When More Generators Hurt: Shellsort on Full Product Grids

[2608.10696] When More Generators Hurt: Shellsort on Full Product Grids Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Computational Complexity arXiv:2608.10696 (cs) [Submitted on 11 Aug 2026 ( v1 ), last revised 15 Aug 2026 (this version, v2)] Title: When More Generators Hurt: Shellsort on Full Product Grids Authors: Ziqi Zhao , Qingjian Ni View a PDF of the paper titled When More Generators Hurt: Shellsort on Full Product Grids, by Ziqi Zhao and Qingjian Ni View PDF HTML (experimental) Abstract: Shellsort repeatedly runs insertion sort with decreasing gaps, so its worst-case cost depends on the gap sequence. Pratt's $2^u3^v$ sequence, one of the few systematic constructions with a proven $O(n\log^2 n)$ bound, includes every product below $n$ of two base numbers, or generators. We ask whether adding more base numbers, and thus more intermediate gaps, can improve this full product grid. We show that it cannot when every product is retained and each base is at most a fixed power of the smallest. With $r$ independent bases (different exponent choices give different products) and $\Theta(\log n)$ gaps, the best possible worst-case cost is $n\exp(\Theta((\log n)^{1-1/r}))$. Thus two bases give the exponent $\sqrt{\log n}$, whereas three give $(\log n)^{2/3}$: more bases are worse. With a budget of $p$ gaps, matching bounds give the factor $\exp(\Theta(\log n/p^{1/r}))$ beyond linear cost. The reason is simple. Few products force the smallest base $m$ to be large, and fullness makes $m$ the next-to-last gap. An input built from reversed blocks is already sorted for every earlier gap, forcing $\Omega(nm)$ work in the final pass. Powers of distinct primes give a matching construction. For arbitrary gaps, we count current-gap multiples that earlier gaps cannot form. This gives upper and lower bounds for individual passes. A Fourier argument gives necessary conditions for small total cost, while short nonnegative sums give sufficient conditions. In both settings, useful distances must be available before they are needed. Subjects: Computational Complexity (cs.CC) Cite as: arXiv:2608.10696 [cs.CC] (or arXiv:2608.10696v2 [cs.CC] for this version) https://doi.org/10.48550/arXiv.2608.10696 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ziqi Zhao [ view email ] [v1] Tue, 11 Aug 2026 09:18:48 UTC (33 KB) [v2] Sat, 15 Aug 2026 12:44:38 UTC (25 KB) Full-text links: Access Paper: View a PDF of the paper titled When More Generators Hurt: Shellsort on Full Product Grids, by Ziqi Zhao and Qingjian Ni View PDF HTML (experimental) TeX Source view license Current browse context: cs.CC < prev | next > new | recent | 2026-08 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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