Efficient Parallel Strategy Improvement for Parity Games

We study strategy improvement algorithms for solving parity games. While these algorithms are known to solve parity games using a very small number of iterations, experimental studies have found that a high step complexity causes them to perform poorly in practice. In this paper we seek to address this situation. Every iteration of the algorithm must compute a best response, and while the standard way of doing this uses the Bellman-Ford algorithm, we give experimental results that show that one-player strategy improvement significantly outperforms this technique in practice. We then study the best way to implement one-player strategy improvement, and we develop an efficient parallel algorithm for carrying out this task, by reducing the problem to computing prefix sums on a linked list. We report experimental results for these algorithms, and we find that a GPU implementation of this algorithm shows a significant speedup over single-core and multi-core CPU implementations.

[1]  Michael Luttenberger,et al.  Solving Parity Games on the GPU , 2013, ATVA.

[2]  Jorne Kandziora.  Playing Parity Games on the Playstation 3 , 2009 .

[3]  E. Allen Emerson,et al.  Tree automata, mu-calculus and determinacy , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[4]  Marcin Jurdzinski,et al.  Small Progress Measures for Solving Parity Games , 2000, STACS.

[5]  Richard Cole,et al.  Faster Optimal Parallel Prefix Sums and List Ranking , 2011, Inf. Comput..

[6]  Thomas Wilke,et al.  Automata Logics, and Infinite Games , 2002, Lecture Notes in Computer Science.

[7]  Jeroen Keiren,et al.  Benchmarks for Parity Games , 2014, FSEN.

[8]  Richard M. Karp,et al.  A characterization of the minimum cycle mean in a digraph , 1978, Discret. Math..

[9]  Marcin Jurdziński,et al.  Deciding the Winner in Parity Games is in UP \cap co-Up , 1998, Inf. Process. Lett..

[10]  Colin Stirling,et al.  Local model checking games (extended abstract) , 1995 .

[11]  Michael Huth,et al.  Concurrent Small Progress Measures , 2011, Haifa Verification Conference.

[12]  Freark van der Berg.  Solving Parity Games on the Playstation 3 , 2010 .

[13]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[14]  Marcin Jurdzinski,et al.  A Discrete Strategy Improvement Algorithm for Solving Parity Games , 2000, CAV.

[15]  Michael Luttenberger,et al.  Solving Mean-Payoff Games on the GPU , 2016, ATVA.

[16]  Henrik Björklund,et al.  A combinatorial strongly subexponential strategy improvement algorithm for mean payoff games , 2007, Discret. Appl. Math..

[17]  Oliver Friedmann,et al.  An Exponential Lower Bound for the Latest Deterministic Strategy Iteration Algorithms , 2011, Log. Methods Comput. Sci..

[18]  Wieslaw Zielonka,et al.  Infinite Games on Finitely Coloured Graphs with Applications to Automata on Infinite Trees , 1998, Theor. Comput. Sci..

[19]  A. Prasad Sistla,et al.  On Model-Checking for Fragments of µ-Calculus , 1993, CAV.

[20]  Michael Luttenberger,et al.  Strategy Iteration using Non-Deterministic Strategies for Solving Parity Games , 2008, ArXiv.

[21]  Jaco van de Pol,et al.  A Multi-Core Solver for Parity Games , 2008, Electron. Notes Theor. Comput. Sci..

[22]  A. Puri.  Theory of hybrid systems and discrete event systems , 1996 .

[23]  Anne Condon,et al.  On Algorithms for Simple Stochastic Games , 1990, Advances In Computational Complexity Theory.

[24]  John Fearnley,et al.  The complexity of all-switches strategy improvement , 2016, SODA 2016.

[25]  Zheng Wei,et al.  Optimization of linked list prefix computations on multithreaded GPUs using CUDA , 2010, 2010 IEEE International Symposium on Parallel & Distributed Processing (IPDPS).

[26]  Micha Sharir,et al.  A subexponential bound for linear programming , 1992, SCG '92.

[27]  Uri Zwick,et al.  The Complexity of Mean Payoff Games on Graphs , 1996, Theor. Comput. Sci..

[28]  Oliver Friedmann,et al.  Solving Parity Games in Practice , 2009, ATVA.

[29]  R. Karp,et al.  On Nonterminating Stochastic Games , 1966 .

[30]  Sven Schewe,et al.  An Optimal Strategy Improvement Algorithm for Solving Parity and Payoff Games , 2008, CSL.

[31]  Joseph JáJá,et al.  Designing Practical Efficient Algorithms for Symmetric Multiprocessors , 1999, ALENEX.