Further results on generalized intersection searching problems: Counting, reporting, and dynamization

Abstract

In a generalized intersection searching problem, a set, S, of colored geometrie objects is to be preprocessed so that given some query object, q, the distinct colors of the objects intersected by q can be reported efficiently or the number of such colors can be counted effi.ciently. In the dynamic setting, colored objects can be inserted into or de1eted from S. These problems generalize the well-studied standard intersection searching problems and are rich in applications. Unfortunate1y, the techniques known for the standard problems do not yie1d efficient solutions for the generalized problems. Moreover, previous work [JL92] on generalized problems applies only to the static reporting problems. In thispaper, a uniform framework is presented to solve efficiently the countingjreportingj dynamic versions of a variety of generalized intersection searching problems, including: 1-, 2-, and 3-dimensional range searching, quadrant searching, interval intersection searching, 1-and 2-dimensional point enclosure searching, and orthogonal segment intersection searching.

Categorical Range Queries in Large Databases

“…Regarding the first issue, a decomposition scheme, based on multi-level B + -trees that first narrow down the search space of the high dimensional space and then apply the solution presented in this work for moderate dimensions, seems a good candidate, demanding polylogarithmic overhead [26]. It is also interesting to consider that research on CRQs can be extended to other analogous queries, like the categorical nearest-neighbor queries [23] or the counting queries [17]. Such extensions can find interesting applications in large databases and, therefore, we consider the problem examined in this paper as a first attempt towards this research direction.…”

“…Solutions for standard problems yield outputinsensitive solutions for their generalized chromatic counterparts. For this reason they attracted a lot of research [2,9,10,17,18,14,23,24]. These papers present a number of efficient, mainly theoretical, algorithms that are tailored to the main-memory context.…”

“…As it will be described in the following, in the context of spatial data this assumption may not hold, and the regular range query can present significant limitations in several cases. [17,20] provided a uniform framework that yields efficient solutions for chromatic problems on iso-oriented objects. Their techniques consist of geometric transformations of generalized problems into equivalent instances of some standard problems and the use of persistence as a method for imposing range restrictions to static problems.…”

“…This type of query is denoted as Categorical Range Query (CRQ) 1 and the attribute that the grouping is based upon is called categorical attribute. CRQs have been comprehensively studied in the research fields of computational geometry and main-memory data structures [2,9,10,11,17,18,20,23]. In contrast, much less attention has been given to supporting CRQs in large databases.…”

Effiiently computing the smallest axis-parallel squares spanning all colors

“…gets minimized or maximized. Beside these two applications, this problem has other applications in statistical clustering, pattern recognition, and generalized range searching [6][7][8].…”