Abstract

We present an algorithm for directed acyclic graphs that breaks through the O(n²) 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 ɛ) worst-case time and perform updates in O(n ω(1,ɛ,1)−ɛ + n 1+ɛ) worst-case time, for any ɛ ∈ [0,1], where ω(1,ɛ,1) is the exponent of the multiplication of an n ×n ɛ matrix by an n ɛ ×n matrix. The current best bounds on ω(1,ɛ,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.

Citations