Online List Labeling: Breaking the [math] Barrier

SIAM Journal on Computing, Ahead of Print.
Abstract. The online list-labeling problem is an algorithmic primitive with a large literature of upper bounds, lower bounds, and applications. The goal is to store a dynamically changing set of [math] items in an array of [math] slots, while maintaining the invariant that the items appear in sorted order and while minimizing the relabeling cost, defined to be the number of items that are moved per insertion/deletion. For the linear regime, where [math], an upper bound of [math] on the relabeling cost has been known since 1981. A lower bound of [math] is known for deterministic algorithms and for so-called smooth algorithms, but the best general lower bound remains [math]. The central open question in the field is whether [math] is optimal for all algorithms. In this paper, we give a randomized data structure that achieves an expected relabeling cost of [math] per operation. More generally, if [math] for [math], the expected relabeling cost becomes [math]. Our solution is history independent, meaning that the state of the data structure is independent of the order in which items are inserted/deleted. For history-independent data structures, we also prove a matching lower bound: for all [math] between [math] and some sufficiently small positive constant, the optimal expected cost for history-independent list-labeling solutions is [math].