Las Vegas versus determinism for one-way communication complexity, finite automata, and polynomial-time computations

  • Pavol Ďuriš
  • Juraj Hromkovič
  • José D. P. Rolim
  • Georg Schnitger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1200)


The study of the computational power of randomized computations is one of the central tasks of complexity theory. The main aim of this paper is the comparison of the power of Las Vegas computation and deterministic respectively nondeterministic computation. An at most polynomial gap has been established for the combinational complexity of circuits and for the communication complexity of two-party protocols. We investigate the power of Las Vegas computation for the complexity measures of one-way communication, finite automata and polynomialtime relativized Turing machine computation.
  1. (i)

    For the one-way communication complexity of two-party protocols we show that Las Vegas communication can save at most one half of the deterministic one-way communication complexity. We also present a language for which this gap is tight.

  2. (ii)

    For the size (i.e., the number of states) of finite automata we show that the size of Las Vegas finite automata recognizing a language L is at least the root of the size of the minimal deterministic finite automaton recognizing L. Using a specific language we verify the optimality of this lower bound.


Note, that this result establishes for the first time an at most polynomial gap between Las Vegas and determinism for a uniform computing model.

  1. (iii)

    For relativized polynomial computations we show that Las Vegas can be even more powerful than nondeterminism with a polynomial restriction on the number of nondeterministic guesses.


On the other hand superlogarithmic many advice bits in nondeterministic computations can be more powerful than Las Vegas (even Monte Carlo) computations in a relativized word.


computational and structural complexity Las Vegas determinism communication complexity automata 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aho, A.V., Hopcroft, J.E., Yannakakis, M.: On notions of information transfer in VLSI circuits. In: Proc. 15th Annual ACM STOC, ACM 1983, 133–139.Google Scholar
  2. 2.
    Bovet,D.P., Crescenzi,P.: Introduction to the Theory of Complexity. Prentice Hall 1994.Google Scholar
  3. 3.
    Diaz, J. Torán, J.: Classes of bounded nondeterminism. Mathematical Systems Theory, 23 (1990), 21–32.CrossRefGoogle Scholar
  4. 4.
    Freivalds, R.: Probabilistic two-way machines. Lecture Notes in Computer Science 118, Springer-Verlag, Berlin 1981, 33–45.Google Scholar
  5. 5.
    Hromkovič, J., Schnitger, G.: On the power of the number of advice bits in non-deterministic computations. Proc. ACM STOC'96, ACM 1996, pp. 551–560.Google Scholar
  6. 6.
    Meyer, A.R., Fischer, M.J.: Economies of description by automata, grammars and formal systems. In: Proceedings 12th SWAT Symp. 1971, 188–191Google Scholar
  7. 7.
    Mehlhorn,K., Schmidt,E.: Las Vegas is better than determinism in VLSI and distributed computing. Proc. 14th ACM STOC'82, ACM 1982, pp. 330–337.Google Scholar
  8. 8.
    Yao, A.C.: Some complexity questions related to distributed computing. In: Proc. 11th Annual ACM STOC, ACM 1981, 308–311.Google Scholar
  9. 9.
    Csiszar, I., Körner, J.: Information theory: coding theorems for discrete memeoryless systems, Academic Press, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Pavol Ďuriš
    • 1
  • Juraj Hromkovič
    • 2
  • José D. P. Rolim
    • 3
  • Georg Schnitger
    • 4
  1. 1.Department of Computer ScienceComenius UniversityBratislavaSlovakia
  2. 2.Institut für InformatikUniversität zu KielKielGermany
  3. 3.Centre Universitaire d'InformatiqueUniversité de GenèveGenéve 4Switzerland
  4. 4.Fachbereich InformatikJohann Wolfgang Goethe-Universität FrankfurtFrankfurt am MainGermany

Personalised recommendations