Online List Labeling: Breaking the log2n Barrier

• Published in: Proceedings / annual Symposium on Foundations of Computer Science pp. 980 - 990
• Main Authors: Bender, Michael A., Conway, Alex, Farach-Colton, Martin, Komlos, Hanna, Kuszmaul, William, Wein, Nicole
• Format: Conference Proceeding
• Language: English
• Published: IEEE 01.10.2022
• Subjects: algorithms; Arrays; Computer science; Costs; data structures; Heuristic algorithms; History; history independence; Labeling; order maintenance; packed memory array; randomized algorithms; Upper bound
• ISSN: 2575-8454
• DOI: 10.1109/FOCS54457.2022.00096

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 n items in an array of m 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 m = (1+\Theta(1))n, an upper bound of O(\log^{2}n) on the relabeling cost has been known since 1981. A lower bound of \Omega(\log^{2}n) is known for deterministic algorithms and for so-called smooth algorithms, but the best general lower bound remains \Omega(\log n). The central open question in the field is whether O(\log^{2}n) is optimal for all algorithms. In this paper, we give a randomized data structure that achieves an expected relabeling cost of O(\log^{3/2}n) per operation. More generally, if m=(1+\varepsilon)n for \varepsilon=O(1), the expected relabeling cost becomes O(\varepsilon^{-1}\log^{3/2}n). 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 \varepsilon between 1/n^{1/3} and some sufficiently small positive constant, the optimal expected cost for history-independent list-labeling solutions is \Theta(\varepsilon^{-1}\log^{3/2}n).