This paper investigates the MaxRS problem in spatial databases. Given a set O of weighted points and a rectangular region r of a given size, the goal of the MaxRS problem is to find a location of r such that the sum of the weights of all the points covered by r is maximized. This problem is useful in many location-based applications such as finding the best place for a new franchise store with a limited delivery range and finding the most attractive place for a tourist with a limited reachable range. However, the problem has been studied mainly in theory, particularly, in computational geometry. The existing algorithms from the computational geometry community are in-memory algorithms which do not guarantee the scalability. In this paper, we propose a scalable external-memory algorithm (ExactMaxRS ) for the MaxRS problem, which is optimal in terms of the I/O complexity. Furthermore, we propose an approximation algorithm (ApproxMaxCRS ) for the MaxCRS problem that is a circle version of the MaxRS problem. We prove the correctness and optimality of the ExactMaxRS algorithm along with the approximation bound of the ApproxMaxCRS algorithm. From extensive experimental results, we show that the ExactMaxRS algorithm is two orders of magnitude faster than methods adapted from existing algorithms, and the approximation bound in practice is much better than the theoretical bound of the ApproxMaxCRS algorithm.

In the maximizing range sum (MaxRS) problem, given (i) a set P of 2D points each of which is associated with a positive weight, and (ii) a rectangle r of specific extents, we need to decide where to place r in order to maximize the covered weight of r-that is, the total weight of the data points covered by r. Algorithms solving the problem exactly entail expensive CPU or I/O cost. In practice, exact answers are often not compulsory in a MaxRS application, where slight imprecision can often be comfortably tolerated, provided that approximate answers can be computed considerably faster. Motivated by this, the present paper studies the (1 − ǫ)-approximate MaxRS problem, which admits the same inputs as MaxRS, but aims instead to return a rectangle whose covered weight is at least (1 − ǫ)m ⋆ , where m ⋆ is the optimal covered weight, and ǫ can be an arbitrarily small constant between 0 and 1. We present fast algorithms that settle this problem with strong theoretical guarantees.

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This article studies the MaxRS problem in spatial databases. Given a set O of weighted points and a rectangle r of a given size, the goal of the MaxRS problem is to find a location of r such that the sum of the weights of all the points covered by r is maximized. This problem is useful in many location-based services such as finding the best place for a new franchise store with a limited delivery range and finding the hotspot with the largest number of nearby attractions for a tourist with a limited reachable range. However, the problem has been studied mainly in the theoretical perspective, particularly in computational geometry. The existing algorithms from the computational geometry community are in-memory algorithms that do not guarantee the scalability. In this article, we propose a scalable external-memory algorithm ( ExactMaxRS ) for the MaxRS problem that is optimal in terms of the I/O complexity. In addition, we propose an approximation algorithm ( ApproxMaxCRS ) for the MaxCRS problem that is a circle version of the MaxRS problem. We prove the correctness and optimality of the ExactMaxRS algorithm along with the approximation bound of the ApproxMaxCRS algorithm. Furthermore, motivated by the fact that all the existing solutions simply assume that there is no tied area for the best location, we extend the MaxRS problem to a more fundamental problem, namely AllMaxRS , so that all the locations with the same best score can be retrieved. We first prove that the AllMaxRS problem cannot be trivially solved by applying the techniques for the MaxRS problem. Then we propose an output-sensitive external-memory algorithm ( TwoPhaseMaxRS ) that gives the exact solution for the AllMaxRS problem through two phases. Also, we prove both the soundness and completeness of the result returned from TwoPhaseMaxRS. From extensive experimental results, we show that ExactMaxRS and ApproxMaxCRS are several orders of magnitude faster than methods adapted from existing algorithms, the approximation bound in practice is much better than the theoretical bound of ApproxMaxCRS, and TwoPhaseMaxRS is not only much faster but also more robust than the straightforward extension of ExactMaxRS.

This paper proposes a group version of the nearest neighbor (NN) query, called the nearest neighborhood (NNH) query, which aims to find the nearest group of points, instead of one nearest point. Given a set O of points, a query point q, and a ρ-radius circle C, the NNH query returns the nearest placement of C to q such that there are at least k points enclosed by C. We present a fast algorithm for processing the NNH query based on the incremental retrieval of nearest neighbors using the Rtree structure on O. Our solution includes several techniques, to efficiently maintain sets of retrieved nearest points and identify their validities in terms of the closeness constraint of their points. These techniques are devised from the unique characteristics of the NNH search problem. As a side product, we solve a new geometric problem, called the nearest enclosing circle (NEC) problem, which is of independent interest. We present a linear expected-time algorithm solving the NEC problem using the properties of the NEC similar to those of the smallest enclosing circle. We provide extensive experimental results, which show that our techniques can significantly improve the query performance.