Joan Antoni Sellarès

Multi-visibility maps of triangulated terrains

Visibility computation on terrain models is an important research topic with many applications in Geographical Information Systems. A multi‐visibility map is the subdivision of the domain of a terrain into regions that, according to different criteria, encodes the visibility with respect to a set of view elements. We present an approach for visualising approximated multi‐visibility maps of a triangulated terrain corresponding to a set of view elements by using graphics hardware. Our method supports heterogeneous sets of view elements containing points, segments, polygonal chains and polygons and works for weak and strong visibility. Moreover, we are also able to efficiently solve approximated point and polygonal region multi‐visibility queries. To illustrate the usefulness of our approach we present results obtained with an implementation of the proposed algorithms.

THE LARGEST EMPTY ANNULUS PROBLEM

Given a set of n points S in the Euclidean plane, we address the problem of computing an annulus A, (open region between two concentric circles) of largest width, that partitions S into a subset of points inside and a subset of points outside the circles, such that no point p∈S lies in the interior of A. This problem can be considered as a maximin facility location problem for n points such that the facility is a circumference. We give a characterization of the centres of annuli which are locally optimal and we show that the problem can be solved in O(n3logn) time and O(n) space. We also consider the case in which the number of points in the inner circle is a fixed value k. When k∈O(n) our algorithm runs in O(n3logn) time and O(n) space, furthermore, we can simultaneously optimize for all values of k within the same time bound. When k is small, that is a fixed constant, we can solve the problem in O(n logn) time and O(n) space.

Higher-order Voronoi diagrams on triangulated surfaces

We study the complexity of higher-order Voronoi diagrams on triangulated surfaces under the geodesic distance, when the sites may be polygonal domains of constant complexity. More precisely, we show that on a surface defined by n triangles the sum of the combinatorial complexities of the order-j Voronoi diagrams of m sites, for j=1,...,k, is O(k^2n^2+k^2m+knm), which is asymptotically tight in the worst case.

Solving the k-influence region problem with the GPU

In this paper we study a problem that arises in the competitive facility location field. Facilities and customers are represented by points of a planar Euclidean domain. We associate a weighted distance to each facility to reflect that customers select facilities depending on distance and importance. We define, by considering weighted distances, the k-influence region of a facility as the set of points of the domain that has the given facility among their k-nearest/farthest neighbors. On the other hand, we partition the domain into subregions so that each subregion has a non-negative weight associated to it which measures a characteristic related to the area of the subregion. Given a weighted partition of the domain, the k-influence region problem finds the points of the domain where are new facility should be opened. This is done considering the known weight associated to the new facility and ensuring a minimum weighted area of its k-influence region. We present a GPU parallel approach, designed under CUDA architecture, for approximately solving the k-influence region problem. In addition, we describe how to visualize the solutions, which improves the understanding of the problem and reveals complicated structures that would be hard to capture otherwise. Integration of computation and visualization facilitates decision makers with an iterative what-if analysis process, to acquire more information to obtain an approximate optimal location. Finally, we provide and discuss experimental results showing the efficiency and scalability of our approach.

Nice Perspective Projections

Abstract A polyhedral object in three-dimensional space is often well represented by a set of points and line segments that act as its features. By a nice perspective projection of an object we mean a projection that gives an image in which the features of the object, relevant for some task, are visible without ambiguity. In this paper we consider the problem of computing a variety of nice perspective projections of three-dimensional objects such as simple polygonal chains, wire-frame drawings of graphs, and geometric rooted trees. These problems arise in areas such as computer vision, computer graphics, graph drawing, knot theory, and computational geometry.

On finding a widest empty 1-corner corridor

A 1-corner corridor through a set S of points is an open subset of CH(S) containing no points from S and bounded by a pair of parallel polygonal lines each of which contains two segments. Given a set of n points in the plane, we consider the problem of computing a widest empty 1-corner corridor. We describe an algorithm that solves the problem in O(n4 log n) time and O(n) space. We also present an approximation algorithm that computes in O(n log n/e 1/2 + n2/e) time a solution with width at least a fraction (1 - e) of the optimal, for any small enough e > 0.

Generalized Higher-Order Voronoi Diagrams on Polyhedral Surfaces

We present an algorithm for computing exact shortest paths, and consequently distances, from a generalized source (point, segment, polygonal chain or polygonal region) on a possibly non-convex polyhedral surface in which polygonal chain or polygon obstacles are allowed. We also present algorithms for computing discrete Voronoi diagrams of a set of generalized sites (points, segments, polygonal chains or polygons) on a polyhedral surface with obstacles. To obtain the discrete Voronoi diagrams our algorithms, exploiting hardware graphics capabilities, compute shortest path distances defined by the sites.

Common influence region problems

We introduce problems related to two competitive sets of collaborative facilities.We solve common influence region queries and location problems.We present algorithms to be run in parallel to solve the introduced problems.We provide the theoretical complexity analysis of the proposed solutions.We experimentally test the algorithms showing their efficiency and scalability. In this paper we propose and solve common influence region problems. These problems are related to the simultaneous influence, or the capacity to attract customers, of two sets of facilities of different types. For instance, while a facility of the first type competes with the other facilities of the first type, it cooperates with several facilities of the second type. The problems studied can be applied, for example, to decision-making support systems for marketing and/or locating facilities. We present parallel algorithms, to be run on a Graphics Processing Unit, for approximately solving the problems considered here. We also provide experimental results and discuss the efficiency and scalability of our approach. Finally, we present the speedup ratios obtained when the running times of the parallel proposed algorithms using a GPU are compared with those obtained from their respective efficient sequential CPU versions.

Multiresolution Approximations of Generalized Voronoi Diagrams

A framework to support multiresolution approximations of planar generalized Voronoi diagrams is presented. Our proposal is: (1) A multiresolution model based on a quadtree data structure which encodes approximations of a generalized Voronoi diagram at different levels of detail. (2) A user driven refinement strategy which generates from the quadtree a continuous polygonal approximation of the Voronoi diagram.

Solving multiple kth smallest dissimilarity queries for non-metric dissimilarities with the GPU

The kth smallest dissimilarity of a query point with respect to a given set is the dissimilarity that ranks number k when we sort, in increasing order, the dissimilarity value of the points in the set with respect to the query point. A multiple kth smallest dissimilarity query determines the kth smallest dissimilarity for several query points simultaneously. Although the problem of solving multiple kth smallest dissimilarity queries is an important primitive operation used in many areas, such as spatial data analysis, facility location, text classification and content-based image retrieval, it has not been previously addressed explicitly in the literature. In this paper we present three parallel strategies, to be run on a Graphics Processing Unit, for computing multiple kth smallest dissimilarity queries when non-metric dissimilarities, that do not satisfy the triangular inequality, are used. The strategies are theoretically and experimentally analyzed and compared among them and with an efficient sequential strategy to solve the problem.