Categorical range maxima queries

Abstract

Given an arrayA[1...n] of n distinct elements from the set {1, 2, ..., n} a range maximum query RMQ(a, b) returns the highest element in A[a...b] along with its position. In this paper, we study a generalization of this classical problem called Categorical Range Maxima Query (CRMQ) problem, in which each element A[i] in the array has an associated category (color) given by C[i] ∈ [σ]. A query then asks to report each distinct color c appearing in C[a...b] along with the highest element (and its position) in A[a...b] with color c. Let pc denote the position of the highest element in A[a...b] with color c. We investigate two variants of this problem: a threshold version and a top-k version. In threshold version, we only need to output the colors with A[pc] more than the input threshold τ , whereas top-k variant asks for k colors with the highest A[pc] values.In the word RAM model, we achieve linear space structure along with O(k) query time, that can report colors in sorted order of A [·]. In external memory, we present a data structure that answers queries in optimal O(1+ k B ) I/O's using almost-linear O(n log * n) space, as well as a linear space data structure with O(log * n + k B ) query I/Os. Here k represents the output size, log * n is the iterated logarithm of n and B is the block size. CRMQ has applications to document retrieval and categorical range reporting -giving a one-shot framework to obtain improved results in both these problems. Our results for CRMQ not only improve the existing best known results for three-sided categorical range reporting but also overcome the hurdle of maintaining color uniqueness in the output set.

Orthogonal Point Location and Rectangle Stabbing Queries in 3-d

“…Lemma 8 (Patil et al [28], Theorem 9) There exists a linear-space data structure that answers the top-k 2-d dominance query in O(log n + k) time. The points are reported in a sorted order.…”

“…When we answer a 6-sided query, we traverse a root-to-leaf path and answer a 5-sided query at every node. We refer the reader to [32] or [29] for a complete description. The height of the tree is O(log w) and, by Theorem 2 we need O(log w n) time to answer a 5-sided query.…”

Further Results on Colored Range Searching

“…Colored range searching (also known as "categorical range searching", or "generalized range searching") have been extensively studied in computational geometry since the 1990s. For example, see the papers [6,12,19,20,21,22,24,25,26,27,29,30,31,32,33,35,37,38,39,41,47] and the survey by Gupta et al [23]. Given a set of n colored data points (where the color of a point represents its "category"), the objective is to build data structures that can provide statistics or some kind of summary about the colors of the points inside a query range.…”

Approximate Range Counting Revisited

“…Colored range searching and its related problems have been studied before [27,35,36,33,28,26,20,12,8,19,23,25,29,30,34,39]. They are known as GROUP-BY queries in the database literature.…”