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arXiv · 2405.00807

Nearly Optimal List Labeling

Michael A. Bender·Alex Conway·Martín Farach-Colton·Hanna Komlós·Michal Koucký·William Kuszmaul·Michael Saks
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Abstract

The list-labeling problem captures the basic task of storing a dynamically changing set of up to $n$ elements in sorted order in an array of size $m = (1 + \Theta(1))n$. The goal is to support insertions and deletions while moving around elements within the array as little as possible. Until recently, the best known upper bound stood at $O(\log^2 n)$ amortized cost. This bound, which was first established in 1981, was finally improved two years ago, when a randomized $O(\log^{3/2} n)$ expected-cost algorithm was discovered. The best randomized lower bound for this problem remains $\Omega(\log n)$, and closing this gap is considered to be a major open problem in data structures. In this paper, we present the See-Saw Algorithm, a randomized list-labeling solution that achieves a nearly optimal bound of $O(\log n \operatorname{polyloglog} n)$ amortized expected cost. This bound is achieved despite at least three lower bounds showing that this type of result is impossible for large classes of solutions.

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Michael A. Bender, Alex Conway, Martín Farach-Colton, Hanna Komlós, Michal Koucký, William Kuszmaul, Michael Saks. 2024-05-01. Nearly Optimal List Labeling. https://arxiv.org/abs/2405.00807

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