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

We prove a 8{loglogn) (i.e. matching upper and lower) bound on the complexity of the Union-Split-Find problem, a variant of the Union-Find problem . OUf lower bound holds for all pointer machine algorithms and does [Jot require the separation a~suIllption used ill the lower bound arguments of Tarjan [T79] and Blum [B86]. We complement this with a 9(logn) bound for the Split-Find problem under the separation assumption. This shows that the separation assumption can imply an exponential loss in efficiency. * This research was supported by the Deutsche FOI'5Chungsgemeinschaft, grant SPP Me 620/&-1