Advertisement

On Routing in Circulant Graphs

  • Jin-Yi Cai
  • George Havas
  • Bernard Mans
  • Ajay Nerurkar
  • Jean-Pierre Seifert
  • Igor Shparlinski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1627)

Abstract

We investigate various problems related to circulant graphs — finding the shortest path between two vertices, finding the shortest loop, and computing the diameter. These problems are related to shortest vector problems in a special class of lattices. We give matching upper and lower bounds on the length of the shortest loop. We claim NP-hardness results, and establish a worst-case/average-case connection for the shortest loop problem. A pseudo-polynomial time algorithm for these problems is also given. Our main tools are results and methods from the geometry of numbers.

Keywords

Circulant graphs Shortest paths Loops Diameter Lattices 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Ajtai. Generating hard instances of lattice problems. In Proc. 28th Annual ACM Symposium on the Theory of Computing (1996) 99–108.Google Scholar
  2. 2.
    M. Ajtai. The shortest vector problem in L 2 is NP-hard for randomized reductions. In Proc. 30th Annual ACM Symposium on the Theory of Computing (1998) 10–19.Google Scholar
  3. 3.
    M. Ajtai and C. Dwork. A public-key cryptosystem with worst-case/average-case equivalence. In Proc. 29th ACM Symposium on Theory of Computing (1997) 284–293.Google Scholar
  4. 4.
    S. Arora, L. Babai, J. Stern, and Z. Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. In Proc. 34th IEEE Symposium on Foundations of Computer Science (FOCS), 724–733, 1993.Google Scholar
  5. 5.
    J.-C. Bermond, F. Comellas and D. F. Hsu. Distributed loop computer networks: A survey. Journal of Parallel and Distributed Computing 24 (1995) 2–10.CrossRefGoogle Scholar
  6. 6.
    J-Y. Cai. A relation of primal-dual lattices and the complexity of shortest lattice vector problem. Theoretical Computer Science 207 (1998) 105–116.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    J-Y. Cai and A. Nerurkar. An improved worst-case to average-case connection for lattice problems. In Proc. 38th IEEE Symposium on Foundations of Computer Science (1997) 468–477.Google Scholar
  8. 8.
    J-Y. Cai and A. Nerurkar. Approximating the SVP to within a factor \( \left( {1 + \frac{1} {{\dim ^ \in }}} \right) \) is NP-hard under randomized reductions. In Proc. 13th Annual IEEE Conference on Computational Complexity (1998) 46–55.Google Scholar
  9. 9.
    J. W. S. Cassels. An introduction to the geometry of numbers. Springer-Verlag, 1959.Google Scholar
  10. 10.
    N. Chalamaiah and B. Ramamurty. Finding shortest paths in distributed loop networks. Inform. Proc. Letters 67 (1998) 157–161.CrossRefGoogle Scholar
  11. 11.
    Y. Cheng and F. K. Hwang. Diameters of weighted loop networks. J. of Algorithms 9 (1988) 401–410.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J. A. Dias da Silva and Y. O. Hamidoune. Cyclic spaces for Grassmann derivatives and additive theory Bull. Lond. Math. Soc. 26 (1994) 140–146.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    I. Dinur, G. Kindler and S. Safra. Approximating-CVP to within almost-polynomial factors is NP-hard. 1998.Google Scholar
  14. 14.
    P. Erdös and H. Heilbronn On the addition of residue classes mod p Acta Arithm. 9 (1964) 149–159.zbMATHGoogle Scholar
  15. 15.
    O. Goldreich and S. Goldwasser. On the limits of non-approximability of lattice problems. In Proc. 30th Annual ACM Symposium on the Theory of Computing (1998) 1–9.Google Scholar
  16. 16.
    O. Goldreich, S. Goldwasser and S. Halevi. Collision-free hashing from lattice problems. 1996. Available as TR96-042 from Electronic Colloquium on Computational Complexity at http://www.eccc.uni-trier.de/eccc/.
  17. 17.
    O. Goldreich, S. Goldwasser and S. Halevi. Eliminating decryption errors in the Ajtai-Dwork cryptosystem. Advances in Cryptology — CRYPTO '97 (editor B. Kaliski Jr.), Lecture Notes in Computer Science 1294 (Springer Verlag, 1997) 105–111.CrossRefGoogle Scholar
  18. 18.
    O. Goldreich, S. Goldwasser and S. Halevi. Public-key cryptosystems from lattice reduction problems. Advances in Cryptology — CRYPTO '97 (editor B. Kaliski Jr.), Lecture Notes in Computer Science 1294 (Springer Verlag, 1997) 112–131.CrossRefGoogle Scholar
  19. 19.
    D. J. Guan. Finding shortest paths in distributed loop networks. Inform. Proc. Letters 65 (1998) 255–260.CrossRefGoogle Scholar
  20. 20.
    J. C. Lagarias. The computational complexity of simultaneous diophantine approximation problems. In Proc. 23rd IEEE Symposium on Foundations of Computer Science (1982) 32–39.Google Scholar
  21. 21.
    F. T. Leighton. Introduction to parallel algorithms and architectures: Arrays, trees, hypercubes. M. Kaufmann, 1992.Google Scholar
  22. 22.
    A. K. Lenstra, H. W. Lenstra and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen 261 (1982) 515–534.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    B. Mans. Optimal Distributed algorithms in unlabeled tori and chordal rings. Journal on Parallel and Distributed Computing 46 (1997) 80–90.zbMATHCrossRefGoogle Scholar
  24. 24.
    D. Micciancio. The shortest vector in a lattice is hard to approximate to within some constant. In Proc. 39th IEEE Symposium on Foundations of Computer Science (1998) 92–98.Google Scholar
  25. 25.
    A. Paz and C. P. Schnorr. Approximating integer lattices by lattices with cyclic factor groups. Automata, Languages and Programming, 14th International Colloquium, Lecture Notes in Computer Science 267 (Springer-Verlag, 1987) 386–393.Google Scholar
  26. 26.
    R. Raz. Personal communication.Google Scholar
  27. 27.
    C. P. Schnorr. A hierarchy of polynomial time basis reduction algorithms. Theoretical Computer Science 53 (1987) 201–224.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    J-P. Seifert and J. Blömer. On the complexity of computing short linearly independent vectors and short bases in a lattice. To appear in the proceedings of STOC 1999.Google Scholar
  29. 29.
    P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in lattices. Technical Report 81-04, Mathematics Department, University of Amsterdam, 1981.Google Scholar
  30. 30.
    J. Žervonik and T. Pisanski. Computing the diameter in multi-loop networks. J. of Algorithms 14 (1993) 226–243.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • George Havas
    • 2
  • Bernard Mans
    • 3
  • Ajay Nerurkar
    • 1
  • Jean-Pierre Seifert
    • 4
  • Igor Shparlinski
    • 3
  1. 1.Department of Computer Science and EngineeringState University of NY at BuffaloBuffaloUSA
  2. 2.Centre for Discrete Mathematics and Computing, Department of Computer Science and Electrical EngineeringThe University of QueenslandQldAustralia
  3. 3.Department of ComputingMacquarie UniversitySydneyAustralia
  4. 4.Department of Mathematics and Computer ScienceUniversity of FrankfurtFrankfurt on the MainGermany

Personalised recommendations