Nearly Optimal List Labeling

The list-labeling problem captures the basic task of storing a dynamically changing set of up to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> elements in sorted order in an array of size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m=(1+\Theta(1))n$</tex> • 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log^{2}n)$</tex> amortized cost. This bound, which was first established in 1981, was finally improved two years ago, when a randomized <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log^{3/2}n)$</tex> expected-cost algorithm was discovered. The best randomized lower bound for this problem remains <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Omega(\log n)$</tex>, 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log n \text{polyloglog}\ n)$</tex> 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.