The @d-annulus of a polygon P is the closed region containing all points in the plane at distance at most @d from the boundary of P. An inner (resp., outer) @d-offset polygon is the polygon defined by the inner (resp., outer) boundary of its @d-annulus. In this paper we address three major problems of covering a given point set S by an offset version or a polygonal annulus of a polygon P. First, the Maximum Cover objective is, given a value of @d, to cover as many points from S as possible by the @d-offset (or by the @d-annulus) of P, allowing translation and rotation. Second, the Containment problem is to minimize the value of @d such that there is a rigid transformation of the @d-offset (or the @d-annulus) of P that covers all points from S. Third, in the Partial Containment problem we seek the minimum offset of P covering k=<|S| points. These problems arise in many applications where one needs to match a given polygonal figure (a known model) to a set of points (usually, obtained measures). We address several variants of these problems, including convex and simple polygons, as well as polygons with holes and sets of polygons, and obtain algorithms with low-degree polynomial running times in all cases.

Offset-polygon annulus placement problems

In this article, we study a class of problems where the sum of truncated convex functions is minimized. In statistical applications, they are commonly encountered when l0-penalized models are fitte...

Minimizing Sum of Truncated Convex Functions and Its Applications

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.

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Maximizing Range Sum in External Memory

Let C be a compact set in R2 and let S be a set of n points in R2. We consider the problem of computing a translate of C that contains the maximum number, ?*, of points of S. It is known that this problem can be solved in a time that is roughly quadratic in n. We show how random-sampling and bucketing techniques can be used to develop a near-linear-time Monte Carlo algorithm that computes a placement of C containing at least (1 - ?)?* points of S, for given ? > 0, with high probability. We also present a deterministic algorithm that solves the ?-approximate version of the optimal-placement problem and runs in O((n1+? + n/?) log m) time, for arbitrary constant ? > 0, if C is a convex m-gon.

Translating a Planar Object to Maximize Point Containment

Given a convex polygon P with m vertices and a set S of n points in the plane, we consider the problem of finding a placement of P (allowing both translation and rotation) that contains the maximum number of points in S. We present first an algorithm requiring O(n^2km^2log(mn)) time and O(n + m) space, where k is the maximum number of points contained. We then give a refinement that makes use of bucketing to improve the running time to O(nk^2c^2m^2log(mk)), where c is the ratio of length to width of the polygon. This provides an improvement over the best previously known algorithm linear in n when k is large (@Q(n)) and a cubic when k is small. We also show that the algorithm can be extended to solve bichromatic and general weighted variants of the problem. The algorithm is self-contained and utilizes the geometric properties of the containing regions in the parameter space of transformations.

Optimal placement of convex polygons to maximize point containment

Given a set S of n points in the plane and a convex polygon P with m vertices, we consider the problem of finding a translation of P that contains the maximum number of points in S . We present two different solutions. Our first algorithm uses standard line-sweep techniques and requires O( nk log( nm ) + m ) time where k is the maximum number of points contained. Our second algorithm requires O( nk log( mk ) + m ) time, which is the asymptotically fastest known algorithm for this problem. Both algorithms require optimal O( m + n ) space. The algorithms also solve in the same running time the bichromatic variant of the problem, where we are given two point sets A and B and the goal is to maximize the number of points covered from A while minimizing the number of points covered from B .

Translating a convex polygon to contain a maximum number of points

Abstract Given a set S of n points in the plane and a convex polygon P with m vertices, we consider the problem of finding a translation of P that contains the maximum number of points in S . We present two different solutions. Our first algorithm uses standard line-sweep techniques and requires O( nk log( nm ) + m ) time where k is the maximum number of points contained. Our second algorithm requires O( nk log( mk ) + m ) time, which is the asymptotically fastest known algorithm for this problem. Both algorithms require optimal O( m + n ) space. The algorithms also solve in the same running time the bichromatic variant of the problem, where we are given two point sets A and B and the goal is to maximize the number of points covered from A while minimizing the number of points covered from B .

Petru Pau

Papers

Translating a convex polygon to contain a maximum number of points

Translating a convex polygon to contain maximum number of points.