Galloping in fast-growth natural merge sorts

TL;DR

This paper analyzes the efficiency of TimSort's galloping routine, showing its limitations and proposing modifications that improve worst-case comparison bounds for arrays with few distinct values.

Contribution

It introduces a modified galloping routine and the concepts of dual runs and growth properties, enhancing understanding of merge sort efficiency on arrays with limited distinct values.

Findings

Original galloping routine does not prevent Θ(n log n) comparisons for arrays with three values.

Modified routine achieves O(n + n log σ) comparisons for arrays with σ distinct values.

New notions of fast- and middle-growth algorithms optimize merge sort performance on low-complexity arrays.

Abstract

We study the impact of merging routines in merge-based sorting algorithms. More precisely, we focus on the galloping routine that TimSort uses to merge monotonic sub-arrays, hereafter called runs, and on the impact on the number of element comparisons performed if one uses this routine instead of a na\"ive merging routine. This routine was introduced in order to make TimSort more efficient on arrays with few distinct values. Alas, we prove that, although it makes TimSort sort array with two values in linear time, it does not prevent TimSort from requiring up to $\Theta(n \log(n))$ element comparisons to sort arrays of length~$n$ with three distinct values. However, we also prove that slightly modifying TimSort's galloping routine results in requiring only $\mathcal{O}(n + n \log(\sigma))$ element comparisons in the worst case, when sorting arrays of length $n$ with $\sigma$ distinct values. We do so by focusing on the notion of dual runs, which was introduced in the 1990s, and on the associated dual run-length entropy. This notion is both related to the number of distinct values and to the number of runs in an array, which came with its own run-length entropy that was used to explain TimSort's otherwise "supernatural" efficiency. We also introduce new notions of fast- and middle-growth for natural merge sorts (i.e., algorithms based on merging runs), which are found in several merge sorting algorithms similar to TimSort. We prove that algorithms with the fast- or middle-growth property, provided that they use our variant of TimSort's galloping routine for merging runs, are as efficient as possible at sorting arrays with low run-induced or dual-run-induced complexities.