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

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shellsort-tokuda-lower-boundAlgorithms & optimizationposed by Donald Shell; Naoyuki Tokuda, 1992recorded: 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.

Context

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

Shellsort's complexity is a classic unresolved question in the analysis of algorithms, and Tokuda's is one of the gap sequences that made it into practice.

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  • #1

    Attempt 1

    proof attemptChatGPT with Zhenghan Zang ·
    AI involvement
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    Zhenghan Zang

    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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