Linear-Space Data Structures for Range Mode Query in Arrays

Abstract

A mode of a multiset S is an element a ∈ S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1 : n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i, j) for which a mode of A[i : j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (ISAAC 2003), requires O( √ n log log n) query time. We improve their result and present an O(n)-space data structure that supports range mode queries in O( p n/ log n) worst-case time. Furthermore, we present strong evidence that a query time significantly below √ n cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two 1998 ACM Subject Classification E.1 DATA STRUCTURES

Linear-Space Data Structures for Range Minority Query in Arrays

“…Using binary rank and select data structures and bit packing, Chan et al [5] reduce the range mode query time from O( √ n) to O( n/ log n) without increasing the data structure's space beyond O(n). Unlike the frequency of the mode, the frequency of the least frequent element does not vary monotonically over a sequence of elements.…”

“…We follow the technique of Chan et al [5] to multiply two n×n boolean matrices L and R via least frequent element range queries. In particular, we build an array A of size n ∈ O(n 2 ), and after preprocessing the array in P (n ) time we perform n 2 least frequent element queries, each in Q(n ) time, to calculate M = LR.…”

“…Most relevant to our low-frequency query problems are results for their high-frequency analogues: an O(n)-space data structure that supports range mode queries in O( n/ log n) time [5], an O(n log(1/β +1))-space data structure that supports β-majority range queries in O(1/β) time [9], and a O(n log n)-space data structure that supports α-majority range queries in O(1/α) time [11]. Related generalizations include examinations of the the β-majority range query problem in the dynamic setting [10] and the α-majority range query problem in two dimensions [11].…”

“…This data structure is a variant of a previous data structure of Chan et al [5] for the range mode problem (which in turn was an improvement of a previous data structure of Krizanc et al [16]). In addition, using an argument similar to that of Chan et al [5], we present a reduction from boolean matrix multiplication to the least frequent element range query problem, showing the hardness of simultaneously improving our preprocessing and query time bounds. Section 3 contains the main result of this paper: an O(n)-space data structure that supports α-minority range queries in O(1/α) time.…”

Low space data structures for geometric range mode query

“…A range mode query Q = [a 1 Although the one-dimensional range query problem has received significant attention [3,8,10,9,6], only limited attention has been paid to the multi-dimensional problem. The first solution for the multi-dimensional case was proposed recently by Chan et al [3]. They gave a data structure that requires O(s n + (n/∆) 2d ) words of space and supports d-dimensional range mode queries in O(∆ · t n ) time for any ∆ ≥ 1, where s n is the space of an orthogonal range counting data structure in d dimensions with query time t n .…”