# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1905.09656