Trade-offs for fully dynamic transitive closure on DAGs: Breaking through the O(n(2)) barrier

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(epsilon)) worst-case time and perform updates in O(n(omega(1, epsilon, 1)-epsilon) + n(1+epsilon)) worst-case time, for any epsilon is an element of [0, 1], where omega(1, epsilon, 1) is the exponent of the multiplication of an n x n(epsilon) matrix by an n(epsilon) x n matrix. The current best bounds on omega(1, epsilon, 1) imply an O(n(0.575)) query time and 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.