Linear-Space Data Structures for Range Minority Query in Arrays

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 ( n 3 / 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.