A Randomized In-Place Algorithm for Positioning the kth Element in a Multiset

Abstract

A variant of the classical selection problem, called the positioning problem, is considered. In this problem we are given a sequence A[1:n] of size n, an integer k, 1 ≤ k ≤ n, and an ordering function ⧀, and the task is to rearrange the elements of the sequence such that A[k] ⧀ A[j] is false for all j, 1 ≤ j < k, and A[l] ⧀ A[k] is false for all l, k < l ≤ n. We present a Las-Vegas algorithm which carries out this rearrangement efficiently using only a constant amount of additional space even if the input contains equal elements and if only pairwise element comparisons are permitted. To be more precise, the algorithm solves the positioning problem in-place in linear time using at most n + k + o(n) element comparisons, k + o(n) element exchanges, and the probability for succeeding within stated time bounds is at least 1 - e ^{-n Ω(1)}

Partially supported by Danish Natural Science Research Council under contract 9701414 (project Experimental Algorithmics) and contract 9801749 (project Performance Engineering).