http://hdl.handle.net/2117/3197 Wed, 22 May 2013 14:19:16 GMT 2013-05-22T14:19:16Z webmaster.bupc@upc.edu Universitat Politècnica de Catalunya. Servei de Biblioteques i Documentació no http://hdl.handle.net/2117/18614 Title: Connecting Red Cells in a Bicolour Voronoi Diagram Authors: Abellanas, Manuel; Bajuelos, Antonio L.; Canales, Santiago; Claverol Aguas, Mercè; Hernández, Gregorio; Pereira de Matos, Inés Abstract: Let S be a set of n + m sites, of which n are red and have weight wR, and m are blue and weigh wB. The objective of this paper is to calculate the minimum value of the red sites’ weight such that the union of the red Voronoi cells in the weighted Voronoi diagram of S is a connected region. This problem is solved for the multiplicativelyweighted Voronoi diagram in O((n+m)2 log(nm)) time and for both the additively-weighted and power Voronoi diagram in O(nmlog(nm)) time Thu, 04 Apr 2013 13:14:09 GMT http://hdl.handle.net/2117/18614 2013-04-04T13:14:09Z Abellanas, Manuel; Bajuelos, Antonio L.; Canales, Santiago; Claverol Aguas, Mercè; Hernández, Gregorio; Pereira de Matos, Inés no Let S be a set of n + m sites, of which n are red and have weight wR, and m are blue and weigh wB. The objective of this paper is to calculate the minimum value of the red sites’ weight such that the union of the red Voronoi cells in the weighted Voronoi diagram of S is a connected region. This problem is solved for the multiplicativelyweighted Voronoi diagram in O((n+m)2 log(nm)) time and for both the additively-weighted and power Voronoi diagram in O(nmlog(nm)) time http://hdl.handle.net/2117/17768 Title: Median trajectories Authors: Buchin, Kevin; Buchin, Maike; Kreveld, Marc van; Löffler, Maarten; Silveira, Rodrigo Ignacio; Wenk, Carola; Wiratma, Lionov Abstract: We investigate the concept of a median among a set of trajectories. We establish criteria that a “median trajectory” should meet, and present two different methods to construct a median for a set of input trajectories. The first method is very simple, while the second method is more complicated and uses homotopy with respect to sufficiently large faces in the arrangement formed by the trajectories. We give algorithms for both methods, analyze the worst-case running time, and show that under certain assumptions both methods can be implemented efficiently. We empirically compare the output of both methods on randomly generated trajectories, and evaluate whether the two methods yield medians that are according to our intuition. Our results suggest that the second method, using homotopy, performs considerably better. Thu, 14 Feb 2013 14:40:06 GMT http://hdl.handle.net/2117/17768 2013-02-14T14:40:06Z Buchin, Kevin; Buchin, Maike; Kreveld, Marc van; Löffler, Maarten; Silveira, Rodrigo Ignacio; Wenk, Carola; Wiratma, Lionov no We investigate the concept of a median among a set of trajectories. We establish criteria that a “median trajectory” should meet, and present two different methods to construct a median for a set of input trajectories. The first method is very simple, while the second method is more complicated and uses homotopy with respect to sufficiently large faces in the arrangement formed by the trajectories. We give algorithms for both methods, analyze the worst-case running time, and show that under certain assumptions both methods can be implemented efficiently. We empirically compare the output of both methods on randomly generated trajectories, and evaluate whether the two methods yield medians that are according to our intuition. Our results suggest that the second method, using homotopy, performs considerably better. http://hdl.handle.net/2117/17651 Title: On the metric dimension of infinite graphs Authors: Cáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, Maria Luz Abstract: A set of vertices Sresolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we study the metric dimension of infinite graphs such that all its vertices have finite degree. We give necessary conditions for those graphs to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some results about the metric dimension of the cartesian product of finite and infinite graphs, and give the metric dimension of the cartesian product of several families of graphs. Description: Infinite graph; Locally finite graph; Resolving set; Metric dimension; Cartesian product Tue, 12 Feb 2013 12:08:09 GMT http://hdl.handle.net/2117/17651 2013-02-12T12:08:09Z Cáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, Maria Luz no A set of vertices Sresolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we study the metric dimension of infinite graphs such that all its vertices have finite degree. We give necessary conditions for those graphs to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some results about the metric dimension of the cartesian product of finite and infinite graphs, and give the metric dimension of the cartesian product of several families of graphs. http://hdl.handle.net/2117/17499 Title: Cusp points in the parameter space of degenerate 3-RPR planar parallel manipulators Authors: Manubens Ferriol, Montserrat; Moroz, Guillaume; Chablat, Damien; Wenger, Philippe; Rouillier, Fabrice Abstract: This paper investigates the conditions in the design parameter space for the existence and distribution of the cusp locus for planar parallel manipulators. Cusp points make possible non-singular assembly-mode changing motion, which increases the maximum singularity-free workspace. An accurate algorithm for the determination is proposed amending some imprecisions done by previous existing algorithms. This is combined with methods of Cylindric Algebraic Decomposition, Gröbner bases and Discriminant Varieties in order to partition the parameter space into cells with constant number of cusp points. These algorithms will allow us to classify a family of degenerate 3-RPR manipulators. Wed, 23 Jan 2013 13:29:30 GMT http://hdl.handle.net/2117/17499 2013-01-23T13:29:30Z Manubens Ferriol, Montserrat; Moroz, Guillaume; Chablat, Damien; Wenger, Philippe; Rouillier, Fabrice no Cusp, Cylindric algebraic decomposition, Degenerate 3-RPR, Discriminant variety, Kinematics, Parallel manipulator, Singularities, Symbolic computation This paper investigates the conditions in the design parameter space for the existence and distribution of the cusp locus for planar parallel manipulators. Cusp points make possible non-singular assembly-mode changing motion, which increases the maximum singularity-free workspace. An accurate algorithm for the determination is proposed amending some imprecisions done by previous existing algorithms. This is combined with methods of Cylindric Algebraic Decomposition, Gröbner bases and Discriminant Varieties in order to partition the parameter space into cells with constant number of cusp points. These algorithms will allow us to classify a family of degenerate 3-RPR manipulators. http://hdl.handle.net/2117/17219 Title: A linear relaxation method for computing workspace slices of the Stewart platform Authors: Bohigas Nadal, Oriol; Manubens Ferriol, Montserrat; Ros Giralt, Lluís Abstract: The workspace of a Stewart platform is a complex sixdimensional volume embedded in the Cartesian space defined by six pose parameters. Because of its large dimension and complex shape, this volume is difficult to compute and represent, and comprehension on its structure is being gained by studying its three-dimensional slices. While successful methods have been given to determine the constantorientation slice, the computation and appropriate visualization of the constant-position slice (also known as the orientation workspace) has proved to be a challenging task. This paper presents a unified method for computing both of such slices, and any other ones defined by fixing three pose parameters, on general Stewart platforms possibly involving mechanical limits on the active and passive joints. Advantages over existing methods include, in addition to the previous, the ability to determine all connected components of the workspace, and any motion barriers present in its interior. Tue, 08 Jan 2013 17:35:44 GMT http://hdl.handle.net/2117/17219 2013-01-08T17:35:44Z Bohigas Nadal, Oriol; Manubens Ferriol, Montserrat; Ros Giralt, Lluís no manipulators robot kinematics PARAULES AUTOR: workspace determination, slices, Stewart platform, branch-and-prune method, linear relaxation, motion barriers, boundaries The workspace of a Stewart platform is a complex sixdimensional volume embedded in the Cartesian space defined by six pose parameters. Because of its large dimension and complex shape, this volume is difficult to compute and represent, and comprehension on its structure is being gained by studying its three-dimensional slices. While successful methods have been given to determine the constantorientation slice, the computation and appropriate visualization of the constant-position slice (also known as the orientation workspace) has proved to be a challenging task. This paper presents a unified method for computing both of such slices, and any other ones defined by fixing three pose parameters, on general Stewart platforms possibly involving mechanical limits on the active and passive joints. Advantages over existing methods include, in addition to the previous, the ability to determine all connected components of the workspace, and any motion barriers present in its interior. http://hdl.handle.net/2117/16981 Title: Improving shortest paths in the Delaunay triangulation Authors: Abellanas, Manuel; Claverol Aguas, Mercè; Hernández-Peñalver, Gregorio; Hurtado Díaz, Fernando Alfredo; Sacristán Adinolfi, Vera; Saumell Mendiola, Maria; Silveira, Rodrigo Ignacio Abstract: We study a problem about shortest paths in Delaunay triangulations. Given two nodes s, t in the Delaunay triangulation of a point set P, we look for a new point p that can be added, such that the shortest path from s to t, in the Delaunay triangulation of P ∪ {p}, improves as much as possible. We study several properties of the problem, and give efficient algorithms to find such point when the graph-distance used is Euclidean and for the link-distance. Several other variations of the problem are also discussed. Tue, 20 Nov 2012 16:25:42 GMT http://hdl.handle.net/2117/16981 2012-11-20T16:25:42Z Abellanas, Manuel; Claverol Aguas, Mercè; Hernández-Peñalver, Gregorio; Hurtado Díaz, Fernando Alfredo; Sacristán Adinolfi, Vera; Saumell Mendiola, Maria; Silveira, Rodrigo Ignacio no We study a problem about shortest paths in Delaunay triangulations. Given two nodes s, t in the Delaunay triangulation of a point set P, we look for a new point p that can be added, such that the shortest path from s to t, in the Delaunay triangulation of P ∪ {p}, improves as much as possible. We study several properties of the problem, and give efficient algorithms to find such point when the graph-distance used is Euclidean and for the link-distance. Several other variations of the problem are also discussed. http://hdl.handle.net/2117/16559 Title: Maximizing maximal angles for plane straight-line graphs Authors: Aichholzer, Oswin; Hackl, Thomas; Hoffmann, Michael; Huemer, Clemens; Pór, Attila; Santos, Francisco; Speckmann, Bettina; Vogtenhuber, Birgit Abstract: Let G=(S,E) be a plane straight-line graph on a finite point set S⊂R2 in general position. The incident angles of a point p∈S in G are the angles between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight-line graph is called φ-open if each vertex has an incident angle of size at least φ. In this paper we study the following type of question: What is the maximum angle φ such that for any finite set S⊂R2 of points in general position we can find a graph from a certain class of graphs on S that is φ-open? In particular, we consider the classes of triangulations, spanning trees, and spanning paths on S and give tight bounds in most cases. Tue, 25 Sep 2012 10:39:59 GMT http://hdl.handle.net/2117/16559 2012-09-25T10:39:59Z Aichholzer, Oswin; Hackl, Thomas; Hoffmann, Michael; Huemer, Clemens; Pór, Attila; Santos, Francisco; Speckmann, Bettina; Vogtenhuber, Birgit no Let G=(S,E) be a plane straight-line graph on a finite point set S⊂R2 in general position. The incident angles of a point p∈S in G are the angles between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight-line graph is called φ-open if each vertex has an incident angle of size at least φ. In this paper we study the following type of question: What is the maximum angle φ such that for any finite set S⊂R2 of points in general position we can find a graph from a certain class of graphs on S that is φ-open? In particular, we consider the classes of triangulations, spanning trees, and spanning paths on S and give tight bounds in most cases. http://hdl.handle.net/2117/16389 Title: Processing aggregated data : the location of clusters in health data Authors: Buchin, Kevin; Buchin, Maike; Kreveld, Marc van; Löffler, Maarten; Luo, Jun; Silveira, Rodrigo Ignacio Abstract: Spatially aggregated data is frequently used in geographical applications. Often spatial data analysis on aggregated data is performed in the same way as on exact data, which ignores the fact that we do not know the actual locations of the data. We here propose models and methods to take aggregation into account. For this we focus on the problem of locating clusters in aggregated data. More specifically, we study the problem of locating clusters in spatially aggregated health data. The data is given as a subdivision into regions with two values per region, the number of cases and the size of the population at risk. We formulate the problem as finding a placement of a cluster window of a given shape such that a cluster function depending on the population at risk and the cases is maximized. We propose area-based models to calculate the cases (and the population at risk) within a cluster window. These models are based on the areas of intersection of the cluster window with the regions of the subdivision. We show how to compute a subdivision such that within each cell of the subdivision the areas of intersection are simple functions. We evaluate experimentally how taking aggregation into account influences the location of the clusters found. Mon, 27 Aug 2012 16:16:37 GMT http://hdl.handle.net/2117/16389 2012-08-27T16:16:37Z Buchin, Kevin; Buchin, Maike; Kreveld, Marc van; Löffler, Maarten; Luo, Jun; Silveira, Rodrigo Ignacio no Aggregated data, Algorithm, Cluster, Public health Spatially aggregated data is frequently used in geographical applications. Often spatial data analysis on aggregated data is performed in the same way as on exact data, which ignores the fact that we do not know the actual locations of the data. We here propose models and methods to take aggregation into account. For this we focus on the problem of locating clusters in aggregated data. More specifically, we study the problem of locating clusters in spatially aggregated health data. The data is given as a subdivision into regions with two values per region, the number of cases and the size of the population at risk. We formulate the problem as finding a placement of a cluster window of a given shape such that a cluster function depending on the population at risk and the cases is maximized. We propose area-based models to calculate the cases (and the population at risk) within a cluster window. These models are based on the areas of intersection of the cluster window with the regions of the subdivision. We show how to compute a subdivision such that within each cell of the subdivision the areas of intersection are simple functions. We evaluate experimentally how taking aggregation into account influences the location of the clusters found. http://hdl.handle.net/2117/16388 Title: Removing local extrema from imprecise terrains Authors: Gray, Chris; Kammer, Frank; Löffler, Maarten; Silveira, Rodrigo Ignacio Abstract: In this paper we consider imprecise terrains, that is, triangulated terrains with a vertical error interval in the vertices. In particular, we study the problem of removing as many local extrema (minima and maxima) as possible from the terrain; that is, fi nding an assignment of one height to each vertex, within its error interval, so that the resulting terrain has minimum number of local extrema. We show that removing only minima or only maxima can be done optimally in O(n log n) time, for a terrain with n vertices. Interestingly, however, the problem of fi nding a height assignment that minimizes the total number of local extrema (minima as well as maxima) is NP-hard, and is even hard to approximate within a factor of O(log log n) unless P = NP. Moreover, we show that even a simpli ed version of the problem where we can have only three di fferent types of intervals for the vertices is already NP-hard, a result we obtain by proving hardness of a special case of 2-Disjoint Connected Subgraphs, a problem that has lately received considerable attention from the graph-algorithms community. Mon, 27 Aug 2012 15:25:48 GMT http://hdl.handle.net/2117/16388 2012-08-27T15:25:48Z Gray, Chris; Kammer, Frank; Löffler, Maarten; Silveira, Rodrigo Ignacio no Data imprecision, Local minima, P2-Con, P2-MaxCon, Terrain analysis In this paper we consider imprecise terrains, that is, triangulated terrains with a vertical error interval in the vertices. In particular, we study the problem of removing as many local extrema (minima and maxima) as possible from the terrain; that is, fi nding an assignment of one height to each vertex, within its error interval, so that the resulting terrain has minimum number of local extrema. We show that removing only minima or only maxima can be done optimally in O(n log n) time, for a terrain with n vertices. Interestingly, however, the problem of fi nding a height assignment that minimizes the total number of local extrema (minima as well as maxima) is NP-hard, and is even hard to approximate within a factor of O(log log n) unless P = NP. Moreover, we show that even a simpli ed version of the problem where we can have only three di fferent types of intervals for the vertices is already NP-hard, a result we obtain by proving hardness of a special case of 2-Disjoint Connected Subgraphs, a problem that has lately received considerable attention from the graph-algorithms community. http://hdl.handle.net/2117/15026 Title: Bounds on the coefficients of tension and flow polynomials Authors: Breuer, Felix; Dall, Aaron Matthew Abstract: The goal of this article is to obtain bounds on the coefficients of modular and integral flow and tension polynomials of graphs. To this end we use the fact that these polynomials can be realized as Ehrhart polynomials of inside-out polytopes. Inside-out polytopes come with an associated relative polytopal complex and, for a wide class of inside-out polytopes, we show that this complex has a convex ear decomposition. This leads to the desired bounds on the coefficients of these polynomials. Thu, 09 Feb 2012 11:09:13 GMT http://hdl.handle.net/2117/15026 2012-02-09T11:09:13Z Breuer, Felix; Dall, Aaron Matthew no The goal of this article is to obtain bounds on the coefficients of modular and integral flow and tension polynomials of graphs. To this end we use the fact that these polynomials can be realized as Ehrhart polynomials of inside-out polytopes. Inside-out polytopes come with an associated relative polytopal complex and, for a wide class of inside-out polytopes, we show that this complex has a convex ear decomposition. This leads to the desired bounds on the coefficients of these polynomials. http://hdl.handle.net/2117/14589 Title: Segmenting trajectories: A framework and algorithms using spatiotemporal criteria Authors: Buchin, Maike; Driemel, Anne; Kreveld, Marc van; Sacristán Adinolfi, Vera Abstract: In this paper we address the problem of segmenting a trajectory based on spatiotemporal criteria. We require that each segment is homogeneous in the sense that a set of spatiotemporal criteria are fulfilled. We define different such criteria, including location, heading, speed, velocity, curvature, sinuosity, curviness, and shape. We present an algorithmic framework that allows us to segment any trajectory into a minimum number of segments under any of these criteria, or any combination of these criteria. In this framework, a segmentation can generally be computed in O(n log n) time, where n is the number of edges of the trajectory to be segmented. We also discuss the robustness of our approach. Tue, 17 Jan 2012 09:29:32 GMT http://hdl.handle.net/2117/14589 2012-01-17T09:29:32Z Buchin, Maike; Driemel, Anne; Kreveld, Marc van; Sacristán Adinolfi, Vera no In this paper we address the problem of segmenting a trajectory based on spatiotemporal criteria. We require that each segment is homogeneous in the sense that a set of spatiotemporal criteria are fulfilled. We define different such criteria, including location, heading, speed, velocity, curvature, sinuosity, curviness, and shape. We present an algorithmic framework that allows us to segment any trajectory into a minimum number of segments under any of these criteria, or any combination of these criteria. In this framework, a segmentation can generally be computed in O(n log n) time, where n is the number of edges of the trajectory to be segmented. We also discuss the robustness of our approach. http://hdl.handle.net/2117/14505 Title: Connect the dot: computing feed-links for network extension Authors: Speckmann, Bettina; Silveira, Rodrigo Ignacio; Aronov, Boris; Buchin, Kevin; Buchin, Maike; Jansen, Bart; De Jong, Tom; Kreveld, Marc van; Loffler, Maarten; Luo, Jun Abstract: Road network analysis can require distance from points that are not on the network themselves. We study the algorithmic problem of connecting a point inside a face (region) of the road network to its boundary while minimizing the detour factor of that point to any point on the boundary of the face. We show that the optimal single connection (feed-link) can be computed in O(λ7(n) log n) time, where n is the number of vertices that bounds the face and λ7(n) is the slightly superlinear maximum length of a Davenport- Schinzel sequence of order 7 on n symbols. We also present approximation results for placing more feed-links, deal with the case that there are obstacles in the face of the road network that contains the point to be connected, and present various related results. Fri, 13 Jan 2012 09:39:29 GMT http://hdl.handle.net/2117/14505 2012-01-13T09:39:29Z Speckmann, Bettina; Silveira, Rodrigo Ignacio; Aronov, Boris; Buchin, Kevin; Buchin, Maike; Jansen, Bart; De Jong, Tom; Kreveld, Marc van; Loffler, Maarten; Luo, Jun no Road network analysis can require distance from points that are not on the network themselves. We study the algorithmic problem of connecting a point inside a face (region) of the road network to its boundary while minimizing the detour factor of that point to any point on the boundary of the face. We show that the optimal single connection (feed-link) can be computed in O(λ7(n) log n) time, where n is the number of vertices that bounds the face and λ7(n) is the slightly superlinear maximum length of a Davenport- Schinzel sequence of order 7 on n symbols. We also present approximation results for placing more feed-links, deal with the case that there are obstacles in the face of the road network that contains the point to be connected, and present various related results. http://hdl.handle.net/2117/13767 Title: Minimizing the range for k-covered paths on sensor networks Authors: Abellanas, Manuel; Bajuelos, Antonio L.; Pereira de Matos, Inés Abstract: Coverage problems are a flourishing topic in optimization, thanks to the recent advances in the field of wireless sensor networks. The main coverage issue centres around critical conditions that require reliable monitoring and prohibit failures. This issue can be addressed by maximal-exposure paths, regarding which this article presents new results. Namely, it shows how to minimize the sensing range of a set of sensors in order to ensure the existence of a k-covered path between two points on a given region. Such a path’s coverage depends on k ≥ 2, which is fixed. The three types of regions studied are: a planar graph, the whole plane and a polygonal region. Tue, 08 Nov 2011 11:09:46 GMT http://hdl.handle.net/2117/13767 2011-11-08T11:09:46Z Abellanas, Manuel; Bajuelos, Antonio L.; Pereira de Matos, Inés no Coverage problems are a flourishing topic in optimization, thanks to the recent advances in the field of wireless sensor networks. The main coverage issue centres around critical conditions that require reliable monitoring and prohibit failures. This issue can be addressed by maximal-exposure paths, regarding which this article presents new results. Namely, it shows how to minimize the sensing range of a set of sensors in order to ensure the existence of a k-covered path between two points on a given region. Such a path’s coverage depends on k ≥ 2, which is fixed. The three types of regions studied are: a planar graph, the whole plane and a polygonal region. http://hdl.handle.net/2117/12490 Title: On computing enclosing isosceles triangles and related problems Authors: Bose, Prosenjit; Mora Giné, Mercè; Seara Ojea, Carlos; Sethia, Saurabh Abstract: Given a set of n points in the plane, we show how to compute various enclosing isosceles triangles where different parameters such as area or perimeter are optimized. We then study a 3-dimensional version of the problem where we enclose a point set with a cone of fixed aperture $\alpha$. Fri, 06 May 2011 09:27:12 GMT http://hdl.handle.net/2117/12490 2011-05-06T09:27:12Z Bose, Prosenjit; Mora Giné, Mercè; Seara Ojea, Carlos; Sethia, Saurabh no Given a set of n points in the plane, we show how to compute various enclosing isosceles triangles where different parameters such as area or perimeter are optimized. We then study a 3-dimensional version of the problem where we enclose a point set with a cone of fixed aperture $\alpha$. http://hdl.handle.net/2117/12026 Title: Fitting a two-joint orthogonal chain to a point set Authors: Díaz-Báñez, José Miguel; López, Mario A.; Mora Giné, Mercè; Seara Ojea, Carlos; Ventura, Inmaculada Abstract: We study the problem of fitting a two-joint orthogonal polygonal chain to a set S of n points in the plane, where the objective function is to minimize the maximum orthogonal distance from S to the chain. We show that this problem can be solved in Θ(n) time if the orientation of the chain is fixed, and in Θ(n log n) time when the orientation is not a priori known. We also consider some variations of the problem in three-dimensions where a polygonal chain is interpreted as a configuration of orthogonal planes. In this case we obtain O(n) and O(n log n) time algorithms depending on which plane orientations are fixed. Wed, 23 Mar 2011 08:53:28 GMT http://hdl.handle.net/2117/12026 2011-03-23T08:53:28Z Díaz-Báñez, José Miguel; López, Mario A.; Mora Giné, Mercè; Seara Ojea, Carlos; Ventura, Inmaculada no We study the problem of fitting a two-joint orthogonal polygonal chain to a set S of n points in the plane, where the objective function is to minimize the maximum orthogonal distance from S to the chain. We show that this problem can be solved in Θ(n) time if the orientation of the chain is fixed, and in Θ(n log n) time when the orientation is not a priori known. We also consider some variations of the problem in three-dimensions where a polygonal chain is interpreted as a configuration of orthogonal planes. In this case we obtain O(n) and O(n log n) time algorithms depending on which plane orientations are fixed.