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

Intersection of paraboloids and application to Minkowski-type problems

In this article, we study the intersection (or union) of the convex hull of N confocal paraboloids (or ellipsoids) of revolution. This study is motivated by a Minkowski-type problem arising in geometric optics. We show that in each of the four cases, the combinatorics is given by the intersection of a power diagram with the unit sphere. We prove the complexity is O(N) for the intersection of paraboloids and Omega(N^2) for the intersection and the union of ellipsoids. We provide an algorithm to compute these intersections using the exact geometric computation paradigm. This algorithm is optimal in the case of the intersection of ellipsoids and is used to solve numerically the far-field reflector problem

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Novel approaches for constructing persistent Delaunay triangulations by applying different equations and different methods

“Delaunay triangulation and data structures are an essential field of study and research in computer science, for this reason, the correct choices, and an adequate design are essential for the development of algorithms for the efficient storage and/or retrieval of information. However, most structures are usually ephemeral, which means keeping all versions, in different copies, of the same data structure is expensive. The problem arises of developing data structures that are capable of maintaining different versions of themselves, minimizing the cost of memory, and keeping the performance of operations as close as possible to the original structure. Therefore, this research aims to aims to examine the feasibility concepts of Spatio-temporal structures such as persistence, to design a Delaunay triangulation algorithm so that it is possible to make queries and modifications at a certain time t, minimizing spatial and temporal complexity. Four new persistent data structures for Delaunay triangulation (Bowyer-Watson, Walk, Hybrid, and Graph) were proposed and developed. The results of using random images and vertex databases with different data (DAG and CGAL), proved that the data structure in its partial version is better than the other data structures that do not have persistence. Also, the full version data structures show an advance in the state of the technique. All the results will allow the algorithms to minimize the cost of memory”--Abstract, page iii

Dispelling the N^3 myth for the Kt jet-finder

At high-energy colliders, jets of hadrons are the observable counterparts of the perturbative concepts of quarks and gluons. Good procedures for identifying jets are central to experimental analyses and comparisons with theory. The Kt family of successive recombination jet finders has been widely advocated because of its conceptual simplicity and flexibility and its unique ability to approximately reconstruct the partonic branching sequence in an event. Until now however, it had been believed that for an ensemble of N particles the algorithmic complexity of the Kt jet finder scaled as N^3, a severe issue in the high multiplicity environments of LHC and heavy-ion colliders. We here show that the computationally complex part of Kt jet-clustering can be reduced to two-dimensional nearest neighbour location for a dynamic set of points. Borrowing techniques developed for this extensively studied problem in computational geometry, Kt jet-finding can then be performed in N ln N time. Code based on these ideas is found to run faster than all other jet finders in current use.Comment: 11 pages, 3 figures; v2, to appear in Phys.Lett.B, includes an extra section briefly discussing the issues of jet areas and pileup subtraction, and also the Cambridge/Aachen jet finde