Linear-Space Data Structures for Range Frequency Queries on Arrays and Trees

Abstract

We present O(n)-space data structures to support various range frequency queries on a given array A[0:n − 1] or tree T with n nodes. Given a query consisting of an arbitrary pair of pre-order rank indices (i,j), our data structures return a least frequent element, mode, or α-minority of the multiset of elements in the unique path with endpoints at indices i and j in A or T. We describe a data structure that supports range least frequent element queries on arrays in $O(\sqrt{n / w})$ time, improving the $\Theta(\sqrt{n})$ worst-case time required by the data structure of Chan et al. (SWAT 2012), where w ∈ Ω(logn) is the word size in bits. We describe a data structure that supports range mode queries on trees in $O(\log\log n \sqrt{n / w})$ time, improving the $\Theta(\sqrt{n} \log n)$ worst-case time required by the data structure of Krizanc et al. (ISAAC 2003). Finally, we describe a data structure that supports range α-minority queries on trees in O(α ^− 1 loglogn) time, where α ∈ [0,1] is specified at query time.

Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and National Science Foundation (NSF) Grants CCF–1017623 (R. Shah and J. S. Vitter) and CCF–1218904 (R. Shah).