, 1992, Vol. 37, P. 95–118.

4. Lakshmivarahan S., Jho J., Dhall S. Symmetry in interconnection networks based on Cayley graphs of permutation groups. A survey. Parallel Computing, 1993, Vol. 19, P. 361–407.

5. Heydemann M. Cayley graphs and interconnection networks, in Graph symmetry: algebraic methods and applications (Editors: Hahnand Sabidussi). Dordrecht: Kluwer Academic Publishers, 1997, P. 167–226.

6. Xu J. Topological Structure and Analysis of Interconnection Networks. Dordrecht: Kluwer Academic Publ., 2001, 352 p.

7. Parhami B. Swapped interconnection networks: Topological, performance, and robustness attributes. Journal of Parallel and Distributed Computing, 2005, Vol. 65, P. 1443–1452.

8. Asai S., Kounoike, Shinano Y., Kaneko K. Computing the diameter of 17-pancake graphs using a PC cluster. LNSC, 2006, Vol. 4128, P. 1114–1124.

9. Chen B., Xiao W., Parhami B. Internode distance and optimal routing in a class of alternating group networks. IEEE Transactionson Computers, 2006, Vol. 55, P. 1645–1648.

10. Wang L., Tang K. The Cayley Graph implementation in TinyOS for dense wireless sensor networks. Proc. of the 6th Wireless Telecommunications Symposium, 2007.

11. Camelo M., Papadimitriou D., Fabrega L., Vila P. Efficient Routing in Data Center with Underlying Cayley Graph. Proceedings of the 5th Workshop on Complex Networks CompleNet, 2014, P. 189–197.

12. Even S., Goldreich O. The Minimum Length Generator Sequence is NP-Hard. Journal of Algorithms, 1981, Vol. 2, P. 311–313.

13. Vaughan-Lee M. The restricted Burnside problem. Oxford: Clarendon Press, 1990, 209 p.

14. Kuznetsov A. A. Relation between growth functions in symmetric groups and tasks of combinatorial optimization. Siberian Electronic Mathematical Reports, 2015, Vol. 12, P. 248–254.

15. Kuznetsov A. A., Kuznetsova A. S. A parallel algorithm for study of the Cayley graphs of permutation groups. Vestnik SibGAU, 2014, № 1 (53), P. 34–39.


Kuznetsov Alexandr Alexeevich – Dr. Sc., professor, Head of Institute of Space Research and High Technologies

of Reshetnev Siberian State Aerospace University. E-mail: kuznetsov@sibsau.ru.

Kuznetsova Alexandra Sergeevna – Cand. Sc., docent of Department of Business Informatics and Computer

Security, Krasnoyarsk State Agrarian University. E-mail: alexakuznetsova85@gmail.com.