Towards Optimal Multiple Selection

The multiple selection problem asks for the elements of rank r1, r2, ..., rk from a linearly ordered set of n elements. Let B denote the information theoretic lower bound on the number of element comparisons needed for multiple selection. We first show that a variant of multiple quickselect — a well known, simple, and practical generalization of quicksort — solves this problem with $B+\mathcal{O}(n)$ expected comparisons. We then develop a deterministic divide-and-conquer algorithm that solves the problem in $\mathcal{O}(B)$ time and $B+o(B)+\mathcal{O}(n)$ element comparisons.