Advertisement

A Partially Synchronizing Coloring

  • Avraham N. Trahtman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6072)

Abstract

Given a finite directed graph, a coloring of its edges turns the graph into a finite-state automaton. A k-synchronizing word of a deterministic automaton is a word in the alphabet of colors at its edges that maps the state set of the automaton at least on k-element subset. A coloring of edges of a directed strongly connected finite graph of a uniform outdegree (constant outdegree of any vertex) is k-synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a k-synchronizing word.

For k = 1 one has the well known road coloring problem. The recent positive solution of the road coloring problem implies an elegant generalization considered first by Béal and Perrin: a directed finite strongly connected graph of uniform outdegree is k-synchronizing iff the greatest common divisor of lengths of all its cycles is k.

Some consequences for coloring of an arbitrary finite digraph are presented. We describe a subquadratic algorithm of the road coloring for the k-synchronization implemented in the package TESTAS. A new linear visualization program demonstrates the obtained coloring. Some consequences for coloring of an arbitrary finite digraph and of such a graph of uniform outdegree are presented.

Keywords

graph algorithm synchronization road coloring deterministic finite automaton 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler, R.L., Weiss, B.: Similarity of automorphisms of the torus, Memoirs of the Amer. Math. Soc., Providence, RI, 98 (1970)Google Scholar
  2. 2.
    Aho, A., Hopcroft, J., Ulman, J.: The Design and Analisys of Computer Algorithms. Addison-Wesley, Reading (1974)Google Scholar
  3. 3.
    Bauer, T., Cohen, N., Trahtman, A.N.: The visualization of digraph based on its structure properties. In: RTAGS, Ramat-Gan, Israel (2007)Google Scholar
  4. 4.
    Béal, M.P., Perrin, D.: A quadratic algorithm for road coloring (2008), arXiv: 0803.0726v2 [cs.DM]Google Scholar
  5. 5.
    Budzban, G., Feinsilver, P.: The Generalized Road Coloring Problem (2008), arXiv:0903.0192v, [cs.DM]Google Scholar
  6. 6.
    Culik, K., Karhumaki, J., Kari, J.: A note on synchronized automata and Road Coloring Problem. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 175–185. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Černy, J.: Poznamka k homogenym eksperimentom s konechnymi automatami. Math.-Fyz. Čas. 14, 208–215 (1964)zbMATHGoogle Scholar
  8. 8.
    Kari, J.: Synchronizing finite automata on Eulerian digraphs. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 432–438. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Mateescu, A., Salomaa, A.: Many-valued truth function. Černy conjuncture and road coloring, EATCS Bulletin 68, 134–150 (1999)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Trahtman, A.N.: Synchronizing Road Coloring. In: 5-th IFIP WCC-TCS, vol. 273, pp. 43–53. Springer, Heidelberg (2008)Google Scholar
  11. 11.
    Trahtman, A.N.: The Road Coloring and Cerny Conjecture. In: Proc. of Prague Stringology Conference, pp. 1–12 (2008)Google Scholar
  12. 12.
    Trahtman, A.N.: The Road Coloring for Mapping on k States (2008), arXiv:0812.4798, [cs.DM]Google Scholar
  13. 13.
    Trahtman, A.N.: The road coloring problem. Israel Journal of Math. 1(172), 51–60 (2009)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Avraham N. Trahtman
    • 1
  1. 1.Dep. of Math.Bar-Ilan UniversityRamat GanIsrael

Personalised recommendations