Ponències/Comunicacions de congressos http://hdl.handle.net/2117/3199 2019-08-18T09:50:12Z Order types of random point sets can be realized with small integer coordinates http://hdl.handle.net/2117/113414 Order types of random point sets can be realized with small integer coordinates Fabila-Monroy, Ruy; Huemer, Clemens Let S := {p1, . . . , pn} be a set of n points chosen independently and uniformly at random from the unit square and let M be a positive integer. For every point pi = (xi , yi) in S, let p 0 i = (bMxic, bMyic). Let S 0 := {p 0 i : 1 = i = n}. We call S 0 the digitization of S by M. In this paper we study the problem: How large does M have to be such that with high probability, S and S 0 have the same order type? 2018-01-30T15:32:16Z Fabila-Monroy, Ruy Huemer, Clemens Let S := {p1, . . . , pn} be a set of n points chosen independently and uniformly at random from the unit square and let M be a positive integer. For every point pi = (xi , yi) in S, let p 0 i = (bMxic, bMyic). Let S 0 := {p 0 i : 1 = i = n}. We call S 0 the digitization of S by M. In this paper we study the problem: How large does M have to be such that with high probability, S and S 0 have the same order type? Matching points with diametral disks http://hdl.handle.net/2117/113382 Matching points with diametral disks Huemer, Clemens; Pérez-Lantero, Pablo; Seara Ojea, Carlos; Silveira, Rodrigo Ignacio We consider matchings between a set R of red points and a set B of blue points with diametral disks. In other words, for each pair of matched points p ¿ R and q ¿ B, we consider the diametral disk defined by p and q. We prove that for any R and B such that |R| = |B|, there exists a perfect matching such that the diametral disks of the matched point pairs have a common intersection. More precisely, we show that a maximum weight perfect matching has this property 2018-01-30T11:46:33Z Huemer, Clemens Pérez-Lantero, Pablo Seara Ojea, Carlos Silveira, Rodrigo Ignacio We consider matchings between a set R of red points and a set B of blue points with diametral disks. In other words, for each pair of matched points p ¿ R and q ¿ B, we consider the diametral disk defined by p and q. We prove that for any R and B such that |R| = |B|, there exists a perfect matching such that the diametral disks of the matched point pairs have a common intersection. More precisely, we show that a maximum weight perfect matching has this property The connectivity of the flip graph of Hamiltonian paths of the grid graph http://hdl.handle.net/2117/113119 The connectivity of the flip graph of Hamiltonian paths of the grid graph Duque, Frank; Fabila Monroy, Ruy; Flores Peñazola, David; Hidalgo Toscano, Carlos; Huemer, Clemens Let Gn,m be the grid graph with n columns and m rows. Let Hn,m be the graph whose vertices are the Hamiltonian paths in Gn,m, where two vertices P1 and P2 are adjacent if we can obtain P2 from P1 by deleting an edge in P1 and adding an edge not in P1. In this paper we show that Hn,2, Hn,3 and Hn,4 are connected 2018-01-23T18:39:15Z Duque, Frank Fabila Monroy, Ruy Flores Peñazola, David Hidalgo Toscano, Carlos Huemer, Clemens Let Gn,m be the grid graph with n columns and m rows. Let Hn,m be the graph whose vertices are the Hamiltonian paths in Gn,m, where two vertices P1 and P2 are adjacent if we can obtain P2 from P1 by deleting an edge in P1 and adding an edge not in P1. In this paper we show that Hn,2, Hn,3 and Hn,4 are connected Location in maximal outerplanar graphs http://hdl.handle.net/2117/107909 Location in maximal outerplanar graphs Claverol Aguas, Mercè; García, Alfredo; Hernández, Gregorio; Hernando Martín, María del Carmen; Maureso Sánchez, Montserrat; Mora Giné, Mercè; Tejel, Javier In this work we study the metric dimension and the location-domination number of maximal outerplanar graphs. Concretely, we determine tight upper and lower bounds on the metric dimension and characterize those maximal outerplanar graphs attaining the lower bound. We also give a lower bound on the location-domination number of maximal outerplanar graphs. 2017-09-22T11:18:42Z Claverol Aguas, Mercè García, Alfredo Hernández, Gregorio Hernando Martín, María del Carmen Maureso Sánchez, Montserrat Mora Giné, Mercè Tejel, Javier In this work we study the metric dimension and the location-domination number of maximal outerplanar graphs. Concretely, we determine tight upper and lower bounds on the metric dimension and characterize those maximal outerplanar graphs attaining the lower bound. We also give a lower bound on the location-domination number of maximal outerplanar graphs. Locating-dominating partitions in graphs http://hdl.handle.net/2117/104422 Locating-dominating partitions in graphs Pelayo Melero, Ignacio Manuel; Hernando Martín, María del Carmen; Mora Giné, Mercè Let G = (V, E) be a connected graph of order n. Let ¿ = {S1, . . . , Sk} be a partition of V . Let r(u|¿) denote the vector of distances between a vertex v ¿ V and the elements of ¿, that is, r(v, ¿) = (d(v, S1), . . . , d(v, Sk)). The partition ¿ is called a locating partition of G if, for every pair of distinct vertices u, v ¿ V , r(u, ¿) 6= r(v, ¿). A locating partition ¿ is called metriclocating-dominating partition (an MLD-partition for short) of G if it is also dominating, 2017-05-15T12:02:03Z Pelayo Melero, Ignacio Manuel Hernando Martín, María del Carmen Mora Giné, Mercè Let G = (V, E) be a connected graph of order n. Let ¿ = {S1, . . . , Sk} be a partition of V . Let r(u|¿) denote the vector of distances between a vertex v ¿ V and the elements of ¿, that is, r(v, ¿) = (d(v, S1), . . . , d(v, Sk)). The partition ¿ is called a locating partition of G if, for every pair of distinct vertices u, v ¿ V , r(u, ¿) 6= r(v, ¿). A locating partition ¿ is called metriclocating-dominating partition (an MLD-partition for short) of G if it is also dominating, Map construction algorithms: an evaluation through hiking data http://hdl.handle.net/2117/103775 Map construction algorithms: an evaluation through hiking data Duran, David; Sacristán Adinolfi, Vera; Silveira, Rodrigo Ignacio We study five existing map construction algorithms, designed and tested with urban vehicle data in mind, and apply them to hiking trajectories with different terrain characteristics. Our main goal is to better understand the existing algorithms and to what extent they apply in a wider context. Indeed, our data differs from the one previously used to evaluate map construction algorithm in several aspects: higher GPS error, narrow and winding paths, and trajectories with its own characteristics in terms of speed or direction. We have chosen four different areas of varied geographic features. For each of them we have considered a set of hiking GPS trajectories, each with a total number of nodes between 38,000 and 288,000. For each algorithm we have analyzed the parameters it uses, and adjusted them to each data set. We present an analysis of the generated maps produced by each algorithm on each data set, and a discussion of the most important artifacts detected. We consider that this analysis sheds new light into the current challenges for map construction algorithms, and will be of help for designing new and better methods. 2017-04-27T10:27:51Z Duran, David Sacristán Adinolfi, Vera Silveira, Rodrigo Ignacio We study five existing map construction algorithms, designed and tested with urban vehicle data in mind, and apply them to hiking trajectories with different terrain characteristics. Our main goal is to better understand the existing algorithms and to what extent they apply in a wider context. Indeed, our data differs from the one previously used to evaluate map construction algorithm in several aspects: higher GPS error, narrow and winding paths, and trajectories with its own characteristics in terms of speed or direction. We have chosen four different areas of varied geographic features. For each of them we have considered a set of hiking GPS trajectories, each with a total number of nodes between 38,000 and 288,000. For each algorithm we have analyzed the parameters it uses, and adjusted them to each data set. We present an analysis of the generated maps produced by each algorithm on each data set, and a discussion of the most important artifacts detected. We consider that this analysis sheds new light into the current challenges for map construction algorithms, and will be of help for designing new and better methods. Implementing data-dependent triangulations with higher order Delaunay triangulations http://hdl.handle.net/2117/103687 Implementing data-dependent triangulations with higher order Delaunay triangulations Rodríguez, Natalia; Silveira, Rodrigo Ignacio The Delaunay triangulation is the standard choice for building triangulated irregular networks (TINs) to represent terrain surfaces. However, the Delaunay triangulation is based only on the 2D coordinates of the data points, ignoring their elevation. It has long been recognized that sometimes it may be beneficial to use other, non-Delaunay, criteria to build TINs. Data-dependent triangulations were introduced decades ago to address this. However, they are rarely used in practice, mostly because the optimization of data- dependent criteria often results in triangulations with many thin and elongated triangles. Recently, in the field of computational geometry, higher order Delaunay triangulations (HODTs) were introduced, trying to tackle both issues at the same time-data-dependent criteria and good triangle shape. Nevertheless, most previous studies about them have been limited to theoretical aspects. In this work we present the first extensive experimental study on the practical use of HODTs, as a tool to build data-dependent TINs. We present experiments with two USGS terrains that show that HODTs can give significant improvements over the Delaunay triangulation for the criteria identified as most important for data-dependent triangulations. The resulting triangulations have data-dependent values comparable to those obtained with pure data-dependent approaches, without compromising the shape of the triangles, and are faster to compute. 2017-04-25T08:25:55Z Rodríguez, Natalia Silveira, Rodrigo Ignacio The Delaunay triangulation is the standard choice for building triangulated irregular networks (TINs) to represent terrain surfaces. However, the Delaunay triangulation is based only on the 2D coordinates of the data points, ignoring their elevation. It has long been recognized that sometimes it may be beneficial to use other, non-Delaunay, criteria to build TINs. Data-dependent triangulations were introduced decades ago to address this. However, they are rarely used in practice, mostly because the optimization of data- dependent criteria often results in triangulations with many thin and elongated triangles. Recently, in the field of computational geometry, higher order Delaunay triangulations (HODTs) were introduced, trying to tackle both issues at the same time-data-dependent criteria and good triangle shape. Nevertheless, most previous studies about them have been limited to theoretical aspects. In this work we present the first extensive experimental study on the practical use of HODTs, as a tool to build data-dependent TINs. We present experiments with two USGS terrains that show that HODTs can give significant improvements over the Delaunay triangulation for the criteria identified as most important for data-dependent triangulations. The resulting triangulations have data-dependent values comparable to those obtained with pure data-dependent approaches, without compromising the shape of the triangles, and are faster to compute. Stabbing circles for some sets of Delaunay segments http://hdl.handle.net/2117/102530 Stabbing circles for some sets of Delaunay segments Claverol Aguas, Mercè; Khramtcova, Elena; Papadopoulou, Evanthia; Saumell, Maria; Seara Ojea, Carlos Let S be a set of n segments in the plane such that, for every segment, its two endpoints are adjacent in the Delaunay triangulation of the set of endpoints of all segments in S. Our goal is to compute all the combinatorially different stabbing circles for S, and the ones with maximum and minimum radius. We exploit a recent result to solve this problem in O(n log n) in two particular cases: (i) all segments in S are parallel; (ii) all segments in S have the same length. We also show that the problem of computing the stabbing circle of minimum radius of a set of n parallel segments of equal length (not necessarily satisfying the Delaunay condition) has an Omega(n log n) lower bound. 2017-03-15T15:55:08Z Claverol Aguas, Mercè Khramtcova, Elena Papadopoulou, Evanthia Saumell, Maria Seara Ojea, Carlos Let S be a set of n segments in the plane such that, for every segment, its two endpoints are adjacent in the Delaunay triangulation of the set of endpoints of all segments in S. Our goal is to compute all the combinatorially different stabbing circles for S, and the ones with maximum and minimum radius. We exploit a recent result to solve this problem in O(n log n) in two particular cases: (i) all segments in S are parallel; (ii) all segments in S have the same length. We also show that the problem of computing the stabbing circle of minimum radius of a set of n parallel segments of equal length (not necessarily satisfying the Delaunay condition) has an Omega(n log n) lower bound. Stabbing circles for sets of segments in the plane http://hdl.handle.net/2117/102514 Stabbing circles for sets of segments in the plane Claverol Aguas, Mercè; Khramtcova, Elena; Papadopoulou, Evanthia; Saumell, Maria; Seara Ojea, Carlos Stabbing a set S of n segments in the plane by a line is a well-known problem. In this paper we consider the variation where the stabbing object is a circle instead of a line. We show that the problem is tightly connected to cluster Voronoi diagrams, in particular, the Hausdorff and the farthest-color Voronoi diagram. Based on these diagrams, we provide a method to compute all the combinatorially different stabbing circles for S, and the stabbing circles with maximum and minimum radius. We give conditions under which our method is fast. These conditions are satisfied if the segments in S are parallel, resulting in a Onlog2n) time algorithm. We also observe that the stabbing circle problem for S can be solved in optimal O(n2) time and space by reducing the problem to computing the stabbing planes for a set of segments in 3D. 2017-03-15T13:04:35Z Claverol Aguas, Mercè Khramtcova, Elena Papadopoulou, Evanthia Saumell, Maria Seara Ojea, Carlos Stabbing a set S of n segments in the plane by a line is a well-known problem. In this paper we consider the variation where the stabbing object is a circle instead of a line. We show that the problem is tightly connected to cluster Voronoi diagrams, in particular, the Hausdorff and the farthest-color Voronoi diagram. Based on these diagrams, we provide a method to compute all the combinatorially different stabbing circles for S, and the stabbing circles with maximum and minimum radius. We give conditions under which our method is fast. These conditions are satisfied if the segments in S are parallel, resulting in a Onlog2n) time algorithm. We also observe that the stabbing circle problem for S can be solved in optimal O(n2) time and space by reducing the problem to computing the stabbing planes for a set of segments in 3D. A new meta-module for efficient reconfiguration of hinged-units modular robots http://hdl.handle.net/2117/100623 A new meta-module for efficient reconfiguration of hinged-units modular robots Parada, Irene; Sacristán Adinolfi, Vera; Silveira, Rodrigo Ignacio We present a robust and compact meta-module for edge-hinged modular robot units such as M-TRAN, SuperBot, SMORES, UBot, PolyBot and CKBot, as well as for central-point-hinged ones such as Molecubes and Roombots. Thanks to the rotational degrees of freedom of these units, the novel meta-module is able to expand and contract, as to double/halve its length in each dimension. Moreover, for a large class of edge-hinged robots the proposed meta-module also performs the scrunch/relax and transfer operations required by any tunneling-based reconfiguration strategy, such as those designed for Crystalline and Telecube robots. These results make it possible to apply efficient geometric reconfiguration algorithms to this type of robots. We prove the size of this new meta-module to be optimal. Its robustness and performance substantially improve over previous results. 2017-02-07T12:20:38Z Parada, Irene Sacristán Adinolfi, Vera Silveira, Rodrigo Ignacio We present a robust and compact meta-module for edge-hinged modular robot units such as M-TRAN, SuperBot, SMORES, UBot, PolyBot and CKBot, as well as for central-point-hinged ones such as Molecubes and Roombots. Thanks to the rotational degrees of freedom of these units, the novel meta-module is able to expand and contract, as to double/halve its length in each dimension. Moreover, for a large class of edge-hinged robots the proposed meta-module also performs the scrunch/relax and transfer operations required by any tunneling-based reconfiguration strategy, such as those designed for Crystalline and Telecube robots. These results make it possible to apply efficient geometric reconfiguration algorithms to this type of robots. We prove the size of this new meta-module to be optimal. Its robustness and performance substantially improve over previous results.