Advertisement

Embedding Exchanged Hypercubes into Rings and Ladders

  • Weibei Fan
  • Jianxi FanEmail author
  • Cheng-Kuan Lin
  • Zhijie Han
  • Peng Li
  • Ruchuan Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11335)

Abstract

Graph embeddings are not only used to study the simulation capabilities of a parallel architecture but also to design its VLSI layout. The n-dimensional hypercube is one of the most popular topological structure for interconnection networks in parallel computing and communication systems. The exchanged hypercube \(EH_{s,t}\) (where \(s\ge 1\) and \(t\ge 1\)) is obtained by systematically deleting edges from a hypercube \(Q_{s+t+1}\), which retains several valuable and desirable properties of the hypercube such as a small diameter, bipancyclicity, and super connectivity. In this paper, we identify maximum induced subgraph of \(EH_{s,t}\) and study embeddings of \(EH_{s,t}\) into a ring and a ladder with minimum wirelength.

Keywords

Interconnection networks \(EH_{s, t}\) Graph embedding Rings Ladders 

Notes

Acknowledgment

We would like to express our sincerest appreciation to Prof. Guoliang Chen for his constructive suggestions. This work is supported by National Key R&D Program of China (2018YFB1003201), Natural Science Foundation of China under grant (No. 61572337, No. 61602333, No. 61672296 and No. 61702351), China Postdoctoral Science Foundation (No. 172985), Scientific & Technological Support Project of Jiangsu Province (No. BE2016777, No. BE2016185), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Nos. 17KJB520036), Jiangsu Planned Projects for Postdoctoral Research Funds under Grant (No. 1701172B) and Jiangsu High Technology Research Key Laboratory for Wireless Sensor Networks Foundation (No. WSNLBKF201701).

References

  1. 1.
    Arockiaraj, M., Abraham, J., Quadras., J.: Linear layout of locally twisted cubes. Int. J. Comput. Math. 94(1), 56–65 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bezrukov, S.L., Das, S.K., Elsasser, R.: An edge-isoperimetric problem for powers of the Petersen graph. Ann. Combinatorics 4(2), 153–169 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bezrukov, S.L., Chavez, J.D., Harper, L.H., Röttger, M., Schroeder, U.P.: Embedding of hypercubes into grids. Mortar Fire Control System, pp. 693–701 (1998)Google Scholar
  4. 4.
    Boals, A.J., Gupta, A.K., Sherwani, N.A.: Incomplete hypercubes: algorithms and embeddings. J. Supercomputing 8(3), 263–294 (1994)CrossRefGoogle Scholar
  5. 5.
    Chen, Y., Shen, H.: Routing and wavelength assignment for hypercube in array-based WDM optical networks. J. Parallel Distrib. Comput. 70(1), 59–68 (2010)CrossRefGoogle Scholar
  6. 6.
    Erbele, J., Chavez, J., Trapp, R.: The cyclic cutwidth of \(Q_{n}\). Technical report, California State UniversitySan Bernardino (CSUSB) (2003)Google Scholar
  7. 7.
    Fan, J., Jia, X., Lin, X.: Complete path embeddings in crossed cubes. Inf. Sci. 176(22), 3332–3346 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fan, J., Jia, X., Lin, X.: Embedding of cycles in twisted cubes with edge-pancyclic. Algorithmica 51(3), 264–282 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Wang, X., Fan, J., Jia, X.: Embedding meshes into twisted-cubes. Inf. Sci. 181(14), 3085–3099 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness (1979)Google Scholar
  11. 11.
    Harper, L.H.: Global Methods for Combinatorial Isoperimetric Problems. Cambridge University Press, UK (2004)Google Scholar
  12. 12.
    Han, Y., Fan, J., Zhang, S.: Embedding meshes into locally twisted cubes. Inf. Sci. 180(19), 3794–3805 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Huang, K.E., Wu, J.: Area efficient layout of balanced hypercubes. Int. J. High Speed Electron. Syst. 6(04), 631–645 (1995)CrossRefGoogle Scholar
  14. 14.
    Hsu, L.-H., Lin, C.-K.: Graph Theory and Interconnection Networks. CRC, Boca Raton (2008)zbMATHGoogle Scholar
  15. 15.
    Liu, Y.-L., Wu, R.-C.: Implementing exchanged hypercube communication patterns on ring-connected WDM optical networks. IEICE Trans. Inf. Syst. 100(12), 2771–2780 (2017)CrossRefGoogle Scholar
  16. 16.
    Loh, P.K.K., Hsu, W.-J., Pan, Y.: The exchanged hypercube. IEEE Trans. Parallel Distrib. Syst. 16(9), 866–874 (2005)CrossRefGoogle Scholar
  17. 17.
    Katseff, H.: Incomplete hypercubes. IEEE Trans. Comput. 37(5), 604–608 (1988)CrossRefGoogle Scholar
  18. 18.
    Ma, M., Liu, B.: Cycles embedding in exchanged hypercubes. Inf. Process. Lett. 110(2), 71–76 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Manuel, P., Rajasingh, I., Rajan, B.: Exact wirelength of hypercubes on a grid. Discrete Appl. Math. 157(7), 1486–1495 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ma, M., Zhu, L.: The super connectivity of exchanged hypercubes. Inf. Process. Lett. 111(8), 360–364 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Miller, M., Rajan, R.S., Parthiban, N.: Minimum linear arrangement of incomplete hypercubes. Comput. J. 58(2), 331–337 (2015)CrossRefGoogle Scholar
  22. 22.
    Nakano, K.: Linear layout of generalized hypercubes. Int. J. Found. Comput. Sci. 14(01), 137–156 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rostami, H., Habibi, J.: Minimum linear arrangement of Chord graphs. Appl. Math. Comput. 203(1), 358–367 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Sýkora, O., Vrt’o, I.: On VLSI layouts of the star graph and related networks. Integr. VLSI J. 17(1), 83–93 (1994)CrossRefGoogle Scholar
  25. 25.
    Wan, L., Liu., Y.: On the embedding genus distribution of ladders and crosses. Appl. Math. Lett. 22(5) 738–742 (2009)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wang, D.: Hamiltonian embedding in crossed cubes with failed links. IEEE Trans. Parallel Distrib. Syst. 23(11), 2117–2124 (2012)CrossRefGoogle Scholar
  27. 27.
    Wang, S., Zhang, S.: Embedding hamiltonian paths in \(k\)-ary \(n\)-cubes with conditional edge faults. Theoret. Comput. Sci. 412(46), 6570–6584 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Yang, Y., Li, J., Wang, S.: Embedding various cycles with prescribed paths into \(k\)-ary \(n\)-cubes. Discrete Appl. Math. 220, 161–169 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Yang, X., David, J.E., Graham, M.: Maximum induced subgraph of a recursive circulant. Inf. Process. Lett. 95(1), 293–298 (2005)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Yeh, C. H., Varvarigos, E. A., Parhami, B.: Multilayer VLSI layout for interconnection networks. In: Proceedings of International Conference on IEEE Parallel Processing, pp. 33–40 (2000)Google Scholar
  31. 31.
    Yu, C., Yang, X.: Routing and wavelength assignment for 3-ary \(n\)-cube in array-based optical network. Inf. Process. Lett. 112(6), 252–256 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Weibei Fan
    • 1
  • Jianxi Fan
    • 1
    Email author
  • Cheng-Kuan Lin
    • 1
  • Zhijie Han
    • 2
  • Peng Li
    • 2
  • Ruchuan Wang
    • 2
  1. 1.School of Computer Science and TechnologySoochow UniversitySuzhouChina
  2. 2.Jiangsu High Technology Research Key Laboratory for Wireless Sensor NetworksNanjingChina

Personalised recommendations