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, α-minority, or top-k colors (values) 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 (Formula Presented) time, improving the (Formula Presented) worst-case time required by the data structure of Chan et al. (SWAT 2012), where (Formula Presented) is the word size in bits. We describe a data structure that supports path mode queries on trees in (Formula Presented) time, improving the (Formula Presented) worst-case time required by the data structure of Krizanc et al. (ISAAC 2003). We describe the first data structures to support path least frequent element queries, path $$\alpha $$α-minority queries, and path top-k color queries on trees in (Formula Presented), and O(k) time, respectively, where α∈[0,1] and k∈{1,…,n} are specified at query time.