Dynamic Range Majority Data Structures

Abstract

Given a set $P$ of coloured points on the real line, we study the problem of answering range $α$-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 $α$-fraction of the points contained in $Q$. We present a new data structure for answering range $α$-majority queries on a dynamic set of points, where $α\in (0,1)$. Our data structure uses O(n) space, supports queries in $O((\lg n) / α)$ time, and updates in $O((\lg n) / α)$ amortized time. If the coordinates of the points are integers, then the query time can be improved to $O(\lg n / (α\lg \lg n) + (\lg(1/α))/α))$. For constant values of $α$, 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.