Range majority in constant time and linear space

Abstract

Given an array A of size n, we consider the problem of answering range majority queries: given a query range $[i..j]$ where $1⩽i⩽j⩽n$, return the majority element of the subarray $A[i..j]$ if it exists. We describe a linear space data structure that answers range majority queries in constant time. We further generalize this problem by defining range α-majority queries: given a query range $[i..j]$, return all the elements in the subarray $A[i..j]$ with frequency greater than $\alpha (j-i+1)$. We prove an upper bound on the number of α-majorities that can exist in a subarray, assuming that query ranges are restricted to be larger than a given threshold. Using this upper bound, we generalize our range majority data structure to answer range α-majority queries in $O(\frac{1}{\alpha })$ time using $O(n\lg (\frac{1}{\alpha }+1))$ space, for any fixed $\alpha \in (0,1)$. This result is interesting since other similar range query problems based on frequency have nearly logarithmic lower bounds on query time when restricted to linear space.