Rank and Select Operations on Binary Strings

Keywords and Synonyms

Binary bit-vector ; Compressed bit-vector, Rank and select dictionary; Fully indexable dictionary (FID)          

Problem Definition

Given a static sequence $\boldsymbol{b} = b_1 \dots b_m$ of m bits, to preprocess the sequence and to create a space-efficient data structure that supports the following operations rapidly:

$\mathsf{rank_1} (i)$ :

takes an index i as input, $1 \le i \le m$, and returns the number of 1s among $b_1 \dots b_i$.

$\mathsf{select_1} (i)$ :

takes an index $i \ge 1$ as input, and returns the position of the ith 1 in $\boldsymbol{b}$, and −1 if i is greater than the number of 1s in $\boldsymbol{b}$.

The operations rank ₀ and select ₀ are defined analogously for the 0s in $\boldsymbol{b}$. As $\mathsf{rank_0} (i) = i - \mathsf{rank_1} (i)$, one considers just rank ₁ (abbreviated to rank), and refers to select ₀ and select ₁ collectively as select. In what follows, |x| denotes the length of a bit sequence x and w(x...