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 + Θ(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 Ω(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 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.