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Worst-Case Complexity of Shellsort with Tokuda's Gap Sequence

Algorithms & optimization · posed by Donald Shell; Naoyuki Tokuda, 1992 · partial

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Statement

Shellsort's worst-case running time is unknown for the gap sequences actually used in practice. Encoding a permutation as the polynomial σ(1)z+⋯+σ(n)zn\sigma(1)z + \cdots + \sigma(n)z^n gives a framework for lower bounds, and yields Ω(N1.26)\Omega(N^{1.26}) for Tokuda's 1992 sequence, extending to any strictly decreasing sequence staying within a fixed distance of a rational geometric one.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Zhenghan Zang, using ChatGPT

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
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    — a person led the work and used a model along the way.

    The acknowledgement credits the model with providing the initial framework of the proof of Lemma 2, which the author then verified, refined and wrote up.

    a lower bound for Tokuda's sequence; the general Shellsort complexity question stays open

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