Shellsort 的新间隔序列：超越 $N^{4/3}$ 的强化学习驱动算法发现
A New Gap Sequence for Shellsort: RL-Driven Algorithm Discovery Beyond $N^{4/3}$
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中文总结 AI 辅助
本文提出强化学习驱动的自监督系统，自动发现Shellsort间隔序列，在保持实用性能的同时，将稀疏序列的最坏情况上界从$N^{4/3}$降至$N^{1.024}$，并匹配上下界。
中文摘要 AI 辅助
选择 Shellsort 的间隔序列是一个著名的开放问题。六十多年来，成功的序列一直依赖于人工设计的公式、数值搜索或数论构造。尽管对于密集或主要是理论性的序列族存在更强的通用界，但对于短、稀疏且实际竞争的构造，其最坏情况上界几十年来一直未超越 $N^{4/3}$。我们提出，序列本身是否可以改为从执行中学习。我们提出了一个强化学习驱动的自监督系统，该系统在可执行的间隔生成器上进行搜索。每个提议在构造上都是有效的，并且执行的候选者返回精确的比较和移动次数；不使用任何经典序列作为目标。在五次独立搜索中，系统发现了一个共同的理性几何序列族。第二个自监督阶段仅调整有限前缀，产生实用序列 $1,3,8,20,47,116,300,585,1416,3303,\ldots$。一旦冻结，它在 25 个 $10^7<N\leq 10^8$ 的大型任务中，在七个经典基线中获得了最低的等任务平均操作数。我们在不改变其实用行为的情况下完成学习到的尾部：仅在 $10^{1000}$ 之外，一个零密度的单位伴随集合 $h_s+1$ 消除了剩余的同余障碍。由此产生的稀疏序列具有匹配的多项式上界和下界指数，直到多对数因子：$\Omega(N^{1.024296451657\ldots}) \leq T(N) \leq O(N^{1.024296451657\ldots}\operatorname{polylog} N)$。下界来自 Zang 最近关于理性几何序列的定理；我们的贡献是匹配的上界。因此，一个精确的序列将自监督发现、大规模实际性能以及相对于稀疏实用 Shellsort 序列的经典 $N^{4/3}$ 界的大幅改进联系起来。
英文摘要
Choosing Shellsort gaps is a well-known open problem. For over sixty years, successful sequences have relied on human-designed formulas, numerical searches, or number-theoretic constructions. Although stronger general bounds exist for dense or mainly theoretical families, the worst-case upper bound for a short, sparse, and practically competitive construction has not advanced beyond $N^{4/3}$ for decades. We ask whether the sequence itself can instead be learned from execution. We present an RL-driven, self-supervised system that searches over executable gap generators. Every proposal is valid by construction, and executed candidates return exact comparison and move counts; no classical sequence is used as a target. Across five independent searches, the system discovers a common rational-geometric family. A second self-supervised stage tunes only a finite prefix, producing the practical sequence $1,3,8,20,47,116,300,585,1416,3303,\ldots$. Once frozen, it obtains the lowest equal-task average operation count among seven classical baselines on 25 large tasks with $10^7<N\leq 10^8$. We complete the learned tail without changing its practical behavior: only beyond $10^{1000}$, a zero-density set of unit companions $h_s+1$ removes the remaining congruence barriers. The resulting sparse sequence has matching polynomial upper and lower exponents, up to polylogarithmic factors: $Ω(N^{1.024296451657\ldots}) \leq T(N) \leq O(N^{1.024296451657\ldots}\operatorname{polylog} N)$. The lower bound follows from Zang's recent theorem for rational-geometric sequences; our contribution is the matching upper bound. Thus one exact sequence connects self-supervised discovery, large-scale practical performance, and a substantial step below the classical $N^{4/3}$ bound for sparse practical Shellsort sequences.
发表机构
- The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
机构由 AI 辅助整理，请以论文原文为准。
