Better Space Bounds for Parameterized Range Majority and Minority

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 O(n lg(1/τ)) space and optimal O(1/τ) query time or linear space and O((1/τ) lg lg σ) query time, where σ is the alphabet size. In this paper we give the first linear-space solution with optimal O(1/τ ) query time. For the case when τ is given at query time, we significantly improve previous bounds, achieving either O(n lg lg σ) space and optimal O(1/τ ) query time or compressed space and O (