# The Strahler number of a parity game

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 parameter: it coincides with its alternative version based on trees of progress measures and 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 ·(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(√( n)) and k = O(√( n)).

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• 5 publications
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research
01/13/2020

### A Universal Attractor Decomposition Algorithm for Parity Games

An attractor decomposition meta-algorithm for solving parity games is gi...
research
04/29/2023

### Improved Complexity Analysis of Quasi-Polynomial Algorithms Solving Parity Games

We improve the complexity of solving parity games (with priorities in ve...
research
03/13/2018

### A pseudo-quasi-polynomial algorithm for solving mean-payoff parity games

In a mean-payoff parity game, one of the two players aims both to achiev...
research
05/02/2022

### Smaller Progress Measures and Separating Automata for Parity Games

Calude et al. have recently shown that parity games can be solved in qua...
research
01/29/2018

### An Optimal Value Iteration Algorithm for Parity Games

The quest for a polynomial time algorithm for solving parity games gaine...
research
04/26/2019

### Improving the complexity of Parys' recursive algorithm

Parys has recently proposed a quasi-polynomial version of Zielonka's rec...
research
09/11/2019

### Quasipolynomial Set-Based Symbolic Algorithms for Parity Games

Solving parity games, which are equivalent to modal μ-calculus model che...