On k-Convex Polygons

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{$2$-convex} objects considered.Comment: 23 pages, 19 figure

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

Stabbers of line segments in the plane

The problem of computing a representation of the stabbing lines of a set S of segments in the plane was solved by Edelsbrunner et al. We provide efficient algorithms for the following problems: computing the stabbing wedges for S, finding a stabbing wedge for a set of parallel segments with equal length, and computing other stabbers for S such as a double-wedge and a zigzag. The time and space complexities of the algorithms depend on the number of combinatorially different extreme lines, critical lines, and the number of different slopes that appear in S.Preprin

Dynamic Planar Orthogonal Point Location in Sublogarithmic Time

We study a longstanding problem in computational geometry: dynamic 2-d orthogonal point location, i.e., vertical ray shooting among n horizontal line segments. We present a data structure achieving O(log n / log log n) optimal expected query time and O(log^{1/2+epsilon} n) update time (amortized) in the word-RAM model for any constant epsilon>0, under the assumption that the x-coordinates are integers bounded polynomially in n. This substantially improves previous results of Giyora and Kaplan [SODA 2007] and Blelloch [SODA 2008] with O(log n) query and update time, and of Nekrich (2010) with O(log n / log log n) query time and O(log^{1+epsilon} n) update time. Our result matches the best known upper bound for simpler problems such as dynamic 2-d dominance range searching. We also obtain similar bounds for orthogonal line segment intersection reporting queries, vertical ray stabbing, and vertical stabbing-max, improving previous bounds, respectively, of Blelloch [SODA 2008] and Mortensen [SODA 2003], of Tao (2014), and of Agarwal, Arge, and Yi [SODA 2005] and Nekrich [ISAAC 2011]

Curve Stabbing Depth: Data Depth for Plane Curves

Measures of data depth have been studied extensively for point data. Motivated by recent work on analysis, clustering, and identifying representative elements in sets of trajectories, we introduce {\em curve stabbing depth} to quantify how deeply a given curve $Q$ is located relative to a given set $\cal C$ of curves in $\mathbb{R}^2$. Curve stabbing depth evaluates the average number of elements of $\cal C$ stabbed by rays rooted along the length of $Q$. We describe an $O(n^3 + n^2 m\log^2m+nm^2\log^2 m)$-time algorithm for computing curve stabbing depth when $Q$ is an $m$-vertex polyline and $\cal C$ is a set of $n$ polylines, each with $O(m)$ vertices.Comment: Preprin

On the hausdorff and other cluster Voronoi diagrams

The Voronoi diagram is a fundamental geometric structure that encodes proximity information. Given a set of geometric objects, called sites, their Voronoi diagram is a subdivision of the underlying space into maximal regions, such that all points within one region have the same nearest site. Problems in diverse application domains (such as VLSI CAD, robotics, facility location, etc.) demand various generalizations of this simple concept. While many generalized Voronoi diagrams have been well studied, many others still have unsettled questions. An example of the latter are cluster Voronoi diagrams, whose sites are sets (clusters) of objects rather than individual objects. In this dissertation we study certain cluster Voronoi diagrams from the perspective of their construction algorithms and algorithmic applications. Our main focus is the Hausdorff Voronoi diagram; we also study the farthest-segment Voronoi diagram, as well as certain special cases of the farthest-color Voronoi diagram. We establish a connection between cluster Voronoi diagrams and the stabbing circle problem for segments in the plane. Our results are as follows. (1) We investigate the randomized incremental construction of the Hausdorff Voronoi diagram. We consider separately the case of non-crossing clusters, when the combinatorial complexity of the diagram is O(n) where n is the total number of points in all clusters. For this case, we present two construction algorithms that require O(n log2 n) expected time. For the general case of arbitrary clusters, we present an algorithm that requires O((m + n log n) log n) expected time and O(m + n log n) expected space, where m is a parameter reflecting the number of crossings between clusters' convex hulls. (2) We present an O(n) time algorithm to construct the farthest-segment Voronoi diagram of n segments, after the sequence of its faces at infinity is known. This augments the well-known linear-time framework for Voronoi diagram of points in convex position, with the ability to handle disconnected Voronoi regions. (3) We establish a connection between the cluster Voronoi diagrams (the Hausdorff and the farthest-color Voronoi diagram) and the stabbing circle problem. This implies a new method to solve the latter problem. Our method results in a near-optimal O(n log2 n) time algorithm for a set of n parallel segments, and in an optimal O(n log n) time algorithm for a set of n segments satisfying some other special conditions. (4) We study the farthest-color Voronoi diagram in special cases considered by the stabbing circle problem. We prove O(n) bound for its combinatorial complexity and present an O(nlogn) time algorithm to construct it