Better Space Bounds for Parameterized Range Majority and Minority

Abstract

Karpinski and Nekrich (2008) introduced the problem of parameterized range majority, which asks to preprocess a string of length $n$ such that, given the endpoints of a range, one can quickly find all the distinct elements whose relative frequencies in that range are more than a threshold $τ$. Subsequent authors have reduced their time and space bounds such that, when $τ$ is given at preprocessing time, we need either $\Oh{n \log (1 / τ)}$ space and optimal $\Oh{1 / τ}$ query time or linear space and $\Oh{(1 / τ) \log \log σ}$ query time, where $σ$ is the alphabet size. In this paper we give the first linear-space solution with optimal $\Oh{1 / τ}$ query time. For the case when $τ$ is given at query time, we significantly improve previous bounds, achieving either $\Oh{n \log \log σ}$ space and optimal $\Oh{1 / τ}$ query time or compressed space and $\Oh{(1 / τ) \log \frac{\log (1 / τ)}{\log w}}$ query time. Along the way, we consider the complementary problem of parameterized range minority that was recently introduced by Chan et al.\ (2012), who achieved linear space and $\Oh{1 / τ}$ query time even for variable $τ$. We improve their solution to use either nearly optimally compressed space with no slowdown, or optimally compressed space with nearly no slowdown. Some of our intermediate results, such as density-sensitive query time for one-dimensional range counting, may be of independent interest.