Improved Time and Space Bounds for Dynamic Range Mode
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Given an array A of n elements, we wish to support queries for the most frequent and least frequent element in a subrange [l, r] of A. We also wish to support updates that change a particular element at index i or insert/ delete an element at index i. For the range mode problem, our data structure supports all operations in O(n^2/3) deterministic time using only O(n) space. This improves two results by Chan et al. C14: a linear space data structure supporting update and query operations in Õ(n^3/4) time and an O(n^4/3) space data structure supporting update and query operations in Õ(n^2/3) time. For the range least frequent problem, we address two variations. In the first, we are allowed to answer with an element of A that may not appear in the query range, and in the second, the returned element must be present in the query range. For the first variation, we develop a data structure that supports queries in Õ(n^2/3) time, updates in O(n^2/3) time, and occupies O(n) space. For the second variation, we develop a Monte Carlo data structure that supports queries in O(n^2/3) time, updates in Õ(n^2/3) time, and occupies Õ(n) space, but requires that updates are made independently of the results of previous queries. The Monte Carlo data structure is also capable of answering k-frequency queries; that is, the problem of finding an element of given frequency in the specified query range. Previously, no dynamic data structures were known for least frequent element or k-frequency queries.
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