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Scalable Sparse Topologies with Small Spectrum

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STACS 2001 (STACS 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2010))

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Abstract

One of the fundamental properties of a graph is the number of distinct eigenvalues of its adjacency or Laplacian matrix. Determining this number is of theoretical interest and also of practical impact. Graphs with small spectra exhibit many symmetry properties and are well suited as interconnection topologies. Es- pecially load balancing can be done on such interconnection topologies in a small number of steps. In this paper we are interested in graphs with maximal degree O(log n), where n is the number of vertices, and with a small number of distinct eigenvalues. Our goal is to find scalable families of such graphs with polyloga- rithmic spectrum in the number of vertices. We present also the eigenvalues of the Butterfly graph.

This work was partially supported by the German Research Association (DFG) within the SFB 376 “Massive Parallelität: Algorithmen, Entwurfsmethoden, Anwendungen”.

The research was done while the second author was visiting the Department of Mathematics and Computer Science at the University of Paderborn.

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References

  1. Akers, S.B., Harel, D., Krishnamurthy, B.: The Star Graph: An Attractive Alternative to the n-Cube, Proc. of the International Conference on Parallel Processing,1987, pp.393–400

    Google Scholar 

  2. Akers, S.B., Krishnamurthy, B.: A Group-Theoretic Model for Symmetric Interconnection Networks, IEEE Transactions on Computers 38, 1989, pp. 555–565

    Article  MATH  MathSciNet  Google Scholar 

  3. Biggs, N. L.:Algebraic Graph Theory, (2nd ed.), Cambridge University Press, Cambridge, 1993

    Google Scholar 

  4. Brouwe, A. E., Cohen, A. M., Neumaier A.: Distance-Regular Graphs, Springer Verlag 1989

    Google Scholar 

  5. Chung, F.R.K.: Spectral Graph Theory, American Mathematical Society, 1994

    Google Scholar 

  6. Cybenko, G: Load balancing for distributed memory multiprocessors J. of Parallel and Distributed Computing 7, 1989, pp. 279–301

    Article  Google Scholar 

  7. Cvetković, D. M., Doob, M., Sachs, H.: Spectra of graphs, Theory and Application, Academic Press, 1980

    Google Scholar 

  8. Decker, T., Monien, B., Preis, R.: Towards Optimal Load Balancing Topologies, Proceedings of the 6th EuroPar Conference, LNCS, 2000, to appear

    Google Scholar 

  9. Delorme, C., Tillich, J. P.: The spectrum of DeBruijn and Kautz Graphs, European Journal of Combinatorics 19, 1998, pp. 307–319

    Article  MATH  MathSciNet  Google Scholar 

  10. Diekmann R., Frommer, A., Monien, B.: Efficient schemes for nearest neighbor load balancing, Parallel Computing 25, 1999, pp. 789–812

    Article  MathSciNet  Google Scholar 

  11. Elsässer, R., Frommer, A., Monien, B., Preis, R.: Optimal and Alternating-Direction Load Balancing, EuroPar’99, LNCS 1685, 1999, pp. 280–290

    Google Scholar 

  12. Flatto, L., Odlyzko, A.M., Wales, D.B.: Random Shuffles and Group Representations, The Annals of Probability 13, 1985 pp. 154–178

    Article  MATH  MathSciNet  Google Scholar 

  13. Gosh, B., Muthukrishnan, S., Schultz, M.H.: First and Second order diffusive methods for rapid, coarse, distributed load balancing, SPAA, 1996, pp. 72–81

    Google Scholar 

  14. Hu, Y.F., Blake, R.J.: An improved diffusion algorithm for dynamic load balancing, Parallel Computing 25, 1999, pp. 417–444

    Article  MATH  MathSciNet  Google Scholar 

  15. Hu, Y.F., Blake, R.J., Emerson, D.R.: An optimal migration algorithm for dynamic load balancing, Concurrency: Prac. and Exp. 10, 1998, pp. 467–483

    Article  MATH  Google Scholar 

  16. Hubaut, X. L.:Strongly Regular Graphs, Discrete Math. 13, 1975, pp. 357–381

    Article  MATH  MathSciNet  Google Scholar 

  17. Tillich, J.-P.: The spectrum of the double-rooted tree, personal communication

    Google Scholar 

  18. Xu, C., Lau, F.C.M.: Load Balancing in Parallel Computers, Kluwer, 1997

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of Paderborn, Germany

    Robert Elsässer & Burkhard Monien

  2. Comenius University, MFF-UK Bratislava, Slovakia

    Rastislav Královič

Authors
  1. Robert Elsässer
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  2. Rastislav Královič
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  3. Burkhard Monien
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Editor information

Editors and Affiliations

  1. I3S & INRIA Sophia Antipolis INRIA, CNRS, 2004 Route des Lucioles, 06902, Sophia-Antipolis, France

    Afonso Ferreira

  2. Institut für Theoretische Informatik, Fakultät Informatik, TU Dresden, 01062, Dresden, Germany

    Horst Reichel

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Elsässer, R., Královič, R., Monien, B. (2001). Scalable Sparse Topologies with Small Spectrum. In: Ferreira, A., Reichel, H. (eds) STACS 2001. STACS 2001. Lecture Notes in Computer Science, vol 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44693-1_19

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  • DOI: https://doi.org/10.1007/3-540-44693-1_19

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41695-1

  • Online ISBN: 978-3-540-44693-4

  • eBook Packages: Springer Book Archive

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