Queries about the largest empty rectangle in large 2-dimensional datasets stored in secondary memory

Abstract

Let be a set of points located in a rectangle  and  is a point that is not in . This article describes the design, implementation, and experimentation of different algorithms to solve the following two problems: ( i ) Maximum Empty Rectangle (MER), which consists in finding an empty rectangle with a maximum area contained in R and does not contain any point from   and ( ii ) Query Maximum Empty Rectangle (QMER), which consists in finding the rectangle with the same restrictions given for the MER problem but must also contain . It is assumed that both problems have insufficient main memory to store all the objects in set . According to the literature, both problems are very practical in fields such as data mining and Geographic Information Systems (GIS). Specifically, the present study proposes two algorithms that assume that  is stored in secondary memory (mainly disk) and that it is impossible to store it completely in main memory. The first algorithm solves the QMER problem and consists of decreasing the size of S by using dominance areas and then processing the points that are not eliminated using an algorithm proposed by Orlowski (1990). The second algorithm solves the MER problem and consists of dividing R into four subrectangles that generate four subsets of similar size; these are processed using an algorithm proposed in Edmons  et al. (2003), and finally the partial solutions are combined to obtain a global solution. For the purpose of verifying algorithm efficiency, results are shown for a series of experiments that consider synthetic and real data.