Strategy Improvement and Randomized Subexponential Algorithms for Stochastic Parity Games

  • Krishnendu Chatterjee
  • Thomas A. Henzinger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3884)


A stochastic graph game is played by two players on a game graph with probabilistic transitions. We consider stochastic graph games with ω-regular winning conditions specified as parity objectives. These games lie in NP ∩ coNP. We present a strategy improvement algorithm for stochastic parity games; this is the first non-brute-force algorithm for solving these games. From the strategy improvement algorithm we obtain a randomized subexponential-time algorithm to solve such games.


Successor State Strategy Improvement Stochastic Game Game Graph Fairness Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bjorklund, H., Sandberg, S., Vorobyov, S.: A discrete subexponential algorithm for parity games. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 663–674. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Chatterjee, K., de Alfaro, L., Henzinger, T.A.: The complexity of stochastic Rabin and Streett games. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 878–890. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Chatterjee, K., Jurdziński, M., Henzinger, T.A.: Simple stochastic parity games. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 100–113. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Chatterjee, K., Jurdziński, M., Henzinger, T.A.: Quantitative stochastic parity games. In: SODA, pp. 114–123. SIAM, Philadelphia (2004)Google Scholar
  5. 5.
    Condon, A.: The complexity of stochastic games. Information and Computation 96, 203–224 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Condon, A.: On algorithms for simple stochastic games. In: Advances in Computational Complexity Theory, American Mathematical Society, pp. 51–73 (1993)Google Scholar
  7. 7.
    Emerson, E.A., Jutla, C.: The complexity of tree automata and logics of programs. In: FOCS, pp. 328–337. IEEE Computer Society Press, Los Alamitos (1988)Google Scholar
  8. 8.
    Hoffman, A., Karp, R.: On nonterminating stochastic games. Management Science 12, 359–370 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jurdziński, M., Paterson, M., Zwick, U.: A deterministic subexponential algorithm for solving parity games. In: SODA (to appear, 2006)Google Scholar
  10. 10.
    Ludwig, W.: A subexponential randomized algorithm for the simple stochastic game problem. Information and Computation 117, 151–155 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Thomas, W.: Languages, automata, and logic. In: Handbook of Formal Languages. Beyond Words, vol. 3, ch. 7, pp. 389–455. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  12. 12.
    Vöge, J., Jurdziński, M.: A discrete strategy improvement algorithm for solving parity games. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 202–215. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Krishnendu Chatterjee
    • 1
  • Thomas A. Henzinger
    • 1
    • 2
  1. 1.EECSBerkeleyUSA
  2. 2.EPFLSwitzerland

Personalised recommendations