The recent breakthrough paper by Calude et al. has given the first algorithm
for solving parity games in quasi-polynomial time, where previously the best
algorithms were mildly subexponential. We devise an alternative
quasi-polynomial time algorithm based on progress measures, which allows us to
reduce the space required from quasi-polynomial to nearly linear. Our key
technical tools are a novel concept of ordered tree coding, and a succinct tree
coding result that we prove using bounded adaptive multi-counters, both of
which are interesting in their own right
