AI 中文总结
研究希尔排序时间复杂度下界，利用参数化映射框架，证明使用特定间隙序列的希尔排序最坏情况时间复杂度至少为$\Omega(N^{1.26})$，推广到特定类型间隙序列，还给出特定时间复杂度的间隙序列项数条件。
AI 中文摘要
在本文中，我们开发了一个框架，使用参数化映射$[\sigma(1), \sigma(2), \cdots, \sigma(n)] \mapsto \sigma(1)z + \sigma(2)z^2 + \cdots \sigma(n)z^n$对希尔排序进行运行时分析。特别地，我们表明1992年提出的使用德田间隙序列的希尔排序的最坏情况时间复杂度至少为$\Omega(N^{1.26})$，并将此结果推广到任何严格递减的间隙序列，其中每个项与有理几何序列的距离至多为固定距离。我们还表明，给出最坏情况希尔排序时间复杂度为$O(N \log^c N)$的严格递减间隙序列必须有$\Omega(\log N / \log \log N)$个阶为$\Omega(N / (\log \log N)^c)$的项。
英文摘要
In this paper we develop the framework of using a parametrized mapping $[σ(1), σ(2), \cdots, σ(n)] \mapsto σ(1)z + σ(2)z^2 + \cdots σ(n)z^n$ 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 $Ω(N^{1.26})$ 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 $Ω(\log N / \log \log N)$ terms of order $Ω(N / (\log N)^c)$.
