How good is the information theory bound in sorting?

AbstractWe define a sorting problem on an n element set S to be a family 〈A1,…,Ar〉 of disjoint subsets of the set of n! linear orderings on S. Given an ordering ω ∈ ∪jAj, we want to determine to which subset Aj the ordering ω belongs by performing a sequence of comparisons between the elements of S. The classical sorting problem corresponds to the case where the subsets Aj comprise the n! singleton sets of orderings.If a sorting problem is defined by r nonempty subsets Aj, then the information theory bound states that at least log2 r comparisons are required to solve that problem in the worst case. The purpose of this paper is to investigate the accuracy of this bound. While we show that it is usually very weak, we are nevertheless able to define a large class of problems for which this bound is good. As an application, we show that if X and Y are n element sets of real numbers, then the n2 element set X + Y can be sorted with O (n2) comparisons, improving upon the n2 log2 n bound established by Harper et al. The problem of sorting X + Y was posed by Berkelamp