Optimization Perspectives on Shellsort

TL;DR

This paper analyzes Shellsort's efficiency by identifying optimal gap sequences through parameterized functions and grid-search, resulting in sequences that outperform existing ones in comparisons and running time for specific array sizes.

Contribution

It introduces a new method for selecting gap sequences in Shellsort using parameterized functions and grid-search, improving performance metrics for certain input sizes.

Findings

Sequences outperform state-of-the-art in comparisons for small arrays.

Sequences outperform Tokuda sequence in running time for large arrays.

No significant improvement in minimizing exchanges.

Abstract

Shellsort is a sorting method that is attractive due to its simplicity, yet it takes effort to analyze its efficiency. The heart of the algorithm is the gap sequence chosen a priori and used during sorting. The selection of this gap sequence affects the efficiency of Shellsort, and thus drives both its theoretical and experimental analysis. We contribute to Shellsort by identifying efficient gap sequences based on new parameterized functions. Specifically, a parameter grid-search identifies optimal parameters for different input sizes for sorting by observing minimal overhead in three categories: number of comparisons, number of exchanges, and running time. We report that our method finds sequences that outperform state-of-the-art gap sequences concerning the number of comparisons for chosen small array sizes. Additionally, our function-based sequences outperform the running time of the Tokuda sequence for chosen large array sizes. However, no substantial improvements were observed when minimizing the number of exchanges.