Improved Time and Space Bounds for Dynamic Range Mode

TL;DR

This paper introduces improved deterministic and probabilistic data structures for dynamic range mode and least frequent element queries, achieving better time and space bounds than previous solutions.

Contribution

It presents the first efficient dynamic data structures for range least frequent and $k$-frequency queries, with improved bounds for range mode queries.

Findings

Range mode queries supported in $O(n^{2/3})$ time with $O(n)$ space.

Range least frequent queries supported in $O(n^{2/3})$ time with $O(n)$ space.

New Monte Carlo data structure for $k$-frequency queries with $O(n^{2/3})$ query and update time.

Abstract

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. \cite{C14}: a linear space data structure supporting update and query operations in $\tilde{O}(n^{3/4})$ time and an $O(n^{4/3})$ space data structure supporting update and query operations in $\tilde{O}(n^{2/3})$ 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 $\tilde{O}(n^{2/3})$ 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 $\tilde{O}(n^{2/3})$ time, and occupies $\tilde{O}(n)$ 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.