Rabin Games and Colourful Universal Trees

TL;DR

This paper introduces an improved algorithm for solving Rabin and Streett games on graphs, utilizing colourful universal trees to achieve better time and space complexity than previous methods.

Contribution

It presents a novel algorithm that leverages colourful universal trees for Rabin game solving, improving runtime and space efficiency over existing approaches.

Findings

Achieves a super quadratic improvement in dependence on k! in runtime.

Reduces space complexity compared to previous algorithms.

Introduces a new characterisation of progress measures using colourful trees.

Abstract

We provide an algorithm to solve Rabin and Streett games over graphs with $n$ vertices, $m$ edges, and $k$ colours that runs in $\tilde{O}\left(mn(k!)^{1+o(1)} \right)$ time and $O(nk\log k \log n)$ space, where $\tilde{O}$ hides poly-logarithmic factors. Our algorithm is an improvement by a super quadratic dependence on $k!$ from the currently best known run time of $O\left(mn^2(k!)^{2+o(1)}\right)$, obtained by converting a Rabin game into a parity game, while simultaneously improving its exponential space requirement. Our main technical ingredient is a characterisation of progress measures for Rabin games using \emph{colourful trees} and a combinatorial construction of succinctly-represented, universal colourful trees. Colourful universal trees are generalisations of universal trees used by Jurdzi\'{n}ski and Lazi\'{c} (2017) to solve parity games, as well as of Rabin progress measures of Klarlund and Kozen (1991). Our algorithm for Rabin games is a progress measure lifting algorithm where the lifting is performed on succinct, colourful, universal trees.