Voronoi diagrams in the max-norm: algorithms, implementation, and applications

Voronoi diagrams and their numerous variants are well-established objects in computational geometry. They have proven to be extremely useful to tackle geometric problems in various domains such as VLSI CAD, Computer Graphics, Pattern Recognition, Information Retrieval, etc. In this dissertation, we study generalized Voronoi diagram of line segments as motivated by applications in VLSI Computer Aided Design. Our work has three directions: algorithms, implementation, and applications of the line-segment Voronoi diagrams. Our results are as follows: (1) Algorithms for the farthest Voronoi diagram of line segments in the Lp metric, 1 ≤ p ≤ ∞. Our main interest is the L2 (Euclidean) and the L∞ metric. We first introduce the farthest line-segment hull and its Gaussian map to characterize the regions of the farthest line-segment Voronoi diagram at infinity. We then adapt well-known techniques for the construction of a convex hull to compute the farthest line-segment hull, and therefore, the farthest segment Voronoi diagram. Our approach unifies techniques to compute farthest Voronoi diagrams for points and line segments. (2) The implementation of the L∞ Voronoi diagram of line segments in the Computational Geometry Algorithms Library (CGAL). Our software (approximately 17K lines of C++ code) is built on top of the existing CGAL package on the L2 (Euclidean) Voronoi diagram of line segments. It is accepted and integrated in the upcoming version of the library CGAL-4.7 and will be released in september 2015. We performed the implementation in the L∞ metric because we target applications in VLSI design, where shapes are predominantly rectilinear, and the L∞ segment Voronoi diagram is computationally simpler. (3) The application of our Voronoi software to tackle proximity-related problems in VLSI pattern analysis. In particular, we use the Voronoi diagram to identify critical locations in patterns of VLSI layout, which can be faulty during the printing process of a VLSI chip. We present experiments involving layout pieces that were provided by IBM Research, Zurich. Our Voronoi-based method was able to find all problematic locations in the provided layout pieces, very fast, and without any manual intervention

L_1 Geodesic Farthest Neighbors in a Simple Polygon and Related Problems

In this paper, we investigate the L_1 geodesic farthest neighbors in a simple polygon P, and address several fundamental problems related to farthest neighbors. Given a subset S subseteq P, an L_1 geodesic farthest neighbor of p in P from S is one that maximizes the length of L_1 shortest path from p in P. Our list of problems include: computing the diameter, radius, center, farthest-neighbor Voronoi diagram, and two-center of S under the L_1 geodesic distance. We show that all these problems can be solved in linear or near-linear time based on our new observations on farthest neighbors and extreme points. Among them, the key observation shows that there are at most four extreme points of any compact subset S subseteq P with respect to the L_1 geodesic distance after removing redundancy

Higher-order Voronoi diagrams of polygonal objects

Higher-order Voronoi diagrams are fundamental geometric structures which encode the k-nearest neighbor information. Thus, they aid in computations that require proximity information beyond the nearest neighbor. They are related to various favorite structures in computational geometry and are a fascinating combinatorial problem to study. While higher-order Voronoi diagrams of points have been studied a lot, they have not been considered for other types of sites. Points lack dimensionality which makes them unable to represent various real-life instances. Points are the simplest kind of geometric object and therefore higher- order Voronoi diagrams of points can be considered as the corner case of all higher-order Voronoi diagrams. The goal of this dissertation is to move away from the corner and bring the higher-order Voronoi diagram to more general geometric instances. We focus on certain polygonal objects as they provide flexibility and are able to represent real-life instances. Before this dissertation, higher-order Voronoi diagrams of polygonal objects had been studied only for the nearest neighbor and farthest Voronoi diagrams. In this dissertation we investigate structural and combinatorial properties and discover that the dimensionality of geometric objects manifests itself in numerous ways which do not exist in the case of points. We prove that the structural complexity of the order-k Voronoi diagram of non-crossing line segments is O(k(n-k)), as in the case of points. We study disjoint line segments, intersecting line segments, line segments forming a planar straight-line graph and extend the results to the Lp metric, 1<=p<=infty. We also establish the connection between two mathematical abstractions: abstract Voronoi diagrams and the Clarkson-Shor framework. We design several construction algorithms that cover the case of non-point sites. While computational geometry provides several approaches to study the structural complexity that give tight realizable bounds, developing an effective construction algorithm is still a challenging problem even for points. Most of the construction algorithms are designed to work with points as they utilize their simplicity and relations with data-structures that work specifically for points. We extend the iterative and the sweepline approaches that are quite efficient in constructing all order-i Voronoi diagrams, for i<=k and we also give three randomized construction algorithms for abstract higher-order Voronoi diagrams that deal specifically with the construction of the order-k Voronoi diagrams

Closest and Farthest-Line Voronoi Diagrams in the Plane

Voronoi diagrams are a geometric structure containing proximity information useful in efficiently answering a number of common geometric problems associated with a set of points in the plane.. They have applications in fields ranging from crystallography to biology. Diagrams of sites other than points and with different distance metrics have been studied. This paper examines the Voronoi diagram of a set of lines, which has escaped study in the computational geometry literature. The combinatorial and topological properties of the closest and farthest Voronoi diagrams are analyzed and O(n^2) and O(n log n) algorithms are presented for their computation respectively

An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams

Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O(k(n-k) log^2 n +n log^3 n) steps, where O(k(n-k)) is the number of faces in the worst case. Due to those axioms, this result applies to disjoint line segments in the L_p norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, this kind of run time with a polylog factor to the number of faces was only achieved for point sites in the L_1 or Euclidean metric before

New Results on Abstract Voronoi Diagrams

Voronoi diagrams are a fundamental structure used in many areas of science. For a given set of objects, called sites, the Voronoi diagram separates the plane into regions, such that points belonging to the same region have got the same nearest site. This definition clearly depends on the type of given objects, they may be points, line segments, polygons, etc. and the distance measure used. To free oneself from these geometric notions, Klein introduced abstract Voronoi diagrams as a general construct covering many concrete Voronoi diagrams. Abstract Voronoi diagrams are based on a system of bisecting curves, one for each pair of abstract sites, separating the plane into two dominance regions, belonging to one site each. The intersection of all dominance regions belonging to one site p defines its Voronoi region. The system of bisecting curves is required to fulfill only some simple combinatorial properties, like Voronoi regions to be connected, the union of their closures cover the whole plane, and the bisecting curves are unbounded. These assumptions are enough to show that an abstract Voronoi diagram of n sites is a planar graph of complexity O(n) and can be computed in expected time O(n log n) by a randomized incremental construction. In this thesis we widen the notion of abstract Voronoi diagrams in several senses. One step is to allow disconnected Voronoi regions. We assume that in a diagram of a subset of three sites each Voronoi region may consist of at most s connected components, for a constant s, and show that the diagram can be constructed in expected time O(s2 n ∑3 ≤ j ≤ n mj / j), where mj is the expected number of connected components of a Voronoi region over all diagrams of a subset of j sites. The case that all Voronoi regions are connected is a subcase, where this algorithm performs in optimal O(n log n) time, because here s = mj =1. The next step is to additionally allow bisecting curves to be closed. We present an algorithm constructing such diagrams which runs in expected time O(s2 n log(max{s,n}) ∑2 ≤ j≤ n mj / j). This algorithm is slower by a log n-factor compared to the one for disconnected regions and unbounded bisectors. The extra time is necessary to be able to handle special phenomenons like islands, where a Voronoi region is completely surrounded by another region, something that can occur only when bisectors are closed. However, this algorithm solves many open problems and improves the running time of some existing algorithms, for example for the farthest Voronoi diagram of n simple polygons of constant complexity. Another challenge was to study higher order abstract Voronoi diagrams. In the concrete sense of an order-k Voronoi diagram points are collected in the same Voronoi region, if they have the same k nearest sites. By suitably intersecting the dominance regions this can be defined also for abstract Voronoi diagrams. The question arising is about the complexity of an order-k Voronoi diagram. There are many subsets of size k but fortunately many of them have an empty order-k region. For point sites it has already been shown that there can be at most O(k (n-k)) many regions and even though order-k regions may be disconnected when considering line segments, still the complexity of the order-k diagram remains O(k(n-k)). The proofs used to show this strongly depended on the geometry of the sites and the distance measure, and were thus not applicable for our abstract higher order Voronoi diagrams. The proofs used to show this strongly depended on the geometry of the sites and the distance measure, and were thus not applicable for our abstract higher order Voronoi diagrams. Nevertheless, we were able to come up with proofs of purely topological and combinatorial nature of Jordan curves and certain permutation sequences, and hence we could show that also the order-k abstract Voronoi diagram has complexity O(k (n-k)), assuming that bisectors are unbounded, and the order-1 regions are connected. Finally, we discuss Voronoi diagrams having the shape of a tree or forest. Aggarwal et. al. showed that if points are in convex position, then given their ordering along the convex hull, their Voronoi diagram, which is a tree, can be computed in linear time. Klein and Lingas have generalized this idea to Hamiltonian abstract Voronoi diagrams, where a curve is given, intersecting each Voronoi region with respect to any subset of sites exactly once. If the ordering of the regions along the curve is known in advance, all Voronoi regions are connected, and all bisectors are unbounded, then the abstract Voronoi diagram can be computed in linear time. This algorithm also applies to diagrams which are trees for all subsets of sites and the ordering of the unbounded regions around the diagram is known. In this thesis we go one step further and allow the diagram to be a forest for subsets of sites as long as the complete diagram is a tree. We show that also these diagrams can be computed in linear time

Convex Hulls: Complexity and Applications (a Survey)

Computational geometry is, in brief, the study of algorithms for geometric problems. Classical study of geometry and geometric objects, however, is not well-suited to efficient algorithms techniques. Thus, for the given geometric problems, it becomes necessary to identify properties and concepts that lend themselves to efficient computation. The primary focus of this paper will be on one such geometric problems, the Convex Hull problem