A Lower Bound for the Complexity of the Union-Split-Find Problem

We prove a Θ(log log n) (i.e. matching upper and lower) bound on the complexity of the Union-Split-Find problem, a variant of the Union-Find problem. Our lower bound holds for all pointer machine algorithms and does not require the separation assumption used in the lower bound arguments of Tarjan [T79] and Blum [B86]. We complement this with a Θ(log n) bound for the Split-Find problem under the separation assumption. This shows that the separation assumption can imply an exponential loss in efficiency.