Empirically Improved Tokuda Gap Sequence in Shellsort

TL;DR

This paper introduces an empirically optimized gap sequence for Shellsort based on a gamma sequence, which reduces comparisons and suggests a potential new fractal pattern.

Contribution

It proposes a new gamma-based gap sequence for Shellsort that improves performance over the Tokuda sequence and hints at a fractal structure in the search process.

Findings

Reduces average comparisons compared to Tokuda sequence

Introduces a gamma-based gap sequence with specific exponential formula

Suggests the existence of a new fractal pattern in the search process

Abstract

Experiments are conducted to improve Tokuda (1992) gap sequence in Shellsort into $\gamma$-sequences, and the best result is the gap sequence in which the $k$-th increment $h_k$ is given by \begin{align} h_k=\left\lceil \frac{\gamma^k-1}{\gamma-1} \right\rceil \end{align} , where $\gamma=2.243609061420001...$ and $k\in\mathbb{N}_1$. The first few increments of the gap sequence are \begin{align} 1,\, 4,\, 9,\, 20,\, 45,\, 102,\, 230,\, 516,\,1158,\,2599,\,5831,\,13082,\,29351,\,65853,\, 147748,\,331490,\,743735,\, ...\end{align}It empirically yields less numbers of comparison on average than Tokuda (1992) gap sequence. In the procedure of search, it reveals the potential existence of a new type of fractal.