Random input helps searching predecessors

We solve the dynamic Predecessor Problem with high probability (whp) in constant time, using only \(n^{1+\delta}\) bits of memory, for any constant \(\delta > 0\). The input keys are random wrt a wider class of the well studied and practically important class of \((f_1, f_2)\)-smooth distributions introduced in \cite{and:mat}. It achieves O(1) whp amortized time. Its worst-case time is \(O(\sqrt{\frac{\log n}{\log \log n}})\). Also, we prove whp \(O(\log \log \log n)\) time using only \(n^{1+ \frac{1}{\log \log n}}= n^{1+o(1)}\) bits. Finally, we show whp \(O(\log \log n)\) time using O(n) space.