Achieving Optimal Redundancy for Small Dynamic Rank/Select Dictionaries

Abstract: In this paper, we study the number of bits required to construct a dynamic dictionary with optimal time for $\texttt{rank}/\texttt{select}$ operations. Using the standard (multiplication) Word-RAM model with $w$-bit words, we construct a data-structure for a dynamic $\texttt{rank}/\texttt{select}$ dictionary for a set $S\subseteq{0,1,\cdots,u-1}$ of $n$ elements that, given a parameter $1\leq k\leq \log<sup>*w$, uses $$\operatorname{lg}\binom{u}{n}+\mathcal{O}(n\log^{{(k)}w)\text{} bits}$$ taking optimal $\mathcal{O}(k+\log_w n)$ time (worst-case) for all operations. We show optimality for $n=w<sup>{\mathcal{O}(1)}$ by extending the lower bound of Li, Liang, Yu, and Zhou [FOCS 2023] to super-polynomial universes: any dynamic dictionary for $n\leq \sqrt{u}$ elements that uses $$\operatorname{lg}\binom{u}{n}+\mathcal{O}(n\log<sup>{(k)}n)$$ bits requires $Ω(k)$ time for operations. Lastly, we extend the data-structure to a dynamic fully indexable dictionary (that also supports $\texttt{rank}/\texttt{select}$ on the complement of $S$).