A Counting Approach to Lower Bounds for Selection Problems

Abstract

Lower bounds are derived on the number of comparisons to solve several well-known selection problems Among the problems are finding the t largest elements of a given set m order (Wt), finding the s smallest and t largest elements in order (We.t), and finding the tth largest element (Vt) The results follow from bounds for more general selection problems, where an arbitrary partml order is given The bounds for Wt and Vt generahze to the case where comparisons between hnear functions of the input are allowed The approach is to show that a comparison tree for a selection problem contains a number of trees for smaller problems, thus estabhshmg a lower bound on the number of leaves An equivalent approach uses an adversary, based on a numerical “chaos” function that measures the number of unknown relations. © 1979, ACM. All rights reserved.