Nearly Optimal List Labeling
Abstract
The listlabeling 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 $O(\log^{3/2} n)$ expectedcost 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 SeeSaw Algorithm, a randomized listlabeling 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.00807
 arXiv:
 arXiv:2405.00807
 Bibcode:
 2024arXiv240500807B
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 39 pages