Optimization Letters

, Volume 4, Issue 1, pp 29–36 | Cite as

An efficient string sorting algorithm for weighing matrices of small weight

  • Ilias S. Kotsireas
  • Christos Koukouvinos
  • Panos M. Pardalos
Original Paper

Abstract

In this paper, we demonstrate that the search for weighing matrices of small weights constructed from two circulants can be viewed as a string sorting problem together with a linear time algorithm to locate common strings in two sorted arrays. We also introduce a sparse encoding of the periodic autocorrelation function vector, based on concepts from Algorithmic Information Theory, also known as Kolmogorov complexity, that allows us to speed up the algorithm considerably. Finally, we use these ideas to find new weighing matrices W(2 · n, 9) constructed from two circulants, for many values of n in the range 100 ≤  n ≤  300. These matrices are given here for the first time. We also discuss briefly a connection with Combinatorial Optimization.

Keywords

Weighing matrices Algorithm Pattern String sorting Sparse encoding 

Mathematics Subject Classification (2000)

05B20 62K05 

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References

  1. 1.
    Chaitin, G.J.: Algorithmic information theory. Cambridge Tracts in Theoretical Computer Science (with a foreword by J.T. Schwartz). Cambridge University Press, Cambridge (1987)Google Scholar
  2. 2.
    Craigen R., Kharaghani H.: Orthogonal designs. In: Colbourn, C.J., Dinitz, J.H. (eds) The CRC Handbook of Combinatorial Designs, 2nd edn, pp. 280–295. CRC Press, Boca Raton (2006)Google Scholar
  3. 3.
    Floudas, C.A., Pardalos, P.M.: Encyclopedia of Optimization. vol. I–VI. Kluwer, Dordrecht (2001)CrossRefGoogle Scholar
  4. 4.
    Georgiou S., Koukouvinos C.: New infinite classes of weighing matrices. Sankhyā Ser. B 64(1), 26–36 (2002)MathSciNetGoogle Scholar
  5. 5.
    Geramita A.V., Seberry J.: Orthogonal Designs. Quadratic forms and Hadamard Matrices. Lecture Notes in Pure and Applied Mathematics, vol. 45. Marcel Dekker, New York (1979)Google Scholar
  6. 6.
    Knuth D.E.: The art of computer programming, vol. 3. Sorting and Searching, Addison-Wesley Series in Computer Science and Information Processing. Addison-Wesley, Reading (1973)Google Scholar
  7. 7.
    Kotsireas, I.S., Koukouvinos, C., Pardalos, P.M., Shylo, O.V.: Periodic complementary binary sequences and Combinatorial Optimization algorithms. J. Comb. Optim. (to appear)Google Scholar
  8. 8.
    Koukouvinos C., Seberry J.: New weighing matrices and orthogonal designs constructed using two sequences with zero autocorrelation function-a review. J. Statist. Plann. Inference 81(1), 153–182 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kreher D.L., Stinson D.R.: Combinatorial Algorithms: Generation, Enumeration and Search. CRC Press, Boca Raton (1998)Google Scholar
  10. 10.
    Li M., Vitányi P.: An Introduction to Kolmogorov Complexity and its Applications, Graduate Texts in Computer Science, 2nd edn. Springer, New York (1997)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Ilias S. Kotsireas
    • 1
  • Christos Koukouvinos
    • 2
  • Panos M. Pardalos
    • 3
  1. 1.Department of Physics and Computer ScienceWilfrid Laurier UniversityWaterlooCanada
  2. 2.Department of MathematicsNational Technical University of AthensZografou, AthensGreece
  3. 3.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

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