A new measure of disorder in sorting - Entropy

Since information theory was used in establishing the lower bound of the complexity of sorting, there have not been many attempts to use information theory in adaptive sorting. In this paper we introduce the measure of entropy H(X) for disorder in a list X of length n. The entropy measure is defined for ascending runs and up-sequences. Then we present new adaptive sorting algorithms, one called minimal merge sort, and its extension. Minimal mergesort merges the ascending runs in the input list from shorter to longer, that is, merging the shortest two lists each time. The extended algorithm first finds the minimum number of up-sequences and calls minimal mergesort. We show that these algorithms run in (O(nH(X)) worst case and expected times respectively, and minmal mergesort is optimal with respect to entropy. Finally we define more entropy measures and discuss their relationships with existing measures in adaptive sorting.