Computing the winning set for Büchi objectives in alternating games on graphs is a central problem in computer aided verification with a large number of applications. The long standing best known upper bound for solving the problem is Õ(n·m), where n is the number of vertices and m is the number of edges in the graph. We are the first to break the Õ(n·m) boundary by presenting a new technique that reduces the running time to O(n²). This bound also leads to O(n²) time algorithms for computing the set of almost sure winning vertices for Büchi objectives (1) in alternating games with probabilistic transitions (improving an earlier bound of Õ(n·m)), (2) in concurrent graph games with constant actions (improving an earlier bound of O(n³)), and (3) in Markov decision processes (improving for m > n⁴⸍³ an earlier bound of O(min(m¹·⁵,m·n²⸍³)). We also show that the same technique can be used to compute the maximal endcomponent decomposition of a graph in time O(n²), which is an improvement over earlier bounds for m > n⁴⸍³. Finally, we show how to maintain the winning set for Büchi objectives in alternating games under a sequence of edge insertions or a sequence of edge deletions in O(n) amortized time per operation. This is the first dynamic algorithm for this problem.

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