We present an algorithm for directed acyclic graphs that breaks through the O(n 2 ) barrier on the single-operation complexity of fully dynamic transitive closure, where n is the number of edges in the graph. We can answer queries in O(n ) worst-case time and perform updates in O(n ω(1, ,1)− + n 1+ ) worst-case time, for any ∈ [0, 1], where ω(1, , 1) is the exponent of the multiplication of an n × n matrix by an n × n matrix. The current best bounds on ω(1, , 1) imply an O(n 0.575 ) query time

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... nd an O(n 1.575 ) update time in the worst case. Our subquadratic algorithm is randomized, and has one-sided error. As an application of this result, we show how to solve single-source reachability in O(n 1.575 ) time per update and constant time per query.