Interval Sorting

Abstract

In the interval sort problem, we are given an array A of n items and p ranges of ranks I ₁ = [ℓ₁,u ₁], ..., I _p  = [ℓ _p ,u _p ]. The goal is to rearrange the array so that A[ℓ _t ..u _t ] contains the ℓ _t -th, ..., u _t -th smallest elements of A in nondecreasing order, for all t, 1 ≤ t ≤ p, and A[u _t  + 1..ℓ _t + 1− 1] contains the (u _t  + 1), ..., (ℓ _t + 1− 1) smallest elements of A, for all t, 0 ≤ t ≤ p. That is, the array is sorted by blocks, with sorted and unsorted blocks alternating. One of the most interesting aspects of this research is the unification of several important and related problems (sorting, selection, multiple selection, partial sorting) under a single framework. Results on interval sorting generalize the results for any of these particular—and fundamental—problems.

We propose a divide-and-conquer algorithm, owing to quicksort and quickselect, named chunksort, to solve the problem. We give an exact expression for the average number of comparisons made by the basic variant of chunksort. Then we consider what is the expected optimal number of comparisons needed to solve an interval sort instance and we design a variant of chunksort that achieves near optimal expected performance, up to n + o(n) comparisons. In fact, we conjecture that the algorithm that we propose has actually optimal expected performance up to o(n) terms and provide some evidence for this conjecture.

This research was supported by the Spanish Min. of Science and Technology project TIN2007–66523 (FORMALISM) and the Catalonian Government program for research groups under contract 2009 SGR 1137.

We thank Amalia Duch for her useful comments and suggestions.