Shellsort with three increments

TL;DR

This paper refines theoretical understanding of Shellsort with three specific increments, demonstrating an average-case running time of O(n^{23/15}) through a novel perturbation approach and analysis of permutation inversions.

Contribution

It introduces a perturbation technique to improve and clarify Yao's theorems for Shellsort with increments (h,g,1), establishing new average-case complexity bounds.

Findings

Average running time is O(n^{23/15}) for specific increments

Perturbation technique simplifies analysis of permutation inversions

Provides sharper bounds on Shellsort performance with three increments

Abstract

A perturbation technique can be used to simplify and sharpen A. C. Yao's theorems about the behavior of shellsort with increments $(h,g,1)$. In particular, when $h=\Theta(n^{7/15})$ and $g=\Theta(h^{1/5})$, the average running time is $O(n^{23/15})$. The proof involves interesting properties of the inversions in random permutations that have been $h$-sorted and $g$-sorted.