Minimal Eulerian Circuit in a Labeled Digraph

  • Eduardo Moreno
  • Martín Matamala
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3887)

Abstract

Let G = (V,A) be an Eulerian directed graph with an arc-labeling. In this work we study the problem of finding an Eulerian circuit of lexicographically minimal label among all Eulerian circuits of the graph. We prove that this problem is NP-hard by showing a reduction from the Directed-Hamiltonian-Circuit problem.

If the labeling of the arcs is such that arcs going out from the same vertex have different labels, the problem can be solved in polynomial time. We present an algorithm to construct the unique Eulerian circuit of lexicographically minimal label starting at a fixed vertex. Our algorithm is a recursive greedy algorithm which runs in \({\mathcal O}\)(|A|) steps.

We also show an application of this algorithm to construct the minimal De Bruijn sequence of a language.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Tutte, W.T.: Graph theory. Encyclopedia of Mathematics and its Applications, vol. 21. Addison-Wesley Publishing Company Advanced Book Program, Reading (1984)MATHGoogle Scholar
  2. 2.
    Cheng, W.C., Pedram, M.: Power-optimal encoding fod DRAM address bus. In: ISLPED, pp. 250–252. ACM, New York (2000)CrossRefGoogle Scholar
  3. 3.
    Kari, J.: Synchronizing finite automata on Eulerian digraphs. Theoret. Comput. Sci. 295(1-3), 223–232 (2003); Mathematical foundations of computer science (Mariánské Lázně, 2001)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Blazewicz, J., Hertz, A., Kobler, D., de Werra, D.: On some properties of DNA graphs. Discrete Appl. Math. 98(1-2), 1–19 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Pevzner, P.A.: L-tuple DNA sequencing: computer analysis. J. Biomol. Struct. Dyn. 7, 63–73 (1989)Google Scholar
  6. 6.
    Pevzner, P.A., Tang, H., Waterman, M.S.: An eulerian path approach to DNA fragment assembly. Proceedings of the National Academy of Sciences 98(17), 9748–9753 (2001)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    de Bruijn, N.G.: A combinatorial problem. Nederl. Akad. Wetensch., Proc. 49, 758–764 (1946)MathSciNetGoogle Scholar
  8. 8.
    Stein, S.K.: The mathematician as an explorer. Sci. Amer. 204(5), 148–158 (1961)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bermond, J.C., Dawes, R.W., Ergincan, F.Ö.: De Bruijn and Kautz bus networks. Networks 30(3), 205–218 (1997)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chung, F., Diaconis, P., Graham, R.: Universal cycles for combinatorial structures. Discrete Math. 110(1-3), 43–59 (1992)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and intractability. W.H. Freeman and Co., San Francisco (1979); A guide to the theory of NP-completeness, A Series of Books in the Mathematical SciencesMATHGoogle Scholar
  12. 12.
    Gibbons, A.: Algorithmic graph theory. Cambridge University Press, Cambridge (1985)MATHGoogle Scholar
  13. 13.
    Moreno, E.: De Bruijn sequences and de Bruijn graphs for a general language. Inf. Process. Lett. 96, 214–219 (2005)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Moreno, E., Matamala, M.: Minimal de Bruijn sequence in a language with forbidden subtrings. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 168–176. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Eduardo Moreno
    • 1
  • Martín Matamala
    • 1
  1. 1.Departamento de Ingeniería Matemática, Centro de Modelamiento MatemáticoUniversidad de Chile, UMR 2071, UCHILE-CNRSSantiagoChile

Personalised recommendations