Funnelselect: Cache-Oblivious Multiple Selection

Abstract

We present the algorithm funnelselect, the first optimal randomized cache-oblivious algorithm for the multiple-selection problem. The algorithm takes as input an unsorted array of N elements and q query ranks r_1 < ⋯ < r_q, and returns in sorted order the q input elements of rank r_1, …, r_q, respectively. The algorithm uses expected and with high probability O(∑_{i = 1}^{q+1} Δ_i/B ⋅ log_{M/B} N/(Δ_i) + N/B) I/Os, where B is the external memory block size, M ≥ B^{1+ε} is the internal memory size, for some constant ε > 0, and Δ_i = r_i - r_{i-1} (assuming r_0 = 0 and r_{q+1} = N + 1). This is the best possible I/O bound in the cache-oblivious and external memory models. The result is achieved by reversing the computation of the cache-oblivious sorting algorithm funnelsort by Frigo, Leiserson, Prokop and Ramachandran [FOCS 1999], using randomly selected pivots for distributing elements, and pruning computations that with high probability are not expected to contain any query ranks.