Linear-Space Data Structures for Range Minority Query in Arrays

Abstract

We consider range queries that search for low-frequency elements (least frequent elements and α-minorities) in arrays. An α-minority of a query range has multiplicity no greater than an α fraction of the elements in the range. Our data structure for the least frequent element range query problem requires O(n) space, O(n3/2) preprocessing time, and O(n) query time. A reduction from boolean matrix multiplication to this problem shows the hardness of simultaneous improvements in both preprocessing time and query time. Our data structure for the α-minority range query problem requires O(n) space, supports queries in O(1/α) time, and allows α to be specified at query time.