  We consider Markov decision processes (MDPs) with \omega-regular
specifications given as parity objectives. We consider the problem of computing
the set of almost-sure winning states from where the objective can be ensured
with probability 1. The algorithms for the computation of the almost-sure
winning set for parity objectives iteratively use the solutions for the
almost-sure winning set for B\"uchi objectives (a special case of parity
objectives). Our contributions are as follows: First, we present the first
subquadratic symbolic algorithm to compute the almost-sure winning set for MDPs
with B\"uchi objectives; our algorithm takes O(n \sqrt{m}) symbolic steps as
compared to the previous known algorithm that takes O(n^2) symbolic steps,
where $n$ is the number of states and $m$ is the number of edges of the MDP. In
practice MDPs have constant out-degree, and then our symbolic algorithm takes
O(n \sqrt{n}) symbolic steps, as compared to the previous known $O(n^2)$
symbolic steps algorithm. Second, we present a new algorithm, namely win-lose
algorithm, with the following two properties: (a) the algorithm iteratively
computes subsets of the almost-sure winning set and its complement, as compared
to all previous algorithms that discover the almost-sure winning set upon
termination; and (b) requires O(n \sqrt{K}) symbolic steps, where K is the
maximal number of edges of strongly connected components (scc's) of the MDP.
The win-lose algorithm requires symbolic computation of scc's. Third, we
improve the algorithm for symbolic scc computation; the previous known
algorithm takes linear symbolic steps, and our new algorithm improves the
constants associated with the linear number of steps. In the worst case the
previous known algorithm takes 5n symbolic steps, whereas our new algorithm
takes 4n symbolic steps.
