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 ) 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 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.
Alternatively, the transitive closure can be maintained dynamically REF , but updating the graph takes O(n 2 ) time.
Trade-offs for fully dynamic transitive closure on DAGs: breaking through the O(n2 barrier
JACM
Computer Science Mathematics