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Bisecting and Gossiping in Circulant Graphs

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LATIN 2004: Theoretical Informatics (LATIN 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2976))

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Abstract

Circulant graphs are popular network topologies that arise in distributed computing. In this paper, we show that, for circulant graphs, a simple condition for isomorphism, combined with lattices reduction algorithms, can be used to develop efficient distributed algorithms. We improve the known upper bounds on the vertex-bisection (respectively the edge-bisection) width of circulant graphs. Our method is novel and provides a polynomial-time algorithm to partition the set of vertices (respectively the set of edges) to obtain these bounds and the respective sets. By exploiting the knowledge of the bisection width of this topology, we introduce generic distributed algorithms to solve the gossip problem in these networks. We present lower and upper bounds of the number of rounds in the vertex-disjoint and the edge-disjoint paths communication models when the number of nodes is prime.

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References

  1. Ádám, A.: Research problem 2-10. J. Combinatorial Theory 3, 393 (1967)

    Google Scholar 

  2. Ajtai, M., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: Proc. 33rd ACM STOC, Crete, Greece, July 6-8, pp. 601–610 (2001)

    Google Scholar 

  3. Ahmadi, A., Belk, R., Tamon, C., Wendler, C.: Mixing in Continuous Quantum Walks on Graphs (April 2003), xxx.arxiv.cornell.edu/pdf/quant-ph/0209106

  4. Annexstein, F., Baumslag, M.: On the diameter and bisector size of Cayley graphs. Mathematical Systems Theory 26, 271–291 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  5. Babai, L.: On Lovász’ lattice reduction and the nearest lattice point problem. Combinatorica 6, 11–13 (1986)

    Article  MathSciNet  Google Scholar 

  6. Babai, L.: Automorphism groups, isomorphism, reconstruction. In: Handbook of Combinatorics, pp. 1749–1783. Elsevier, Amsterdam (1995)

    Google Scholar 

  7. Barrière, L., Cohen, J., Mitjana, M.: Gossiping in chordal rings under the line model. Theoretical Computer Science 264(1), 53–64 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Barrière, L., Fàbrega, J.: Edge-bisection of chordal rings. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 162–171. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  9. Blackburn, S.R.: Node Bisectors of Cayley Graphs. Mathematical Systems Theory 29, 589–598 (1996)

    MATH  MathSciNet  Google Scholar 

  10. Bermond, J.-C., Comellas, F., Hsu, D.F.: Distributed loop computer networks: A survey. Journal of Parallel and Distributed Computing 24, 2–10 (1995)

    Article  Google Scholar 

  11. Cai, J.-Y., Havas, G., Mans, B., Nerurkar, A., Seifert, J.-P., Shparlinski, I.: On routing in circulant graphs. In: Asano, T., Imai, H., Lee, D.T., Nakano, S.-i., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, pp. 360–378. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  12. Elspas, B., Turner, J.: Graphs with circulant adjacency matrices. J. Comb. Theory 9, 229–240 (1970)

    MathSciNet  Google Scholar 

  13. Farley, A.: Minimum-time line broadcast networks. Networks 10, 57–70 (1980)

    Article  MathSciNet  Google Scholar 

  14. Gruber, R., Gunzinger, A.: The Swiss-Tx Supercomputer Project. Speedup 11(2), 20–26 (1997)

    Google Scholar 

  15. Hamidoune, Y.O., Serra, O.: On small cuts separating an Abelian Cayley graph into two equal parts. Mathe. Systems Theory 29(4), 407–409 (1996)

    MATH  MathSciNet  Google Scholar 

  16. Hromkovič, J., Klasing, R., Stöhr, E.A.: Dissemination of information in generalized communication modes. Comp. Art. Intell. 15(4), 295–318 (1996)

    Google Scholar 

  17. Hromkovič, J., Klasing, R., Stöhr, E.A., Wagener, H.: Gossiping in vertexdisjoint paths mode in d-dimensional grids and planar graphs. Information and Computation 123, 17–28 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hromkovič, J., Klasing, R., Unger, W., Wagener, H.: Optimal algorithms for broadcast and gossip in the edge-disjoint path modes. Information and Computation 133, 1–33 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Klasing, R.: The relationship between the gossip complexity in the vertex-disjoint paths mode and the vertex bisection width. D.A.M. 83, 229–246 (1998)

    MATH  MathSciNet  Google Scholar 

  20. Leighton, F.T.: Introduction to parallel algorithms and architectures: Arrays, trees, hypercubes. M. Kaufmann, San Francisco (1992)

    MATH  Google Scholar 

  21. Lenstra, K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mans, B., Pappalardi, F., Shparlinski, I.: On the spectral Ádám property for circulant graphs. Discrete Math. 254(1-3), 309–329 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Rogers, C.A.: Packing and covering. Cambridge Univ. Press, NY (1964)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computing, Macquarie University, Sydney, NSW, 2109, Australia

    Bernard Mans & Igor Shparlinski

Authors
  1. Bernard Mans
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  2. Igor Shparlinski
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Rutgers University, NJ 08854, Piscataway

    Martín Farach-Colton

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© 2004 Springer-Verlag Berlin Heidelberg

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Mans, B., Shparlinski, I. (2004). Bisecting and Gossiping in Circulant Graphs. In: Farach-Colton, M. (eds) LATIN 2004: Theoretical Informatics. LATIN 2004. Lecture Notes in Computer Science, vol 2976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24698-5_61

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  • DOI: https://doi.org/10.1007/978-3-540-24698-5_61

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-21258-4

  • Online ISBN: 978-3-540-24698-5

  • eBook Packages: Springer Book Archive

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