On the Average Case of MergeInsertion

TL;DR

This paper analyzes the average case complexity of the MergeInsertion sorting algorithm, providing new bounds, exact distributions, and experimental insights into its performance and variations.

Contribution

It establishes an upper bound on average comparisons, describes the distribution of insertion chain lengths, and compares different insertion orders experimentally.

Findings

Upper bound of n log n - 1.4005n + o(n) comparisons.

Exact average comparisons computed for n up to 148.

Different insertion orders can improve average case performance.

Abstract

MergeInsertion, also known as the Ford-Johnson algorithm, is a sorting algorithm which, up to today, for many input sizes achieves the best known upper bound on the number of comparisons. Indeed, it gets extremely close to the information-theoretic lower bound. While the worst-case behavior is well understood, only little is known about the average case. This work takes a closer look at the average case behavior. In particular, we establish an upper bound of $n \log n - 1.4005n + o(n)$ comparisons. We also give an exact description of the probability distribution of the length of the chain a given element is inserted into and use it to approximate the average number of comparisons numerically. Moreover, we compute the exact average number of comparisons for $n$ up to 148. Furthermore, we experimentally explore the impact of different decision trees for binary insertion. To conclude, we conduct experiments showing that a slightly different insertion order leads to a better average case and we compare the algorithm to the recent combination with (1,2)-Insertionsort by Iwama and Teruyama.