Improved Average Complexity for Comparison-Based Sorting

TL;DR

This paper improves the theoretical understanding of the average comparison complexity in comparison-based sorting algorithms, achieving a tighter lower bound on the linear term constant through a novel two-element insertion method.

Contribution

It introduces a new insertion technique that refines the linear term constant in the average comparison complexity, narrowing the gap to the information-theoretic lower bound.

Findings

Achieved a new constant of -1.4106 in the average comparison complexity.

Developed a two-element insertion method that complements binary insertion.

Narrowed the gap to the theoretical lower bound by approximately 25%.

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 "two-element 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.