Improved average complexity for comparison-based sorting

Abstract

This paper studies the average complexity on the number of comparisons for sorting algorithms. Its information-theoretic lower bound is n lg n − 1.4427n + O(log n). For many efficient algorithms, the first n lg n term is easy to achieve and our focus is on the (negative) constant factor of the linear term. The current best value is −1.3999 for the MergeInsertion sort. Our new value is −1.4106, narrowing the gap by some 25%. An important building block of our algorithm is “twoelement insertion,” which inserts two numbers A and B, A < B, into a sorted sequence T. This insertion algorithm is still sufficiently simple for rigorous mathematical analysis and works well for a certain range of the length of T for which the simple binary insertion does not, thus allowing us to take a complementary approach with the binary insertion.