Improved lower bounds of the time complexity of shellsort

Zhenghan Zang

Improved lower bounds of the time complexity of shellsort

Data Structures and Algorithms 2026-07-10 v1

Authors: Zhenghan Zang

Abstract

In this paper we develop the framework of using a parametrized mapping $[\sigma(1), \sigma(2), \cdots, \sigma(n)] \mapsto \sigma(1)z + \sigma(2)z^2 + \cdots \sigma(n)z^n$ to perform runtime analysis on Shellsort. In particular, we show that the worst-case time complexity of Shellsort using Tokuda's gap sequence proposed in 1992 is at least $\Omega(N^{1.26})$ with a generalisation of this result to any strictly decreasing gap sequence where each term at most a fixed distance away from a rational geometric sequence, and we also show that strictly decreasing gap sequences giving worst-case Shellsort time complexities of $O(N \log^c N)$ must have $\Omega(\log N / \log \log N)$ terms of order $\Omega(N / (\log \log N)^c)$ .

Cite

@article{arxiv.2607.08997,
  title  = {Improved lower bounds of the time complexity of shellsort},
  author = {Zhenghan Zang},
  journal= {arXiv preprint arXiv:2607.08997},
  year   = {2026}
}