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(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.

Encodings for Range Majority Queries

“…In our example, 1 is a majority in A [5,7], and it holds A 1 [5,7] = 1 1 1 and [2,4] = 0 1 1 , where 1 is a majority. Thus, with A x and M x we can determine whether x is a majority in any range.…”

“…All these results assume τ is fixed during the construction time. For the case where τ is also a part of the query input, a data structure of O(n log n) words was proposed by Chan et al [5]. Very recently, Belazzougui et al [1] brought down the space occupancy to O(n log log n) words.…”

“…Now consider the set of segments S x within [1, n] [6,6], [7,7], [6,7], [5,7]}, [4,4], [5,5], [4,5], [2,4], [3,5], [4,6], [2,5], [1,5], [2,6] We recall operation rank(B, i) in bitmaps B [1, m], which returns the number of 1s in B [1, i]. Operation rank can be implemented using o(m) bits on top of B and in constant time [12].…”

“…That is, C x is obtained by merging all the segments of S x that touch or overlap. In our example, C 1 = {[1, 1], [5,7]}, C 2 = {[3, 3]}, and C 3 = {[1, 6]}. Note that segments of C x do not overlap, unlike those of S x .…”

Low space data structures for geometric range mode query

“…Chan et al [4] gave the first solution with linear space and O( √ n) query time. Later, Durocher et al [5] improved the query time to O( n/w).…”