Optimal Query Time for Encoding Range Majority

Abstract

We revisit the range $\tau$-majority problem, which asks us to preprocess an array $\mathsf {A}[1..n]$ for a fixed value of $\tau \in (0,\frac{1}{2}]$, such that for any query range [i, j] we can return a position in $\mathsf {A}$ of each distinct $\tau$-majority element. A $\tau$-majority element is one that has relative frequency at least $\tau$ in the range [i, j]: i.e., frequency at least $\tau (j-i+1)$. Belazzougui et al. [WADS 2013] presented a data structure that can answer such queries in $\mathcal {O}(1/\tau )$ time, which is optimal, but the space can be as much as $\Theta (n \lg n)$ bits. Recently, Navarro and Thankachan [Algorithmica 2016] showed that this problem could be solved using an $\mathcal {O}(n \lg (1/\tau ))$ bit encoding, which is optimal in terms of space, but has suboptimal query time. In this paper, we close this gap and present a data structure that occupies $\mathcal {O}(n \lg (1/\tau ))$ bits of space, and has $\mathcal {O}(1/\tau )$ query time. We also show that this space bound is optimal, even for the much weaker query in which we must decide whether the query range contains at least one $\tau$-majority element.