Random input helps searching predecessors

We solve the dynamic Predecessor Problem with high probability (whp) in constant time, using only n bits of memory, for any constant δ > 0. The input keys are random wrt a wider class of the well studied and practically important class of (f1, f2)-smooth distributions introduced in [3]. It achieves O(1) whp amortized time. Its worst-case time is O( √ logn log logn ). Also, we prove whp O(log log log n) time using only n 1 log log n = n bits. Finally, we show whp O(log log n) time using O(n) space.