More Haste, Less Waste: Lowering the Redundancy in Fully Indexable Dictionaries

Abstract

We consider the problem of representing, in a compressed format, a bit-vector~$S$ of $m$ bits with $n$ $\mathbf{1}$s, supporting the following operations, where $b \in \{ \mathbf{0}, \mathbf{1} \}$:
\begin{itemize}
\item $\mathtt{rank}_b(S,i)$ returns the number of occurrences of bit $b$ in the prefix $S\left[1..i\right]$;
\item $\mathtt{select}_b(S,i)$ returns the position of the $i$th occurrence of bit $b$ in $S$.
\end{itemize}
Such a data structure is called \emph{fully indexable dictionary (\textsc{fid})} [Raman, Raman, and Rao, 2007], and is at least as powerful as predecessor data structures. Viewing $S$ as a set $X = \{ x_1, x_2, \ldots, x_n \}$ of $n$ distinct integers drawn from a universe $[m] = \{1, \ldots, m\}$, the predecessor of integer $y \in [m]$ in $X$ is given by $\ensuremath{\mathtt{select}^{}_1}(S, \ensuremath{\mathtt{rank}_1}(S,y-1))$. {\textsc{fid}}s have many applications in succinct and compressed data structures, as they are often involved in the construction of succinct representation for a variety of abstract data types.

Our focus is on space-efficient {\textsc{fid}}s on the \textsc{ram} model with word size $\Theta(\lg m)$ and constant time for all operations, so that the time cost is independent of the input size.

Given the bitstring $S$ to be encoded, having length $m$ and containing $n$ ones, the minimal amount of information that needs to be stored is $B(n,m) = \lceil \log {{m}\choose{n}} \rceil$. The state of the art in building a \textsc{fid}\ for~$S$ is given in~\mbox{}[P\v{a}tra\c{s}cu, 2008] using $B(m,n)+O( m / ( (\log m/ t) ^t) ) + O(m^{3/4}) $ bits, to support the operations in $O(t)$ time.

Here, we propose a parametric data structure exhibiting a time/space trade-off such that, for any real constants $0 < \delta \leq 1/2$, $0 < \varepsilon \leq 1$, and integer $s > 0$, it uses
\[ B(n,m) + O\left(n^{1+\delta} + n \left(\frac{m}{n^s}\right)^\varepsilon\right) \]
bits and performs all the operations in time $O(s\delta^{-1} + \varepsilon^{-1})$. The improvement is twofold: our redundancy can be lowered parametrically and, fixing $s = O(1)$, we get a constant-time \textsc{fid}\ whose space is $B(n,m) + O(m^\varepsilon/\mathrm{poly}(n))$ bits, for sufficiently large $m$. This is a significant improvement compared to the previous bounds for the general case.