A Universal Attractor Decomposition Algorithm for Parity Games
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An attractor decomposition meta-algorithm for solving parity games is given that generalizes the classic McNaughton-Zielonka algorithm and its recent quasi-polynomial variants due to Parys (2019), and to Lehtinen, Schewe, and Wojtczak (2019). The central concepts studied and exploited are attractor decompositions of dominia in parity games and the ordered trees that describe the inductive structure of attractor decompositions. The main technical results include the embeddable decomposition theorem and the dominion separation theorem that together help establish a precise structural condition for the correctness of the universal algorithm: it suffices that the two ordered trees given to the algorithm as inputs embed the trees of some attractor decompositions of the largest dominia for each of the two players, respectively. The universal algorithm yields McNaughton-Zielonka, Parys's, and Lehtinen-Schewe-Wojtczak algorithms as special cases when suitable universal trees are given to it as inputs. The main technical results provide a unified proof of correctness and deep structural insights into those algorithms. A symbolic implementation of the universal algorithm is also given that improves the symbolic space complexity of solving parity games in quasi-polynomial time from O(d n)—achieved by Chatterjee, Dvořák, Henzinger, and Svozil (2018)—down to O( d), where n is the number of vertices and d is the number of distinct priorities in a parity game. This not only exponentially improves the dependence on d, but it also entirely removes the dependence on n.
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