Compressing Dynamic Fully Indexable Dictionaries in Word-RAM

Abstract

We study the problem of constructing a dynamic fully indexable dictionary (FID) in the Word-RAM model using space close to the information-theoretic lower bound. A FID is a data-structure that encodes a bit-vector $B$ of length $u$ and answers, for $b\in\{0,1\}$, $\texttt{rank}_b(B, x)=|{\{y\leq x~|~B[y]=b\}}|$ and $\texttt{select}_b(B, r)=\min\{0\leq x<u~|~\texttt{rank}_b(B, x)=r\}$ ($-1$ if empty). A dynamic FID supports updates that modify a single bit of $B$, i.e., $B[i]\gets b$. We work in the Word-RAM model with $w$-bit words, assuming $w\geq \operatorname{lg} u$. Integer multiplication takes $\mathcal{O}(1)$ time. Our memory model is $\mathcal{M}_B$, allowing access to a fixed precomputed table of $τ=\operatorname{polylog}(w)$ words, which can be computed in $\mathcal{O}(wτ)$ time. In this paper, we show a dynamic FID based on the famous fusion-tree data-structure of P{ă}tra{ş}cu and Thorup [FOCS 2014], modified to use fewer bits and to support $\texttt{select}_0$. Let $n$ denote the number of ones in $B$. We describe a parametric construction: for every $ε\leq 1/2$, there is a dynamic FID using $$\operatorname{lg}\binom{u}{n}+\mathcal{O}(nw^ε/ε)\text{ bits}$$ taking $\mathcal{O}({1/ε+\log_w(n)})$ time for $\texttt{rank}_0/\texttt{rank}_1/\texttt{select}_0$ and updates, and $\mathcal{O}({\log_w(n)})$ time for $\texttt{select}_1$. All time bounds are worst-case. For $ε={1/\sqrt{\operatorname{lg} w}}$, we reduce the space to $\operatorname{lg}\binom{u}{n}+\mathcal{O}(n\log w)$ bits. For $ε=Θ(1)$, the running time matches the lower bound of Fredman and Saks [STOC 1989]. This is the first deterministic dynamic FID in the standard Word-RAM model that achieves $o(n\sqrt{w})$ bits of redundancy in $\mathcal{M}_B$ (e.g., $ε=1/4$), and optimal worst-case time.