Finding the αn-th largest element

Abstract

We describe an algorithm for selecting the αn-th largest element (where 0<α<1), from a totally ordered set ofn elements, using at most (1+(1+o(1))H(α))·n comparisons whereH(α) is the binary entropy function and theo(1) stands for a function that tends to 0 as α tends to 0. For small values of α this is almost the best possible as there is a lower bound of about (1+H(α))·n comparisons. The algorithm obtained beats the global 3n upper bound of Schönhage, Paterson and Pippenger for α<1/3.