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The Strahler Number of a Parity Game

Authors Laure Daviaud , Marcin Jurdziński , K. S. Thejaswini



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Author Details

Laure Daviaud
  • CitAI, Department of Computer Science, City, University of London, UK
Marcin Jurdziński
  • Department of Computer Science, University of Warwick, Coventry, UK
K. S. Thejaswini
  • Department of Computer Science, University of Warwick, Coventry, UK

Acknowledgements

We thank colleagues in the following human chain for contributing to our education on the Strahler number of a tree: the first and the second authors have learnt it from the third author, who has learnt it from K Narayan Kumar, who has learnt it from Dmitry Chistikov, who has learnt it from Radu Iosif and Stefan Kiefer at a workshop on Infinite-State Systems, which has been organized and hosted by Joël Ouaknine and Prakash Panangaden at Bellairs Research Institute.

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Laure Daviaud, Marcin Jurdziński, and K. S. Thejaswini. The Strahler Number of a Parity Game. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 123:1-123:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.123

Abstract

The Strahler number of a rooted tree is the largest height of a perfect binary tree that is its minor. The Strahler number of a parity game is proposed to be defined as the smallest Strahler number of the tree of any of its attractor decompositions. It is proved that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices n and linear in (d/(2k))^k, where d is the number of priorities and k is the Strahler number. This complexity is quasi-polynomial because the Strahler number is at most logarithmic in the number of vertices. The proof is based on a new construction of small Strahler-universal trees. It is shown that the Strahler number of a parity game is a robust, and hence arguably natural, parameter: it coincides with its alternative version based on trees of progress measures and - remarkably - with the register number defined by Lehtinen (2018). It follows that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices and linear in (d/(2k))^k, where k is the register number. This significantly improves the running times and space achieved for parity games of bounded register number by Lehtinen (2018) and by Parys (2020). The running time of the algorithm based on small Strahler-universal trees yields a novel trade-off k ⋅ lg(d/k) = O(log n) between the two natural parameters that measure the structural complexity of a parity game, which allows solving parity games in polynomial time. This includes as special cases the asymptotic settings of those parameters covered by the results of Calude, Jain Khoussainov, Li, and Stephan (2017), of Jurdziński and Lazić (2017), and of Lehtinen (2018), and it significantly extends the range of such settings, for example to d = 2^O(√{lg n}) and k = O(√{lg n}).

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Theory and algorithms for application domains
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Logic and verification
Keywords
  • parity game
  • attractor decomposition
  • progress measure
  • universal tree
  • Strahler number

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