Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings

We provide linear-time algorithms for geometric graphs with sublinearly many crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an iterated logarithmic factor. Specific problems we study include Voronoi diagrams and single-source shortest paths. Our algorithms all run in linear time in the standard comparison-based computational model; hence, we make no assumptions about the distribution or bit complexities of edge weights, nor do we utilize unusual bit-level operations on memory words. Instead, our algorithms are based on a planarization method that "zeroes in" on edge crossings, together with methods for extending planar separator decompositions to geometric graphs with sublinearly many crossings. Incidentally, our planarization algorithm also solves an open computational geometry problem of Chazelle for triangulating a self-intersecting polygonal chain having n segments and k crossings in linear time, for the case when k is sublinear in n by an iterated logarithmic factor.Comment: Expanded version of a paper appearing at the 20th ACM-SIAM Symposium on Discrete Algorithms (SODA09

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

External polygon containment problems

AbstractGiven a convex polygonal object P with k vertices and an environment consisting of polygonal obstacles with a total of n corners, we seek a placement for the largest copy of P that does not intersect any of the obstacles, allowing translation, rotation and scaling. We employ the parametric search technique of Megiddo (1983), and the fixed size polygon placement algorithms developed by Leven and Sharir (1987), to obtain an algorithm that runs in time O(k2nλ6(kn)log3(kn)loglog(kn)). We also present several other efficient algorithms for restricted variants of the extremal polygon containment problem, using the same ideas. These variants include: placement of the largest homothetic copies of one or two convex polygons in another convex polygon and placement of the largest similar copy of a triangle in a convex polygon

Efficient Parallel and Distributed Algorithms for GIS Polygon Overlay Processing

Polygon clipping is one of the complex operations in computational geometry. It is used in Geographic Information Systems (GIS), Computer Graphics, and VLSI CAD. For two polygons with n and m vertices, the number of intersections can be O(nm). In this dissertation, we present the first output-sensitive CREW PRAM algorithm, which can perform polygon clipping in O(log n) time using O(n + k + k\u27) processors, where n is the number of vertices, k is the number of intersections, and k\u27 is the additional temporary vertices introduced due to the partitioning of polygons. The current best algorithm by Karinthi, Srinivas, and Almasi does not handle self-intersecting polygons, is not output-sensitive and must employ O(n^2) processors to achieve O(log n) time. The second parallel algorithm is an output-sensitive PRAM algorithm based on Greiner-Hormann algorithm with O(log n) time complexity using O(n + k) processors. This is cost-optimal when compared to the time complexity of the best-known sequential plane-sweep based algorithm for polygon clipping. For self-intersecting polygons, the time complexity is O(((n + k) log n log log n)/p) using p In addition to these parallel algorithms, the other main contributions in this dissertation are 1) multi-core and many-core implementation for clipping a pair of polygons and 2) MPI-GIS and Hadoop Topology Suite for distributed polygon overlay using a cluster of nodes. Nvidia GPU and CUDA are used for the many-core implementation. The MPI based system achieves 44X speedup while processing about 600K polygons in two real-world GIS shapefiles 1) USA Detailed Water Bodies and 2) USA Block Group Boundaries) within 20 seconds on a 32-node (8 cores each) IBM iDataPlex cluster interconnected by InfiniBand technology

On Geometric Range Searching, Approximate Counting and Depth Problems

In this thesis we deal with problems connected to range searching, which is one of the central areas of computational geometry. The dominant problems in this area are halfspace range searching, simplex range searching and orthogonal range searching and research into these problems has spanned decades. For many range searching problems, the best possible data structures cannot offer fast (i.e., polylogarithmic) query times if we limit ourselves to near linear storage. Even worse, it is conjectured (and proved in some cases) that only very small improvements to these might be possible. This inefficiency has encouraged many researchers to seek alternatives through approximations. In this thesis we continue this line of research and focus on relative approximation of range counting problems. One important problem where it is possible to achieve significant speedup through approximation is halfspace range counting in 3D. Here we continue the previous research done and obtain the first optimal data structure for approximate halfspace range counting in 3D. Our data structure has the slight advantage of being Las Vegas (the result is always correct) in contrast to the previous methods that were Monte Carlo (the correctness holds with high probability). Another series of problems where approximation can provide us with substantial speedup comes from robust statistics. We recognize three problems here: approximate Tukey depth, regression depth and simplicial depth queries. In 2D, we obtain an optimal data structure capable of approximating the regression depth of a query hyperplane. We also offer a linear space data structure which can answer approximate Tukey depth queries efficiently in 3D. These data structures are obtained by applying our ideas for the approximate halfspace counting problem. Approximating the simplicial depth turns out to be much more difficult, however. Computing the simplicial depth of a given point is more computationally challenging than most other definitions of data depth. In 2D we obtain the first data structure which uses near linear space and can answer approximate simplicial depth queries in polylogarithmic time. As applications of this result, we provide two non-trivial methods to approximate the simplicial depth of a given point in higher dimension. Along the way, we establish a tight combinatorial relationship between the Tukey depth of any given point and its simplicial depth. Another problem investigated in this thesis is the dominance reporting problem, an important special case of orthogonal range reporting. In three dimensions, we solve this problem in the pointer machine model and the external memory model by offering the first optimal data structures in these models of computation. Also, in the RAM model and for points from an integer grid we reduce the space complexity of the fastest known data structure to optimal. Using known techniques in the literature, we can use our results to obtain solutions for the orthogonal range searching problem as well. The query complexity offered by our orthogonal range reporting data structures match the most efficient query complexities known in the literature but our space bounds are lower than the previous methods in the external memory model and RAM model where the input is a subset of an integer grid. The results also yield improved orthogonal range searching in higher dimensions (which shows the significance of the dominance reporting problem). Intersection searching is a generalization of range searching where we deal with more complicated geometric objects instead of points. We investigate the rectilinear disjoint polygon counting problem which is a specialized intersection counting problem. We provide a linear-size data structure capable of counting the number of disjoint rectilinear polygons intersecting any rectilinear polygon of constant size. The query time (as well as some other properties of our data structure) resembles the classical simplex range searching data structures

Graph problems arising from parameter identification of discrete dynamical systems

This paper focuses on combinatorial feasibility and optimization problems that arise in the context of parameter identification of discrete dynamical systems. Given a candidate parametric model for a physical system and a set of experimental observations, the objective of parameter identification is to provide estimates of the parameter values for which the model can reproduce the experiments. To this end, we define a finite graph corresponding to the model, to each arc of which a set of parameters is associated. Paths in this graph are regarded as feasible only if the sets of parameters corresponding to the arcs of the path have nonempty intersection. We study feasibility and optimization problems on such feasible paths, focusing on computational complexity. We show that, under certain restrictions on the sets of parameters, some of the problems become tractable, whereas others are NP-hard. In a similar vein, we define and study some graph problems for experimental design, whose goal is to support the scientist in optimally designing new experiment