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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 end-component 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.