A New Gap Sequence for Shellsort: RL-Driven Algorithm Discovery Beyond $N^{4/3}$

2026-09-24

AI Summary

This study addresses the long-standing challenge that the complexity bound of Shellsort gap sequences has remained constrained at $N^{4/3}$ for six decades, with sparse practical sequences struggling to surpass this theoretical bottleneck. Departing from conventional manual design and number-theoretic constructions, this work introduces a reinforcement learning-based self-supervised framework to autonomously search for executable generators. By integrating precise operation-count evaluation with Zang’s theorem for theoretical lower-bound proofs, it achieves a paradigm shift toward “learning from execution.” The approach discovers novel rational geometric gap sequences and establishes matching polynomial upper and lower bounds with an exponent of approximately 1.024, significantly improving upon classical limits. Large-scale experiments further validate that these new sequences yield the lowest average operation counts, thereby achieving concurrent breakthroughs in both theory and practice.

Abstract

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.

Problem

• Shellsort
• Gap sequence
• Worst-case complexity
• Algorithm discovery

Innovation

• Reinforcement Learning
• Shellsort
• Gap Sequence
• Self-supervised Algorithm Discovery
• Computational Complexity