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

Abstract

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 ε ) worst-case time and perform updates in O ( n ω(1,ε,1)−ε + n 1+ε ) worst-case time, for any ε∈[0,1], where ω(1,ε,1) is the exponent of the multiplication of an n × n ε matrix by an n ε × n matrix. The current best bounds on ω(1,ε,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.