Compressing Dynamic Fully Indexable Dictionaries in Word-RAM

TL;DR

This paper introduces a new dynamic fully indexable dictionary (FID) data structure in the Word-RAM model that uses space close to the theoretical minimum and supports efficient updates and queries, improving over previous methods.

Contribution

It presents the first deterministic dynamic FID in the Word-RAM model with near-optimal space and worst-case time bounds, based on a modified fusion-tree structure.

Findings

Achieves space close to the information-theoretic lower bound.

Supports rank, select, and update operations in worst-case time.

Reduces space redundancy to o(n√w) bits with optimal time.

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 $\tau=\operatorname{polylog}(w)$ words, which can be computed in $\mathcal{O}(w\tau)$ time. In this paper, we show a dynamic FID based on the famous fusion-tree data-structure of P{\u{a}}tra{\c{s}}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 $\epsilon\leq 1/2$, there is a dynamic FID using $$\operatorname{lg}\binom{u}{n}+\mathcal{O}(nw^{\epsilon}/\epsilon)\text{ bits}$$ taking $\mathcal{O}({1/\epsilon+\log_w(n)})$ 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 $\epsilon={1/\sqrt{\operatorname{lg} w}}$, we reduce the space to $\operatorname{lg}\binom{u}{n}+\mathcal{O}(n\log w)$ bits. For $\epsilon=\Theta(1)$, 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., $\epsilon=1/4$), and optimal worst-case time.