Weighted skeletons and fixed-share decomposition

AbstractWe introduce the concept of weighted skeleton of a polygon and present various decomposition and optimality results for this skeletal structure when the underlying polygon is convex

Voronoi Diagrams for Parallel Halflines and Line Segments in Space

We consider the Euclidean Voronoi diagram for a set of $n$ parallel halflines in 3-space. A relation of this diagram to planar power diagrams is shown, and is used to analyze its geometric and topological properties. Moreover, an easy-to-implement space sweep algorithm is proposed that computes the Voronoi diagram for parallel halflines at logarithmic cost per face. Previously only an approximation algorithm for this problem was known. Our method of construction generalizes to Voronoi diagrams for parallel line segments, and to higher dimensions

Piecewise-Linear Farthest-Site Voronoi Diagrams

Voronoi diagrams induced by distance functions whose unit balls are convex polyhedra are piecewise-linear structures. Nevertheless, analyzing their combinatorial and algorithmic properties in dimensions three and higher is an intriguing problem. The situation turns easier when the farthest-site variants of such Voronoi diagrams are considered, where each site gets assigned the region of all points in space farthest from (rather than closest to) it. We give asymptotically tight upper and lower worst-case bounds on the combinatorial size of farthest-site Voronoi diagrams for convex polyhedral distance functions in general dimensions, and propose an optimal construction algorithm. Our approach is uniform in the sense that (1) it can be extended from point sites to sites that are convex polyhedra, (2) it covers the case where the distance function is additively and/or multiplicatively weighted, and (3) it allows an anisotropic scenario where each site gets allotted its particular convex distance polytope

Partially Walking a Polygon

Deciding two-guard walkability of an n-sided polygon is a well-understood problem. We study the following more general question: How far can two guards reach from a given source vertex while staying mutually visible, in the (more realistic) case that the polygon is not entirely walkable? There can be Theta(n) such maximal walks, and we show how to find all of them in O(n log n) time

Towards compatible triangulations

AbstractWe state the following conjecture: any two planar n-point sets that agree on the number of convex hull points can be triangulated in a compatible manner, i.e., such that the resulting two triangulations are topologically equivalent. We first describe a class of point sets which can be triangulated compatibly with any other set (that satisfies the obvious size and shape restrictions). The conjecture is then proved true for point sets with at most three interior points. Finally, we demonstrate that adding a small number of extraneous points (the number of interior points minus three) always allows for compatible triangulations. The linear bound extends to point sets of arbitrary size and shape

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

Optimizing the Access to Healthcare Services in Dense Refugee Hosting Urban Areas: A Case for Istanbul

With over 3.5 million refugees, Turkey continues to host the world's largest refugee population. This introduced several challenges in many areas including access to healthcare system. Refugees have legal rights to free healthcare services in Turkey's public hospitals. With the aim of increasing healthcare access for refugees, we looked at where the lack of infrastructure is felt the most. Our study attempts to address these problems by assessing whether Migrant Health Centers' locations are optimal. The aim of this study is to improve refugees' access to healthcare services in Istanbul by improving the locations of health facilities available to them. We used call data records provided by Turk Telekom.Comment: version to submit for D4R competitio

3-colorability of pseudo-triangulations

Electronic version of an article published as International Journal of Computational Geometry & Applications, Vol. 25, No. 4 (2015) 283–298 DOI: 10.1142/S0218195915500168 © 2015 World Scientific Publishing Company. http://www.worldscientific.com/worldscinet/ijcgaDeciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.Postprint (author's final draft