Nearly Optimal List Labeling
<|reference_start|>Nearly Optimal List Labeling: 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.<|reference_end|>
@article{bender2024nearly,
    title={Nearly Optimal List Labeling},
    author={Michael A. Bender, Alex Conway, Mart'in Farach-Colton, Hanna
  Koml'os, Michal Kouck'y, William Kuszmaul, Michael Saks},
    journal={arXiv preprint arXiv:2405.00807},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.00807},
    primaryClass={cs.DS}
}