Searching edges in the overlap of two plane graphs

Consider a pair of plane straight-line graphs, whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. This includes: (1) a solution to the batched red-blue search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an algorithm to compute the maximum vertical distance between a pair of 3D polyhedral terrains one of which is convex in O(n log n) time, where n is the total complexity of both terrains; (3) an algorithm to construct the Hausdorff Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n) time and O(n+m) space, where n is the total number of points in all clusters and m is the number of crossings between all clusters; (4) an algorithm to construct the farthest-color Voronoi diagram of the corners of n axis-aligned rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle problem for n parallel line segments in the plane in optimal O(n log n) time. All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure

On the Complexity of Randomly Weighted Voronoi Diagrams

In this paper, we provide an $O(n \mathrm{polylog} n)$ bound on the expected complexity of the randomly weighted Voronoi diagram of a set of $n$ sites in the plane, where the sites can be either points, interior-disjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, not their location. This compares favorably with the worst case complexity of these diagrams, which is quadratic. As a consequence we get an alternative proof to that of Agarwal etal [AHKS13] of the near linear complexity of the union of randomly expanded disjoint segments or convex sets (with an improved bound on the latter). The technique we develop is elegant and should be applicable to other problems

Learned navigation in unknown terrains: A retraction method

The problem of learned navigation of a circular robot R, of radius delta (is greater than or equal to 0), through a terrain whose model is not a-priori known is considered. Two-dimensional finite-sized terrains populated by an unknown (but, finite) number of simple polygonal obstacles are also considered. The number and locations of the vertices of each obstacle are unknown to R. R is equipped with a sensor system that detects all vertices and edges that are visible from its present location. In this context two problems are covered. In the visit problem, the robot is required to visit a sequence of destination points, and in the terrain model acquisition problem, the robot is required to acquire the complete model of the terrain. An algorithmic framework is presented for solving these two problems using a retraction of the freespace onto the Voronoi diagram of the terrain. Algorithms are then presented to solve the visit problem and the terrain model acquisition problem

Facility location on terrains

Given a terrain defined as a piecewise-linear function with n triangles, and m point sites on it, we would like to identify the location on the terrain that minimizes the maximum distance to the sites. The distance is measured as the length of the Euclidean shortest path along the terrain. To simplify the problem somewhat, we extend the terrain to (the surface of) a polyhedron. To compute the optimum placement, we compute the furthest-site Voronoi diagram of the sites on the polyhedron. The diagram has maximum combinatorial complexity Q(mn2), and the algorithm runs in O(mn² log²m log n) time

09111 Abstracts Collection -- Computational Geometry

From March 8 to March 13, 2009, the Dagstuhl Seminar 09111 ``Computational Geometry \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Linear Expected Complexity for Directional and Multiplicative Voronoi Diagrams

While the standard unweighted Voronoi diagram in the plane has linear worst-case complexity, many of its natural generalizations do not. This paper considers two such previously studied generalizations, namely multiplicative and semi Voronoi diagrams. These diagrams both have quadratic worst-case complexity, though here we show that their expected complexity is linear for certain natural randomized inputs. Specifically, we argue that the expected complexity is linear for: (1) semi Voronoi diagrams when the visible direction is randomly sampled, and (2) for multiplicative diagrams when either weights are sampled from a constant-sized set, or the more challenging case when weights are arbitrary but locations are sampled from a square

In pursuit of linear complexity in discrete and computational geometry

Many computational problems arise naturally from geometric data. In this thesis, we consider three such problems: (i) distance optimization problems over point sets, (ii) computing contour trees over simplicial meshes, and (iii) bounding the expected complexity of weighted Voronoi diagrams. While these topics are broad, here the focus is on identifying structure which implies linear (or near linear) algorithmic and descriptive complexity. The first topic we consider is in geometric optimization. More specifically, we define a large class of distance problems, for which we provide linear time exact or approximate solutions. Roughly speaking, the class of problems facilitate either clustering together close points (i.e. netting) or throwing out outliers (i.e pruning), allowing for successively smaller summaries of the relevant information in the input. A surprising number of classical geometric optimization problems are unified under this framework, including finding the optimal k-center clustering, the kth ranked distance, the kth heaviest edge of the MST, the minimum radius ball enclosing k points, and many others. In several cases we get the first known linear time approximation algorithm for a given problem, where our approximation ratio matches that of previous work. The second topic we investigate is contour trees, a fundamental structure in computational topology. Contour trees give a compact summary of the evolution of level sets on a mesh, and are typically used on massive data sets. Previous algorithms for computing contour trees took Θ(n log n) time and were worst-case optimal. Here we provide an algorithm whose running time lies between Θ(nα(n)) and Θ(n log n), and varies depending on the shape of the tree, where α(n) is the inverse Ackermann function. In particular, this is the first algorithm with O(nα(n)) running time on instances with balanced contour trees. Our algorithmic results are complemented by lower bounds indicating that, up to a factor of α(n), on all instance types our algorithm performs optimally. For the final topic, we consider the descriptive complexity of weighted Voronoi diagrams. Such diagrams have quadratic (or higher) worst-case complexity, however, as was the case for contour trees, here we push beyond worst-case analysis. A new diagram, called the candidate diagram, is introduced, which allows us to bound the complexity of weighted Voronoi diagrams arising from a particular probabilistic input model. Specifically, we assume weights are randomly permuted among fixed Voronoi sites, an assumption which is weaker than the more typical sampled locations assumption. Under this assumption, the expected complexity is shown to be near linear