# An Ordered Approach to Solving Parity Games in Quasi Polynomial Time and Quasi Linear Space

## TL;DR

This paper presents an efficient implementation of a quasi-polynomial time algorithm for solving parity games, improving practical performance and providing new complexity bounds and algorithmic insights.

## Contribution

It introduces a backward implementation of Calude et al.'s algorithm using a progress measure, with modifications enabling efficient solving and new complexity results.

## Key findings

- First implementation of a quasi-polynomial parity game solver
- Achieved quasi bi-linear complexity for fixed colors
- Provided matching lower bounds for existing algorithms

## Abstract

Parity games play an important role in model checking and synthesis. In their paper, Calude et al. have shown that these games can be solved in quasi-polynomial time. We show that their algorithm can be implemented efficiently: we use their data structure as a progress measure, allowing for a backward implementation instead of a complete unravelling of the game. To achieve this, a number of changes have to be made to their techniques, where the main one is to add power to the antagonistic player that allows for determining her rational move without changing the outcome of the game. We provide a first implementation for a quasi-polynomial algorithm, test it on small examples, and provide a number of side results, including minor algorithmic improvements, a quasi bi-linear complexity in the number of states and edges for a fixed number of colours, and matching lower bounds for the algorithm of Calude et al.