Strategies for Stable Merge Sorting

TL;DR

This paper introduces new stable merge sort algorithms, $2$-merge and $eta$-merge sorts, providing tight bounds on their merge costs, which outperform existing sorts like Timsort in worst-case scenarios and are easier to implement.

Contribution

The paper presents two novel stable merge sort algorithms with optimal bounds and improved practical performance, expanding the theoretical understanding of merge costs in sorting.

Findings

$2$-merge sort has approximately 1.089 n log m merge cost.

New algorithms outperform Timsort in worst-case merge cost.

Experimental results show better performance of new sorts over existing algorithms.

Abstract

We introduce new stable natural merge sort algorithms, called $2$-merge sort and $\alpha$-merge sort. We prove upper and lower bounds for several merge sort algorithms, including Timsort, Shivers' sort, $\alpha$-stack sorts, and our new $2$-merge and $\alpha$-merge sorts. The upper and lower bounds have the forms $c \cdot n \log m$ and $c \cdot n \log n$ for inputs of length~$n$ comprising $m$~monotone runs. For Timsort, we prove a lower bound of $(1.5 - o(1)) n \log n$. For $2$-merge sort, we prove optimal upper and lower bounds of approximately $(1.089 \pm o(1))n \log m$. We prove similar asymptotically matching upper and lower bounds for $\alpha$-merge sort, when $\varphi < \alpha < 2$, where $\varphi$ is the golden ratio. Our bounds are in terms of merge cost; this upper bounds the number of comparisons and accurately models runtime. The merge strategies can be used for any stable merge sort, not just natural merge sorts. The new $2$-merge and $\alpha$-merge sorts have better worst-case merge cost upper bounds and are slightly simpler to implement than the widely-used Timsort; they also perform better in experiments. We report also experimental comparisons with algorithms developed by Munro-Wild and Jug\'e subsequently to the results of the present paper.