Optimal Dynamic Sequence Representations

TL;DR

This paper introduces a new data structure for dynamic sequences that supports efficient access, rank, select, insertions, and deletions, achieving optimal or near-optimal time complexities with entropy-based space usage.

Contribution

It presents a novel dynamic sequence data structure with optimal worst-case query times and improved update times, surpassing previous methods in efficiency.

Findings

Supports access, rank, select in O(log n / log log n) time

Supports insertions and deletions with amortized O(log n) time

Uses space close to the zero-order entropy of the sequence

Abstract

We describe a data structure that supports access, rank and select queries, as well as symbol insertions and deletions, on a string $S[1,n]$ over alphabet $[1..\sigma]$ in time $O(\lg n/\lg\lg n)$, which is optimal even on binary sequences and in the amortized sense. Our time is worst-case for the queries and amortized for the updates. This complexity is better than the best previous ones by a $\Theta(1+\lg\sigma/\lg\lg n)$ factor. We also design a variant where times are worst-case, yet rank and updates take $O(\lg n)$ time. Our structure uses $nH_0(S)+o(n\lg\sigma) + O(\sigma\lg n)$ bits, where $H_0(S)$ is the zero-order entropy of $S$. Finally, we pursue various extensions and applications of the result.