An efficient string sorting algorithm for weighing matrices of small weight
- 70 Downloads
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.
KeywordsWeighing matrices Algorithm Pattern String sorting Sparse encoding
Mathematics Subject Classification (2000)05B20 62K05
Unable to display preview. Download preview PDF.
- 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.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
- 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.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.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
- 9.Kreher D.L., Stinson D.R.: Combinatorial Algorithms: Generation, Enumeration and Search. CRC Press, Boca Raton (1998)Google Scholar
- 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