Dynamic Range Majority Data Structures

TL;DR

This paper introduces a new dynamic data structure for efficiently answering range alpha-majority queries on colored points, supporting fast queries and updates with optimal bounds for constant alpha.

Contribution

The authors present a novel dynamic data structure for range alpha-majority queries that achieves optimal query times for constant alpha and extends to multi-dimensional data and dynamic arrays.

Findings

Supports queries in O((log n)/alpha) time

Supports updates in O((log n)/alpha) amortized time

Achieves optimal query time for constant alpha

Abstract

Given a set $P$ of coloured points on the real line, we study the problem of answering range $\alpha$-majority (or "heavy hitter") queries on $P$. More specifically, for a query range $Q$, we want to return each colour that is assigned to more than an $\alpha$-fraction of the points contained in $Q$. We present a new data structure for answering range $\alpha$-majority queries on a dynamic set of points, where $\alpha \in (0,1)$. Our data structure uses O(n) space, supports queries in $O((\lg n) / \alpha)$ time, and updates in $O((\lg n) / \alpha)$ amortized time. If the coordinates of the points are integers, then the query time can be improved to $O(\lg n / (\alpha \lg \lg n) + (\lg(1/\alpha))/\alpha))$. For constant values of $\alpha$, this improved query time matches an existing lower bound, for any data structure with polylogarithmic update time. We also generalize our data structure to handle sets of points in d-dimensions, for $d \ge 2$, as well as dynamic arrays, in which each entry is a colour.