Cell Probe Lower Bounds and Approximations for Range Mode

Abstract

Abstract. The mode of a multiset of labels, is a label that occurs at least as often as any other label. The input to the range mode problem is an array A of size n. A range query [i, j] must return the mode of the subarray. We prove that any data structure that uses S memory cells of w bits needs Ω( log n log(Sw/n) ) time to answer a range mode query. Secondly, we consider the related range k-frequency problem. The input to this problem is an array A of size n, and a query [i, j] must return whether there exists a label that occurs precisely k times in the subarray. We show that for any constant k > 1, this problem is equivalent to 2D orthogonal rectangle stabbing, and that for k = 1 this is no harder than four-sided 3D orthogonal range emptiness. Finally, we consider approximate range mode queries. A c-approximate range mode query must return a label that occurs at least 1/c times that of the mode. We describe a linear space data structure that supports 3-approximate range mode queries in constant time, and a data structure that uses O( n ε ) space and supports (1 + ε)-approximation queries in O(log 1 ε ) time.

Low space data structures for geometric range mode query

“…Using reductions from boolean matrix multiplication, they showed show that query times significantly lower than √ n are unlikely for this problem with linear space [3]. Finally, Greve et al [6] proved a lower bound of Ω(log n/log(s · w/n)) time for any data structure that supports range mode query on arrays using s memory cells of w bits in the cell probe model.…”

“…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].…”

Colored Range Queries and Document Retrieval

“…Greve, Jørgensen, Larsen and Truelsen [17] recently gave a data structure that, for any > 0, stores S in O((n/ ) log n) bits such that we can find an element such that no element is more than 1 + times more frequent in S[i, j], in O(log(1/ )) time. Thus, their data structure solves the approximate CRTK problem for k = 1, which is called the approximate range-mode problem.…”

“…This information could be, for example, the minimum or maximum value in S[i, j] [12], the element with a specified rank in sorted order [15] (e.g., the median [7]), the mode [17], a complete list of the distinct elements [31], the frequencies of the elements [35], a list of the k most frequent elements for a given k [20], or the number of distinct elements [6]. In this paper, motivated by problems in document retrieval, we consider the latter three kinds of problems, which are often referred to as "colored" range queries: colored range listing (with or without color frequencies), colored range top-k queries, and colored range counting.…”

Linear-Space Data Structures for Range Minority Query in Arrays

“…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]. Greve et al [13] give a lower bound of Ω(log n/ log(s · w/n)) on the range mode query time for any data structure that uses s memory cells of w bits in the cell probe model; they show the same bound applies to the problem of determining whether any element in a given query range has frequency exactly k, for any k given at query time. Consequently, no O(n)-space data structure can support constant-time (independent of α) α-minority queries.…”