Optimal Query Time for Encoding Range Majority

We revisit the range τ -majority problem, which asks us to preprocess an array A[1..n] for a fixed value of τ ∈ (0, 1/2], such that for any query range [i, j] we can return a position in A of each distinct τ majority element. A τ -majority element is one that has relative frequency at least τ in the range [i, j]: i.e., frequency at least τ(j − i + 1). Belazzougui et al. [WADS 2013] presented a data structure that can answer such queries in O(1/τ) time, which is optimal, but the space can be as much as Θ(n lgn) bits. Recently, Navarro and Thankachan [Algorithmica 2016] showed that this problem could be solved using an O(n lg(1/τ)) 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 O(n lg(1/τ)) bits of space, and has O(1/τ) 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 τ -majority element.