Adaptive Range Counting and Other Frequency-Based Range Query Problems

Abstract

We consider variations of range searching in which, given a query range, our goal is to compute some function based on frequencies of points that lie in the range. The most basic such computation involves counting the number of points in a query range. Data structures that compute this function solve the well-studied range counting problem. We consider adaptive and approximate data structures for the 2-D orthogonal range counting problem under the w-bit word RAM model. The query time of an adaptive range counting data structure is sensitive to k, the number of points being counted. We give an adaptive data structure that requires O(n loglog n) space and O(loglog n + log_w k) query time. Non-adaptive data structures on the other hand require Ω(log_w n) query time (Pătraşcu, 2007). Our specific bounds are interesting for two reasons. First, when k=O(1), our bounds match the state of the art for the 2-D orthogonal range emptiness problem (Chan et al., 2011). Second, when k=Θ(n), our data structure is tight to the aforementioned Ω(log_w n) query time lower bound.

We also give approximate data structures for 2-D orthogonal range counting whose bounds match the state of the art for the 2-D orthogonal range emptiness problem. Our first data structure requires O(n loglog n) space and O(loglog n) query time. Our second data structure requires O(n) space and O(log^ε n) query time for any fixed constant ε>0. These data structures compute an approximation k' such that (1-δ)k≤k'≤(1+δ)k for any fixed constant δ>0.

The range selection query problem in an array involves finding the kth lowest element in a given subarray. Range selection in an array is very closely related to 3-sided 2-D orthogonal range counting. An extension of our technique for 3-sided 2-D range counting yields an efficient solution to adaptive range selection in an array. In particular, we present an adaptive data structure that requires O(n) space and O(log_w k) query time, exactly matching a recent lower bound (Jørgensen and Larsen, 2011).

We next consider a variety of frequency-based range query problems in arrays. We give efficient data structures for the range mode and least frequent element query problems and also exhibit the hardness of these problems by reducing Boolean matrix multiplication to the construction and use of a range mode or least frequent element data structure. We also give data structures for the range α-majority and α-minority query problems. An α-majority is an element whose frequency in a subarray is greater than an α fraction of the size of the subarray; any other element is an α-minority. Surprisingly, geometric insights prove to be useful even in the design of our 1-D range α-majority and α-minority data structures.