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 α( 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 ( 1 α ) time using O (n lg( 1 α + 1)) space, for any fixed α ∈ (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.

Optimal Query Time for Encoding Range Majority

“…As observed by Durocher et al [6], for every query range [i, j] there exists a unique level k in the tree such that [i, j] contains at least one and at most two consecutive blocks in T (k), and, if [i, j] contains two blocks, then the nodes representing these blocks are not siblings in the tree T . Thus, based on our quadruple decomposition, for every query range [i, j] we can associate it with…”

“…The upper bound makes use of the quadruple decomposition of Durocher et al [6]. For ease of description, we assume that n is a power of 2, but note that decomposition works in general.…”

“…Moreover, Durocher et al [6] proved the following lemma: Furthermore, if we consider any arbitrary query range [i, j] that is associated with a quadruple D, there are at most 4/τ elements in the range represented by D that could be τ -majorities for the query range. Following Durocher et al, we refer to these elements as candidates for the quadruple D.…”

“…In the last few years, this problem has received a lot of attention [2,6,9,13], finally leading to a recent result of Belazzougui et al [1,2]: these queries can be supported in O(1/τ ) time, using (1 + ε)nH 0 + o(n) bits of space, where H 0 is the zero-th order empirical entropy of the array A, and ε is an arbitrary positive constant. 1 Since, for an arbitrary τ -majority query, there can be 1/τ answers, there is not much hope for significantly improving the query time of O(1/τ ), except perhaps to make the time bound output-sensitive on the number of results returned [1,Sec.7].…”

“…In terms of techniques, our approach uses the level-based decomposition of Durocher et al [6], but with three significant improvements. We define two new methods for pruning their data structure to reduce space, and one method to speed up queries.…”