Compressed Data Structures for Dynamic Sequences

TL;DR

This paper introduces a fully-dynamic compressed data structure for large alphabets that supports efficient updates and queries, achieving optimal time complexities and minimal space usage based on the string's entropy.

Contribution

It presents the first fully-dynamic structure with optimal query times and worst-case update guarantees for large alphabets, using space close to the entropy bound.

Findings

Supports insertions and deletions with worst-case guarantees

Achieves optimal query times for access, rank, and select

Uses space close to the entropy of the string

Abstract

We consider the problem of storing a dynamic string $S$ over an alphabet $\Sigma=\{\,1,\ldots,\sigma\,\}$ in compressed form. Our representation supports insertions and deletions of symbols and answers three fundamental queries: $\mathrm{access}(i,S)$ returns the $i$-th symbol in $S$, $\mathrm{rank}_a(i,S)$ counts how many times a symbol $a$ occurs among the first $i$ positions in $S$, and $\mathrm{select}_a(i,S)$ finds the position where a symbol $a$ occurs for the $i$-th time. We present the first fully-dynamic data structure for arbitrarily large alphabets that achieves optimal query times for all three operations and supports updates with worst-case time guarantees. Ours is also the first fully-dynamic data structure that needs only $nH_k+o(n\log\sigma)$ bits, where $H_k$ is the $k$-th order entropy and $n$ is the string length. Moreover our representation supports extraction of a substring $S[i..i+\ell]$ in optimal $O(\log n/\log\log n + \ell/\log_{\sigma}n)$ time.