Two problems in computational geometry

En aquesta tesi s'estudien dos problemes del camp de la geometria computacional. El primer problema és: donat un set S de n punts en el pla en posició general, com de prop són quatre punts de S de ser cocirculars. Definim tres mesures per estudiar aquesta qüestió, la mesura de Tales, la mesura de Voronoi, i la mesura del Determinant. Presentem cotes per la mesura de Tales, i algoritmes per computar aquestes mesures de cocircularitat. També reduïm el problema de computar la cocircularitat emprant la mesura del Determinant al problema de 4SUM. El segon problema és: donat dos sets R i B de punts rojos i blaus respectivament, com computar la discrepància bicromàtica amb caixes i cercles. La discrepància bicromàtica és definida com la diferència entre el nombre de punts vermells i blaus que són a l'interior de la figura examinada. Presentem una comparativa entre algoritmes ja existents per les dues figures. També comparem la discrepància bicromàtica de caixes orientades en els eixos vs. d'orientació general. A més a més, també presentem un nou algoritme per la discrepància en esferes/discs per a altes dimensions, basat en literatura ja existent. També relacionem altres problemes en el tema de separabilitat amb algoritmes sensitius a l'output per la discrepància amb caixes.En esta tesis se estudian dos problemas del campo de la geometría computacional. El primer problema es: dado un set S de n puntos en el plan en posición general, como de cerca son cuatro puntos de S de ser cocirculares. Definimos tres medidas para estudiar esta cuestión, la medida de Tales, la medida de Voronoi, y la medida del Determinante. Presentamos cotas por la medida de Tales, y algoritmos para computar estas medidas de cocircularidad. También reducimos el problema de computar la cocircularidad usando la medida del Determinante al problema de 4SUM. El segundo problema es: dado dos sets R y B de puntos rojos y azules respectivamente, como computar la discrepancia bicromática con cajas y círculos. La discrepancia bicromática es definida como la diferencia entre el número de puntos rojos y azules que están en el interior de la figura examinada. Presentamos una comparativa entre algoritmos ya existentes por las dos figuras. También comparamos la discrepancia bicromática de cajas orientadas en los ejes vs. de orientación general. Además, también presentamos un nuevo algoritmo por la discrepancia en esferas/discos para altas dimensiones, basado en literatura ya existente. También relacionamos otros problemas en el tema de separabilidad con algoritmos sensitivos al output por la discrepancia con cajas.Two different problems belonging to computational geometry are studied in this thesis. The first problem studies: given a set S of n points in the plane in general position, how close are four points of S to being cocircular. We define three measures to study this question, the Thales, Voronoi and Determinant measures. We present bounds on the Thales almost-cocircularity measure over a point set. Algorithms for computing these measures of cocircularity are presented as well. We give a reduction from computing cocircularity using the Determinant measure to the 4SUM problem. The second problem studies: given two sets R and B of red and blue points respectively, how to compute the bichromatic discrepancy using boxes and circles. The bichromatic discrepancy is defined as the difference between the number of red points and blue points inside the shape. We present a comparison of algorithms in the existing literature for the two shapes. Bichromatic discrepancy in axis-parallel boxes .vs non-axis-parallel boxes is also compared. Furthermore, we also present a new algorithm for disk discrepancy in higher dimensions, based on existing literature. We also relate existing problems in separability with existing output sensitive algorithms for bichromatic discrepancy using boxes

Reverse Nearest Neighbor Heat Maps: A Tool for Influence Exploration

We study the problem of constructing a reverse nearest neighbor (RNN) heat map by finding the RNN set of every point in a two-dimensional space. Based on the RNN set of a point, we obtain a quantitative influence (i.e., heat) for the point. The heat map provides a global view on the influence distribution in the space, and hence supports exploratory analyses in many applications such as marketing and resource management. To construct such a heat map, we first reduce it to a problem called Region Coloring (RC), which divides the space into disjoint regions within which all the points have the same RNN set. We then propose a novel algorithm named CREST that efficiently solves the RC problem by labeling each region with the heat value of its containing points. In CREST, we propose innovative techniques to avoid processing expensive RNN queries and greatly reduce the number of region labeling operations. We perform detailed analyses on the complexity of CREST and lower bounds of the RC problem, and prove that CREST is asymptotically optimal in the worst case. Extensive experiments with both real and synthetic data sets demonstrate that CREST outperforms alternative algorithms by several orders of magnitude.Comment: Accepted to appear in ICDE 201

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

Graphene under bichromatic driving: Commensurability and spatio-temporal symmetries

We study the non-linear current response of a Dirac model that is coupled to two time-periodic electro-magnetic fields with different frequencies. We distinguish between incommensurable and commensurable frequencies, the latter characterized by their ratio p/q with co-prime integers p and q. Coupling the (effective) two-level system to a dissipative bath ensures a well-defined long-time solution for the reduced density operator and, thus, the current. We then analyze the spatio-temporal symmetries that force certain current components to vanish and close with conclusions for directed average currents.Comment: 8 pages, 5 figure

Diseño de un algoritmo de minería metaheurístico para separar puntos de dos colores en un entorno bidimensional

The separation of color points is one of the important issues in computational geometry, which is used in various parts of science; it can be used in facility locating, image processing and clustering. Among these, one of the most widely used computational geometry in the real-world is the problem of covering and separating points with rectangles. In this paper, we intend to consider the problemof separating the two-color points sets, using three rectangles. In fact, our goal is to separate desired blue points from undesired red points by three rectangles, in such a way that these three rectangles contain the most desire points. For this purpose, we provide a metaheuristic algorithm based on the simulated annealing method that could separates blue points from input points, , in time order O (n) with the help of three rectangles. The algorithm is executed with C# and also it has been compared and evaluated with the optimum algorithm results. The results show that our recommended algorithm responses is so close to optimal responses, and also in some cases we obtains the exact optimal response