External-Memory Computational Geometry

(c) 1993 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.In this paper we give new techniques for designing e cient algorithms for computational geometry prob- lems that are too large to be solved in internal mem- ory. We use these techniques to develop optimal and practical algorithms for a number of important large- scale problems. We discuss our algorithms primarily in the context of single processor/single disk machines, a domain in which they are not only the rst known optimal results but also of tremendous practical value. Our methods also produce the rst known optimal al- gorithms for a wide range of two-level and hierarchical multilevel memory models, including parallel models. The algorithms are optimal both in terms of I/O cost and internal computation

Discretization of Planar Geometric Cover Problems

We consider discretization of the 'geometric cover problem' in the plane: Given a set $P$ of $n$ points in the plane and a compact planar object $T_0$, find a minimum cardinality collection of planar translates of $T_0$ such that the union of the translates in the collection contains all the points in $P$. We show that the geometric cover problem can be converted to a form of the geometric set cover, which has a given finite-size collection of translates rather than the infinite continuous solution space of the former. We propose a reduced finite solution space that consists of distinct canonical translates and present polynomial algorithms to find the reduce solution space for disks, convex/non-convex polygons (including holes), and planar objects consisting of finite Jordan curves.Comment: 16 pages, 5 figure

Maximum Matchings in Geometric Intersection Graphs

Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in O(ρ3ω/2nω/2) time with high probability, where ρ is the density of the geometric objects and ω>2 is a constant such that n×n matrices can be multiplied in O(nω) time. The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O(nω/2) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [1,Ψ] can be found in O(Ψ6log11n+Ψ12ωnω/2) time with high probability

Algorithms for Triangles, Cones & Peaks

Three different geometric objects are at the center of this dissertation: triangles, cones and peaks. In computational geometry, triangles are the most basic shape for planar subdivisions. Particularly, Delaunay triangulations are a widely used for manifold applications in engineering, geographic information systems, telecommunication networks, etc. We present two novel parallel algorithms to construct the Delaunay triangulation of a given point set. Yao graphs are geometric spanners that connect each point of a given set to its nearest neighbor in each of $k$ cones drawn around it. They are used to aid the construction of Euclidean minimum spanning trees or in wireless networks for topology control and routing. We present the first implementation of an optimal $\mathcal{O}(n \log n)$-time sweepline algorithm to construct Yao graphs. One metric to quantify the importance of a mountain peak is its isolation. Isolation measures the distance between a peak and the closest point of higher elevation. Computing this metric from high-resolution digital elevation models (DEMs) requires efficient algorithms. We present a novel sweep-plane algorithm that can calculate the isolation of all peaks on Earth in mere minutes

A Randomized Incremental Algorithm for the Hausdorff Voronoi Diagram of Non-crossing Clusters

In the Hausdorff Voronoi diagram of a family of \emph{clusters of points} in the plane, the distance between a point $t$ and a cluster $P$ is measured as the maximum distance between $t$ and any point in $P$, and the diagram is defined in a nearest-neighbor sense for the input clusters. In this paper we consider %El."non-crossing" \emph{non-crossing} clusters in the plane, for which the combinatorial complexity of the Hausdorff Voronoi diagram is linear in the total number of points, $n$, on the convex hulls of all clusters. We present a randomized incremental construction, based on point location, that computes this diagram in expected $O(n\log^2{n})$ time and expected $O(n)$ space. Our techniques efficiently handle non-standard characteristics of generalized Voronoi diagrams, such as sites of non-constant complexity, sites that are not enclosed in their Voronoi regions, and empty Voronoi regions. The diagram finds direct applications in VLSI computer-aided design.Comment: arXiv admin note: substantial text overlap with arXiv:1306.583

Abstracts for the twentyfirst European workshop on Computational geometry, Technische Universiteit Eindhoven, The Netherlands, March 9-11, 2005

This volume contains abstracts of the papers presented at the 21st European Workshop on Computational Geometry, held at TU Eindhoven (the Netherlands) on March 9–11, 2005. There were 53 papers presented at the Workshop, covering a wide range of topics. This record number shows that the field of computational geometry is very much alive in Europe. We wish to thank all the authors who submitted papers and presented their work at the workshop. We believe that this has lead to a collection of very interesting abstracts that are both enjoyable and informative for the reader. Finally, we are grateful to TU Eindhoven for their support in organizing the workshop and to the Netherlands Organisation for Scientific Research (NWO) for sponsoring the workshop

Geometric algorithms for geographic information systems

A geographic information system (GIS) is a software package for storing geographic data and performing complex operations on the data. Examples are the reporting of all land parcels that will be flooded when a certain river rises above some level, or analyzing the costs, benefits, and risks involved with the development of industrial activities at some place. A substantial part of all activities performed by a GIS involves computing with the geometry of the data, such as location, shape, proximity, and spatial distribution. The amount of data stored in a GIS is usually very large, and it calls for efficient methods to store, manipulate, analyze, and display such amounts of data. This makes the field of GIS an interesting source of problems to work on for computational geometers. In chapters 2-5 of this thesis we give new geometric algorithms to solve four selected GIS problems.These chapters are preceded by an introduction that provides the necessary background, overview, and definitions to appreciate the following chapters. The four problems that we study in chapters 2-5 are the following: Subdivision traversal: we give a new method to traverse planar subdivisions without using mark bits or a stack. Contour trees and seed sets: we give a new algorithm for generating a contour tree for d-dimensional meshes, and use it to determine a seed set of minimum size that can be used for isosurface generation. This is the first algorithm that guarantees a seed set of minimum size. Its running time is quadratic in the input size, which is not fast enough for many practical situations. Therefore, we also give a faster algorithm that gives small (although not minimal) seed sets. Settlement selection: we give a number of new models for the settlement selection problem. When settlements, such as cities, have to be displayed on a map, displaying all of them may clutter the map, depending on the map scale. Choices have to be made which settlements are selected, and which ones are omitted. Compared to existing selection methods, our methods have a number of favorable properties. Facility location: we give the first algorithm for computing the furthest-site Voronoi diagram on a polyhedral terrain, and show that its running time is near-optimal. We use the furthest-site Voronoi diagram to solve the facility location problem: the determination of the point on the terrain that minimizes the maximal distance to a given set of sites on the terrain