 Research group
 DCCG  Research group on discrete, combinatorial and computational geometry
 Department
 Department of Applied Mathematics II
 rodrigo.silveiraupc.edu
 Contact details
 UPC directory
 Orcid
 0000000302024543
 Scopus Author ID
 23398643600
Scientific and technological production


Flow computations on imprecise terrains
Driemel, Anne; Haverkort, Herman; Löffler, Maarten; Silveira, Rodrigo Ignacio
Journal of Computational Geometry
Date of publication: 2013
Journal article
Read the abstract Access to the full text Share Reference managersWe study water flow computation on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predened graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x; y)coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NPhard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, or a set of vertices, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time.
We study water flow computation on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a prede ned graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x; y)coordinates are fi xed. For the fi rst model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NPhard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, or a set of vertices, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time. 
Stabbing Segments with Rectilinear Objects
Claverol Aguas, Merce; Seara Ojea, Carlos; Garijo, Delia; Korman Cozzetti, Matias; Silveira, Rodrigo Ignacio
Mexican Conference on Discrete Mathematics and Computational Geometry
Presentation's date: 20131113
Presentation of work at congresses
Read the abstract View Share Reference managersGiven a set of n line segments in the plane, we say that a region R of the plane is a stabber if R contains exactly one end point of each segment of the set. In this paper we provide efficient algorithms for determining wheter or not a stabber exists for several shapes of stabbers. Specially, we consider the case in which the stabber can be described as the intersecction of isothetic halfplanes (thus the stabbers are halfplanes, strips, quadrants, 3sided rectangles, or rectangles). We provided efficient algorithms reporting all combinatorially different stabbers of the shape. The algorithms run in O(n) time (for the halfplane case), O(n logn) time (for strips and quadrants), O(n^2) (for 3sided rectangles), or O(n^3) time (for rectangles).
Given a set of n line segments in the plane, we say that a region R of the plane is a stabber if R contains exactly one end point of each segment of the set. In this paper we provide efficient algorithms for determining wheter or not a stabber exists for several shapes of stabbers. Specially, we consider the case in which the stabber can be described as the intersecction of isothetic halfplanes (thus the stabbers are halfplanes, strips, quadrants, 3sided rectangles, or rectangles). We provided efficient algorithms reporting all combinatorially different stabbers of the shape. The algorithms run in O(n) time (for the halfplane case), O(n logn) time (for strips and quadrants), O(n^2) (for 3sided rectangles), or O(n^3) time (for rectangles). 
Colored spanning graphs for set visualization
Hurtado Diaz, Fernando Alfredo; Korman Cozzetti, Matias; Van Kreveld, Matias; Löffler, Maarten; Sacristán Adinolfi, Vera; Silveira, Rodrigo Ignacio; Speckmann, Bettina
Symposium on Graph Drawing
Presentation's date: 201309
Presentation of work at congresses
Read the abstract View Share Reference managersWe study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A redbluepurple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem is NPhard. Hence we give an (1/2¿+1)approximation, where ¿ is the Steiner ratio. We also present efficient exact solutions if the points are located on a line or a circle. Finally we consider extensions to more than two sets.
We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A redbluepurple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem is NPhard. Hence we give an (1/2¿+1)approximation, where ¿ is the Steiner ratio. We also present efficient exact solutions if the points are located on a line or a circle. Finally we consider extensions to more than two sets. 
New results on stabbing segments with a polygon
Díaz Bañez, José Miguel; Korman Cozzetti, Matias; Pérez Lantero, Pablo; Pilz, Alexander; Seara Ojea, Carlos; Silveira, Rodrigo Ignacio
International Conference on Algorithms and Complexity
Presentation's date: 201305
Presentation of work at congresses
Read the abstract View Share Reference managersWe consider a natural variation of the concept of stabbing a segment by a simple polygon: a segment is stabbed by a simple polygon P if at least one of its two endpoints is contained in P. A segment set S is stabbed by P if every segment of S is stabbed by P. We show that if S is a set of pairwise disjoint segments, the problem of computing the minimum perimeter polygon stabbing S can be solved in polynomial time. We also prove that for general segments the problem is NPhard. Further, an adaptation of our polynomialtime algorithm solves an open problem posed by Löffler and van Kreveld [Algorithmica 56(2), 236269 (2010)] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments.
We consider a natural variation of the concept of stabbing a segment by a simple polygon: a segment is stabbed by a simple polygon P if at least one of its two endpoints is contained in P. A segment set S is stabbed by P if every segment of S is stabbed by P. We show that if S is a set of pairwise disjoint segments, the problem of computing the minimum perimeter polygon stabbing S can be solved in polynomial time. We also prove that for general segments the problem is NPhard. Further, an adaptation of our polynomialtime algorithm solves an open problem posed by Löffler and van Kreveld [Algorithmica 56(2), 236269 (2010)] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments. 
Median trajectories
Buchin, Kevin; Buchin, Maike; Kreveld, Marc van; Löffler, Maarten; Silveira, Rodrigo Ignacio; Wenk, Carola; Wiratma, Lionov
Algorithmica
Date of publication: 201205
Journal article
Read the abstract Access to the full text Share Reference managersWe 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 worstcase 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.
Postprint (author’s final draft) 
Drawing (complete) binary tanglegrams hardness, approximation, fixedparameter tractability
Buchin, Kevin; Buchin, Maike; Byrka, Jaroslaw; Noellenburg, Martin; Okamoto, Yoshio; Silveira, Rodrigo Ignacio; Wolff, Alexander
Algorithmica
Date of publication: 201202
Journal article
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Improving shortest paths in the Delaunay triangulation
Claverol Aguas, Merce; Hernández Peñalver, Gregorio; Hurtado Diaz, Fernando Alfredo; Sacristán Adinolfi, Vera; Saumell, Maria; Silveira, Rodrigo Ignacio; Abellanas, Manuel
International journal of computational geometry and applications
Date of publication: 2012
Journal article
Read the abstract Access to the full text Share Reference managersWe study a problem about shortest paths in Delaunay triangulations. Given two nodes s, t in the Delaunay triangulation of a point set S, we look for a new point p ¿ S that can be added, such that the shortest path from s to t, in the Delaunay triangulation of S¿{p}, improves as much as possible. We study several properties of the problem, and give efficient algorithms to find such a point when the graphdistance used is Euclidean and for the linkdistance. Several other variations of the problem are also discussed.
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 graphdistance used is Euclidean and for the linkdistance. Several other variations of the problem are also discussed. 
Processing aggregated data : the location of clusters in health data
Buchin, Kevin; Buchin, Maike; Kreveld, Marc van; Löffler, Maarten; Luo, Jun; Silveira, Rodrigo Ignacio
Geoinformatica
Date of publication: 201207
Journal article
Read the abstract View Share Reference managersSpatially 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 areabased 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. 
Removing local extrema from imprecise terrains
Gray, Chris; Kammer, Frank; Löffler, Maarten; Silveira, Rodrigo Ignacio
Computational geometry: theory and applications
Date of publication: 201208
Journal article
Read the abstract Access to the full text Share Reference managersIn 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 NPhard, 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 NPhard, a result we obtain by proving hardness of a special case of 2Disjoint Connected Subgraphs, a problem that has lately received considerable attention from the graphalgorithms community. 
Bichromatic 2center of pairs of points
Arkin, Esther M.; Díaz Bañez, José Miguel; Hurtado Diaz, Fernando Alfredo; Kumar, Piyush; Mitchell, Joseph S. B.; Palop, Belén; Pérez Lantero, Pablo; Saumell, Maria; Silveira, Rodrigo Ignacio
Latin American Symposium on Theoretical Informatics
Presentation's date: 20120419
Presentation of work at congresses
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Connect the dot: computing feedlinks for network extension
Aronov, Boris; Buchin, Kevin; Buchin, Maike; Jansen, Bart; De Jong, Tom; Kreveld, Marc van; Loffler, Maarten; Luo, Jun; Silveira, Rodrigo Ignacio; Speckmann, Bettina
Journal of Spatial Information Science
Date of publication: 20111220
Journal article
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On the number of higher order Delaunay triangulations
Mitsche, Dieter Wilhelm; Saumell Mendiola, Maria; Silveira, Rodrigo Ignacio
Theoretical computer science
Date of publication: 20110701
Journal article
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Embedding rivers in triangulated irregular networks with linear programming
Kreveld, Marc van; Silveira, Rodrigo Ignacio
International journal of geographical information science
Date of publication: 2011
Journal article
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Peeling meshed potatoes
Aronov, Boris; Kreveld, Marc van; Löffler, Maarten; Silveira, Rodrigo Ignacio
Algorithmica
Date of publication: 2011
Journal article
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Puntos y grafos: puentes geométricos (IP04 en CRP Comb. of points sets, ComPoSe,EuroGIGA ESF)
Claverol Aguas, Merce; Dall, Aaron Matthew; Silveira, Rodrigo Ignacio; Huemer, Clemens; Mora Gine, Mercè; Sacristán Adinolfi, Vera; Hernando Martin, Maria Del Carmen; Seara Ojea, Carlos; Montes Lozano, Antonio; Hurtado Diaz, Fernando Alfredo
Participation in a competitive project
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MATHEMATICAL FOUNDATIONS OF HIGH QUALITY TERRAIN MODELS
Silveira, Rodrigo Ignacio; Hurtado Diaz, Fernando Alfredo
Participation in a competitive project
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Adjacencypreserving spatial treemaps
Buchin, Kevin; Eppstein, David; Löffler, Maarten; Nöllenburg, Martin; Silveira, Rodrigo Ignacio
Workshop on Algorithms and Data Structures
Presentation's date: 20110816
Presentation of work at congresses
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Improving shortest paths in the Delaunay triangulation
Claverol Aguas, Merce; Hernández Peñalver, Gregorio; Hurtado Diaz, Fernando Alfredo; Sacristán Adinolfi, Vera; Saumell Mendiola, Maria; Silveira, Rodrigo Ignacio; Abellanas, Manuel
European Workshop on Computational Geometry
Presentation's date: 20110328
Presentation of work at congresses
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Computing a visibility polygon using few variables
Barba, Luis; Korman Cozzetti, Matias; Langerman, Stefan; Silveira, Rodrigo Ignacio
International Symposium on Algorithms and Computation
Presentation's date: 20111205
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Flow computations on imprecise terrains
Driemel, Anne; Haverkort, Herman; Löffler, Maarten; Silveira, Rodrigo Ignacio
Workshop on Algorithms and Data Structures
Presentation's date: 20110815
Presentation of work at congresses
Read the abstract View Share Reference managersWe study water flow computation on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x, y)coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NPhard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time. 
Flow computations on imprecise terrains
Driemel, Anne; Haverkort, Herman; Loffler, Maarten; Silveira, Rodrigo Ignacio
European Workshop on Computational Geometry
Presentation's date: 20110328
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Improving shortest paths in the Delaunay triangulation
Claverol Aguas, Merce; Hernández, Gregorio; Hurtado Diaz, Fernando Alfredo; Sacristán Adinolfi, Vera; Saumell Mendiola, Maria; Silveira, Rodrigo Ignacio; Abellanas, Manuel
Spanish Meeting on Computational Geometry
Presentation's date: 20110628
Presentation of work at congresses
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Finding the most relevant fragments in networks
Buchin, Kevin; Cabello, Sergio; Gudmundsson, Joachim; Löffler, Maarten; Luo, Jun; Rote, Günter; Silveira, Rodrigo Ignacio; Speckmann, Bettina; Wolle, Thomas
Journal of graph algorithms and applications
Date of publication: 201006
Journal article
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Smoothing imprecise 1.5D terrains
Gray, Chris; Löffler, Maarten; Silveira, Rodrigo Ignacio
International journal of computational geometry and applications
Date of publication: 201008
Journal article
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Flooding countries and destroying dams
Silveira, Rodrigo Ignacio; Van Oostrum, René
International journal of computational geometry and applications
Date of publication: 201006
Journal article
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Optimization for first order Delaunay triangulations
Kreveld, Marc van; Löffler, Maarten; Silveira, Rodrigo Ignacio
Computational geometry: theory and applications
Date of publication: 201005
Journal article
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Removing local extrema from imprecise terrains
Gray, Chris; Kammer, Frank; Löffler, Maarten; Silveira, Rodrigo Ignacio
European Workshop on Computational Geometry
Presentation's date: 20100324
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Median trajectories
Buchin, Kevin; Buchin, M.; Kreveld, Marc van; Löffler, Maarten; Silveira, Rodrigo Ignacio; Wenk, Carola; Wiratma, Lionov
European Symposium on Algorithms
Presentation's date: 20100906
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Computing similarity between piecewiselinear functions
Agarwal, Pankaj; Aronov, Boris; Kreveld, Marc van; Löffler, Maarten; Silveira, Rodrigo Ignacio
ACM Annual Symposium on Computational Geometry
Presentation's date: 20100616
Presentation of work at congresses
Read the abstract Access to the full text Share Reference managersWe study the problem of computing the similarity between two piecewiselinear bivariate functions de ned over a common domain, where the surfaces they de ne in 3Dpolyhedral terrainscan be transformed vertically by a linear transformation of the third coordinate (scaling and translation). We present a randomized algorithm that minimizes the maximum vertical distance between the graphs of the two functions, over all linear transformations of one of the terrains, in O(n4=3 polylog n) expected time, where n is the total number of vertices in the graphs of the two functions. We also study the computation of similarity between two univariate or bivariate functions by minimizing the area or volume between their graphs. For univariate functions we give a (1+")approximation algorithm for minimizing the area that runs in O(n=p") time, for any xed " > 0. The (1 + ") approximation algorithm for the bivariate version, where volume is minimized, runs in O(n="2) time, for any xed " > 0, provided the two functions are de ned over the same triangulation of their domain.
Postprint (author’s final draft) 
On the number of higher order Delaunay triangulations
Mitsche, Dieter Wilhelm; Saumell Mendiola, Maria; Silveira, Rodrigo Ignacio
International Conference on Algorithms and Complexity
Presentation's date: 20100527
Presentation of work at congresses
Read the abstract View Share Reference managersHigher order Delaunay triangulations are a generalization of the Delaunay triangulation which provides a class of wellshaped triangulations, over which extra criteria can be optimized. A triangulation is orderk Delaunay if the circumcircle of each triangle of the triangulation contains at most k points. In this paper we study lower and upper bounds on the number of higher order Delaunay triangulations, as well as their expected number for randomly distributed points. We show that arbitrarily large point sets can have a single higher order Delaunay triangulation, even for large orders, whereas for first order Delaunay triangulations, the maximum number is 2n−3. Next we show that uniformly distributed points have an expected number of at least 2ρ1n(1+o(1)) first order Delaunay triangulations, where ρ1 is an analytically defined constant (ρ1 ≈ 0.525785), and for k > 1, the expected number of orderk Delaunay triangulations (which are not orderi for any i < k) is at least 2ρkn(1+o(1)), where ρk can be calculated numerically. 
Towards a definition of higher order constrained Delaunay triangulations
Silveira, Rodrigo Ignacio; Kreveld, Marc van
Computational geometry: theory and applications
Date of publication: 2009
Journal article
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Planar bichromatic minimum spanning trees
Borgelt, Magdalene; Kreveld, Marc van; Löffler, Maarten; Luo, Jun; Merrick, Damian; Silveira, Rodrigo Ignacio; Vahedi, Mostafa
Journal of discrete algorithms
Date of publication: 2009
Journal article
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Optimal higher order Delaunay triangulations of polygons
Silveira, Rodrigo Ignacio; Kreveld, Marc van
Computational geometry: theory and applications
Date of publication: 2009
Journal article
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Optimization of polyhedral terrains
Silveira, Rodrigo Ignacio
Defense's date: 20090708
Utrecht University
Theses
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Matching terrains under a linear transformation
Agarwal, Pankaj; Aronov, Boris; Kreveld, Marc van; Löffler, Maarten; Silveira, Rodrigo Ignacio
European Workshop on Computational Geometry
Presentation's date: 2009
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Best Paper Award
Silveira, Rodrigo Ignacio; Buchin, Kevin; Cabello, Sergio; Gudmundsson, Joachim; Loffler, Maarten; Luo, Jun; Rote, Günter; Speckmann, Bettina; Wolle, Thomas
Award or recognition
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Connect the dot: computing feedlinks with minimum dilation
Aronov, Boris; Buchin, Kevin; Buchin, Maike; Kreveld, Marc van; Löffler, Maarten; Luo, Jun; Silveira, Rodrigo Ignacio; Speckmann, Bettina
Workshop on Algorithms and Data Structures
Presentation's date: 20090822
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Smoothing imprecise 1.5D terrains
Gray, Chris; Löffler, Maarten; Silveira, Rodrigo Ignacio
Workshop on Approximation and Online Algorithms
Presentation's date: 2009
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Embedding rivers in polyhedral terrains
Kreveld, Marc van; Silveira, Rodrigo Ignacio
ACM Annual Symposium on Computational Geometry
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Detecting hotspots in geographic networks
Buchin, Kevin; Cabello, Sergio; Gudmundsson, Joachim; Löffler, Maarten; Luo, Jun; Rote, Günter; Silveira, Rodrigo Ignacio; Speckmann, Bettina; Wolle, Thomas
AGILE International Conference on Geographic Information Science
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Minimizing slope change in imprecise 1.5D terrains
Gray, Chris; Löffler, Maarten; Silveira, Rodrigo Ignacio
Canadian Conference on Computational Geometry
Presentation's date: 2009
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Feedlinks for network extensions
Aronov, Boris; Buchin, Kevin; Buchin, Maike; Jansen, Bart; De Jong, Tom; Kreveld, Marc van; Löffler, Maarten; Luo, Jun; Silveira, Rodrigo Ignacio; Speckmann, Bettina
ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
Presentation's date: 2008
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Smoothing imprecise 1dimensional terrains
Gray, Chris; Löffler, Maarten; Silveira, Rodrigo Ignacio
European Workshop on Computational Geometry
Presentation's date: 2008
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Drawing (complete) binary tanglegrams: hardness, approximation, fixedparameter tractability
Buchin, Kevin; Buchin, M.; Byrka, Jaroslaw; Nöllenburg, Martin; Okamoto, Yoshio; Silveira, Rodrigo Ignacio; Wolff, Alexander
Symposium on Graph Drawing
Presentation's date: 2008
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Clusters in aggregated health data
Buchin, Kevin; Buchin, M.; Kreveld, Marc van; Löffler, Maarten; Luo, Jun; Silveira, Rodrigo Ignacio
International Symposium on Spatial Data Handling
Presentation's date: 2008
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Optimal higher order delaunay triangulations of polygons
Silveira, Rodrigo Ignacio; Kreveld, Marc van
Latin American Theoretical Informatics Symposium
Presentation's date: 2008
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Optimal higherorder delaunay triangulations of polygons
Silveira, Rodrigo Ignacio; Kreveld, Marc van
European Workshop on Computational Geometry
Presentation's date: 2007
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Towards a definition of higher order constrained delaunay triangulations
Silveira, Rodrigo Ignacio; Kreveld, Marc van
Canadian Conference on Computational Geometry
Presentation's date: 2007
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Optimization for first order delaunay triangulations
Kreveld, Marc van; Löffler, Maarten; Silveira, Rodrigo Ignacio
Workshop on Algorithms and Data Structures
Presentation's date: 2007
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Flooding countries and destroying dams
Silveira, Rodrigo Ignacio; Van Oostrum, René
Workshop on Algorithms and Data Structures
Presentation's date: 2007
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