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Karpinski and Nekrich (2008) introduced the problem of parameterized rangemajority, which asks us to preprocess a string of length $n$ such that, giventhe endpoints of a range, one can quickly find all the distinct elements whoserelative frequencies in that range are more than a threshold $\tau$. Subsequentauthors have reduced their time and space bounds such that, when $\tau$ isfixed at preprocessing time, we need either $O(n \log (1 / \tau))$ space andoptimal $O(1 / \tau)$ query time or linear space and $O((1 / \tau) \log \log\sigma)$ query time, where $\sigma$ is the alphabet size. In this paper we givethe first linear-space solution with optimal $O(1 / \tau)$ query time, evenwith variable $\tau$ (i.e., specified with the query). For the case when$\sigma$ is polynomial on the computer word size, our space is optimallycompressed according to the symbol frequencies in the string. Otherwise, eitherthe compressed space is increased by an arbitrarily small constant factor orthe time rises to any function in $(1/\tau)\cdot\omega(1)$. We obtain the sameresults on the complementary problem of parameterized range minority introducedby Chan et al. (2015), who had achieved linear space and $O(1 / \tau)$ querytime with variable $\tau$.

Range Majorities and Minorities in Arrays