  Progress-measure lifting algorithms for solving parity games have the best
worst-case asymptotic runtime, but are limited by their asymmetric nature, and
known from the work of Czerwi\'nski et al. (2018) to be subject to a matching
quasi-polynomial lower bound inherited from the combinatorics of universal
trees. Parys (2019) has developed an ingenious quasi-polynomial McNaughton-
Zielonka-style algorithm, and Lehtinen et al. (2019) have improved its
worst-case runtime. Jurdzi\'nski and Morvan (2020) have recently brought
forward a generic attractor-based algorithm, formalizing a second class of
quasi-polynomial solutions to solving parity games, which have runtime
quadratic in the size of universal trees. First, we adapt the framework of
iterative lifting algorithms to computing attractor-based strategies. Second,
we design a symmetric lifting algorithm in this setting, in which two lifting
iterations, one for each player, accelerate each other in a recursive fashion.
The symmetric algorithm performs at least as well as progress-measure liftings
in the worst-case, whilst bypassing their inherent asymmetric limitation.
Thirdly, we argue that the behaviour of the generic attractor-based algorithm
of Jurdzinski and Morvan (2020) can be reproduced by a specific deceleration of
our symmetric lifting algorithm, in which some of the information collected by
the algorithm is repeatedly discarded. This yields a novel interpretation of
McNaughton-Zielonka-style algorithms as progress-measure lifting iterations
(with deliberate set-backs), further strengthening the ties between all known
quasi-polynomial algorithms to date.
