J. Antoni Sellarès

Efficient computation of location depth contours by methods of computational geometry

The concept of location depth was introduced as a way to extend the univariate notion of ranking to a bivariate configuration of data points. It has been used successfully for robust estimation, hypothesis testing, and graphical display. The depth contours form a collection of nested polygons, and the center of the deepest contour is called the Tukey median. The only available implemented algorithms for the depth contours and the Tukey median are slow, which limits their usefulness. In this paper we describe an optimal algorithm which computes all bivariate depth contours in O(n2) time and space, using topological sweep of the dual arrangement of lines. Once these contours are known, the location depth of any point can be computed in O(log2n) time with no additional preprocessing or in O(log n) time after O(n2) preprocessing. We provide fast implementations of these algorithms to allow their use in everyday statistical practice.

Approximations of 2D and 3D generalized Voronoi diagrams

We propose a new approach for computing in an efficient way polygonal approximations of generalized 2D/3D Voronoi diagrams. The method supports distinct site shapes (points, line-segments, curved-arc segments, polygons, spheres, lines, polyhedra, etc.), different distance functions (Euclidean distance, convex distance functions, etc.) and is restricted to diagrams with connected Voronoi regions. The presented approach constructs a tree (a quadtree in 2D/an octree in 3D) which encodes in its nodes and in a compact way all the information required for generating an explicit representation of the boundaries of the Voronoi diagram approximation. Then, by using this hierarchical data structure a reconstruction strategy creates the diagram approximation. We also present the algorithms required for dynamically maintaining under the insertion or deletion of sites the Voronoi diagram approximation. The main features of our approach are its generality, efficiency, robustness and easy implementation.

Mesh Modification Under Local Domain Changes

We propose algorithms to incrementally modify a mesh of a planar domain by interactively inserting and removing elements (points, segments, polygonal lines, etc.) into or from the planar domain, keeping the quality of the mesh during the process. Our algorithms, that combine mesh improvement techniques, achieve quality by deleting, moving or inserting Steiner points from or into the mesh. The changes applied to the mesh are local and the number of Steiner points added during the process remains low. Moreover, our approach can also be applied to the directly generation of refined Delaunay quality meshes.

Finding extremal sets on the GPU

The extremal sets of a family F of sets consist of all sets of F that are maximal or minimal with respect to the partial order induced by the subset relation in F . In this paper we present efficient parallel GPU-based algorithms, designed under CUDA architecture, for finding the extremal sets of a family F of sets. The complexity analysis of the presented algorithms together with experimental results showing the efficiency and scalability of the approach is provided. We present parallel algorithms designed under CUDA for finding the extremal sets of a family F of sets.We solve either the maximal or the minimal sets problem.The complexity analysis of the presented algorithms is provided.Algorithms are experimentally tested and several results are provided.

Finding influential location regions based on reverse k -neighbor queries

In this paper we introduce and solve several problems that arise in the single facility location field. A reverse k-influential location problem finds a region such that the location of a new facility, desirable or obnoxious, in the region guarantees a minimum k-influential value associated to the importance, attractiveness or repulsiveness, of the facility as a solution to a reverse k-nearest or farthest neighbor query. Solving reverse k-influential location problems help decision makers to progress towards suitable locations for a new facility. We present a parallel approach, to be ran on a graphics processing unit, for approximately solving reverse k-influential location problems, and also provide and discuss experimental results showing the efficiency and scalability of our approach.

Computing and visualizing popular places

Data analysis and knowledge discovery in trajectory databases is an emerging field with a growing number of applications such as managing traffic, planning tourism infrastructures, analyzing professional sport matches or better understanding wildlife. A well-known collection of patterns which can occur for a subset of trajectories of moving objects exists. In this paper, we study the popular places pattern, that is, locations that are visited by many moving objects. We consider two criteria, strong and weak, to establish either the exact number of times that an object has visited a place during its complete trajectory or whether it has visited the place, or not. To solve the problem of reporting popular places, we introduce the popularity map. The popularity of a point is a measure of how many times the moving objects of a set have visited that point. The popularity map is the subdivision, into regions, of a plane where all the points have the same popularity. We propose different algorithms to efficiently compute and visualize popular places, the so-called popular regions and their schematization, by taking advantage of the parallel computing capabilities of the graphics processing units. Finally, we provide and discuss the experimental results obtained with the implementation of our algorithms.

A parallel GPU-based approach for reporting flock patterns

Data analysis and knowledge discovery in trajectory databases is an emerging field with a growing number of applications such as managing traffic, planning tourism infrastructures or better understanding wildlife. In this paper, we study the problem of finding flock patterns in trajectory databases. A flock refers to a large enough subset of entities that move close to each other for, at least, a given time interval. We present parallel algorithms, to be run on a Graphics Processing Unit, for reporting three different variants of the flock pattern: (1) all maximal flocks, (2) the largest flock and (3) the longest flock. We also provide their complexity analysis together with experimental results showing the efficiency and scalability of our approach.

GPU-Based influence regions optimization

In this paper we introduce an optimization problem, that arises in the competitive facility location area, which involves the maximization of the weighted area of the region where a new facility has influence. We consider a finite set of points S in a bounded polygonal region domain D subdivided into several non-negative weighted regions according to a weighted domain partition

Computing Popular Places Using Graphics Processors

\mathcal{P}

Solving Multiple Bichromatic Mutual Nearest Neighbor Queries with the GPU

. For each point in S we define its k-nearest/farthest neighbor influence region as the region containing all the points of D having the considered point as one of their k-nearest/farthest neighbors in S. We want to find a new point s in D whose k-influence region is maximal in terms of weighted area according to the weighted partition