Nearly Optimal List Labeling

TL;DR

This paper introduces the See-Saw Algorithm, a randomized list-labeling method that nearly matches the theoretical lower bound, significantly improving the efficiency of maintaining sorted dynamic sets.

Contribution

The paper presents a novel randomized algorithm that achieves near-optimal amortized costs for list labeling, surpassing previous bounds and addressing longstanding open problems.

Findings

Achieves $O( ext{log} n ext{polyloglog} n)$ expected amortized cost.

Breaks through multiple lower bounds for this problem class.

Advances understanding of dynamic sorted set data structures.

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.