Visibility and Ray Shooting Queries in Polygonal Domains

The support of range, segment and rectangle queries are fundamental problems in computational geometry, and have extensive applications in many domains. Despite significant theoretical work on these problems, efficient implementations can be complicated, and most implementations do not have useful theoretical bounds. In this paper, we focus on simple and efficient parallel algorithms and implementations for range, segment and rectangle queries, which have worst-case bounds in theory and good performance in practice, both sequentially and in parallel. We propose to use a framework based on the abstract data type augmented map, to model the problems. Based on the augmented map interface, we develop both multi-level tree structures and sweepline algorithms supporting range, segment and rectangle queries in two dimensions. For the sweepline algorithms, we also propose a parallel paradigm and show corresponding cost bounds. Theoretically, the construction algorithms of all of our data structures are work-efficient and highly parallelized.We have implemented all the data structures described in the paper, ten in all, using a parallel augmented map library. Based on the library, each data structure only requires about 100 lines of C++ code. We test their performance on large data sets (up to 108 elements) and a machine with 72-cores (144 hyperthreads). The parallel construction achieves 32-68x speedup, and the speedup numbers for queries are up to 126-fold. Sequentially, each of our implementations outperforms the CGAL library by at least 2x in both construction and queries. Our sequential implementation has approximately the same construction time as the R-tree in the Boost library, but has significantly better query performance (1.6-1200x). We believe this paper provides the most comprehensive experimental study of data structures on range, segment and rectangle queries, both in parallel and sequential setting.

In this work we study quantum algorithms for Hopcroft’s problem which is a fundamental problem in computational geometry. Given n points and n lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in O ( n 4/3 ) time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity \(\widetilde{O}(n^{5/6}) \) . The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems. Finally, we examine the connections between Hopcroft’s problem and other computational problems via fine-grained complexity. For example, we show a conditional Ω ( n 3/4 ) time lower bound on Hopcroft’s problem in 5 dimensions based on the quantum analogue of a classical hardness conjecture, which is stronger than the (optimal) Θ ( n 2/3 ) query complexity bounds.

Spatial data structures form a core ingredient of many geometric algorithms, both in theory and in practice. Many of these data structures, especially the ones used in practice, are based on partitioning the underlying space (examples are binary space partitions and decompositions of polygons) or partitioning the set of objects (examples are bounding-volume hierarchies). The efficiency of such data structures---and, hence, of the algorithms that use them---depends on certain characteristics of the partitioning. For example the performance of many algorithms that use binary space partitions (BSPs) depends on the size of the BSPs. Similarly, the performance of answering range queries using bounding-volume hierarchies (BVHs) depends on the so-called crossing number that can be associated with the partitioning on which the BVH is based. Much research has been done on the problem of computing partitioning whose characteristics are good in the worst case. In this thesis, we studied the problem from a different point of view, namely instance-optimality. In particular, we considered the following question: given a class of geometric partitioning structures---like BSPs, simplicial partitions, polygon triangulations, …---and a cost function---like size or crossing number---can we design an algorithm that computes a structure whose cost is optimal or close to optimal for any input instance (rather than only worst-case optimal). We studied the problem of finding optimal data structures for some of the most important spatial data structures. As an example having a set of n points and an input parameter r, It has been proved that there are input sets for which any simplicial partitions has crossing number ¿(vr). It has also been shown that for any set of n input points and the parameter r one can make a simplicial partition with stabbing number O(vr). However, there are input point sets for which one can make simplicial partition with lower stabbing number. As an example when the points are on a diagonal, one can always make a simplicial partition with stabbing number 1. We started our research by studying BSPs for line segments in the plane, where the cost function is the size of the BSPs. A popular type of BSPs for line segments are the so-called auto-partitions. We proved that finding an optimal auto-partition is NP-hard. In fact, finding out if a set of input segments admits an auto-partition without any cuts is already NP-hard. We also studied the relation between two other types of BSPs, called free and restricted BSPs, and showed that the number of cuts of an optimal restricted BSP for a set of segments in R2 is at most twice the number of cuts of an optimal free BSP for that set. The details are being represented in Chapter 1 of the thesis. Then we turned our attention to so-called rectilinear r-partitions for planar point sets, with the crossing number as cost function. A rectilinear r-partition of a point set P is a partitioning of P into r subsets, each having roughly |P|/r points. The crossing number of the partition is defined using the bounding boxes of the subsets; in particular, it is the maximum number of bounding boxes that can be intersected by any horizontal or vertical line. We performed some theoretical as well as experimental studies on rectilinear r-partitions. On the theoretical side, we proved that computing a rectilinear r-partition with optimal stabbing number for a given set of points and parameter r is NP-hard. We also proposed an exact algorithm for finding optimal rectilinear r-partitions whose running time is polynomial when r is a constant, and a faster 2-approximation algorithm. Our last theoretical result showed that considering only partitions whose bounding boxes are disjoint is not sufficient for finding optimal rectilinear r-partitions. On the experimental side, we performed a comparison between four different heuristics for constructing rectilinear r-partitions. The so-called windmill KD-tree gave the best results. Chapter 2 of the thesis describes all the details of our research on rectilinear r-partitions. We studied another spatial data structure in Chapter 3 of the thesis. Decomposition of the interior of polygons is one of the fundamental problems in computational geometry. In case of a simple polygon one usually wants to make a Steiner triangulation of it, and when we have a rectilinear polygon at hand, one typically wants to make a rectilinear decomposition for it. Due to this reason there are algorithms which make Steiner triangulations and rectangular decompositions with low stabbing number. These algorithms are worst-case optimal. However, similar to the two previous data structures, there are polygons for which one can make decompositions with lower stabbing numbers. In 3 we proposed a 3-approximation for finding an optimal rectangular decomposition of a rectilinear polygon. We also proposed an O(1)-approximation for finding optimal Steiner triangulation of a simple polygon. Finally, in Chapter 4 of the thesis, we considered another optimization problem, namely how to approximate a piecewise-linear function F: R ¿R with another piecewise-linear function with fewer pieces. Here one can distinguish two versions of the problem. The first one is called the min-k problem; the goal is then to approximate the function within a given error e such that the resulting function has the minimum number of links. The second one is called the min-e problem; here the goal is to find an approximation with at most k links (for a given k) such that the error is minimized. These problems have already been studied before. Our contribution is to consider the problem for so-called uncertain functions, where the value of the input function F at its vertices is given as a discrete set of different values, each with an associated probability. We show how to compute an approximation that minimizes the expected error.

In this paper, we propose streaming algorithms for approximating the smallest intersecting ball of a set of disjoint balls in \(\mathbb {R}^d\). This problem is a generalization of the \(1\)-center problem, one of the most fundamental problems in computational geometry. We consider the single-pass streaming model; only one-pass over the input stream is allowed and a limited amount of information can be stored in memory. We introduce three approximation algorithms: one is an algorithm for the problem in arbitrarily dimensions, but in the other two we assume \(d\) is a constant. The first algorithm guarantees a \((2+\sqrt{2}+\varepsilon ^*)\)-factor approximation using \(O(d^2)\) space and \(O(d)\) update time where \(\varepsilon ^*\) is an arbitrarily small positive constant. The second algorithm guarantees an approximation factor \(3\) using \(O(1)\) space and \(O(1)\) update time (assuming constant \(d\)). The third one is a \((1+\varepsilon )\)-approximation algorithm that uses \(O(1/\varepsilon ^{d})\) space and \(O(1/\varepsilon ^{(d-1)/2})\) amortized update time. They are the first approximation algorithms for the problem, and also the first results in the streaming model.KeywordsDisjoint BallsStreaming ModelApproximation FactorBalanced InputFactor Analysis EstimationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The parametric minimax problem, which finds the parameter value minimizing the weight of a solution of a combinatorial maximization problem, is a fundamental problem in sensitivity analysis. Moreover, several problems in computational geometry can be formulated as parametric minimax problems. The parametric search paradigm gives an efficient sequential algorithm for a convex parametric minimax problem with one parameter if the original non-parametric problem has an efficient parallel algorithm. We consider the parametric minimax problem with d parameters for a constant d, and solve it by using multidimensional version of the parametric search paradigm. As a new feature, we give a feasible region in the parameter space in which the parameter vector must be located.Typical results obtained as applications are: (1) Efficient solutions for some geometric problems, including theoretically efficient solutions for the minimum diameter bridging problem in d-dimensional space between convex polytopes. (2) Parametric polymatroid optimization, for example, O(n log n) time algorithm to compute the parameter vector minimizing k-largest linear parametric elements with d dimensions.

Geostationary (GEO) satellites used for civilian, military and various other strategic requirements are critical space assets for a country. Given a satellite of interest, an assessment of the area of influence of neighborhood satellite operations is critical for the safety of the satellite asset. Nearest neighbor search is a fundamental problem in computational geometry. Voronoi diagram is a geometric structures that is based on proximity and can be used for nearest neighbour identification. A solution methodology for assessment of threat perception is presented using Voronoi diagrams, Delaunay triangulation, and power diagram. The intuitive properties based on the topology and geometry of Voronoi diagrams, and the adjacency relations of Delaunay triangulation help in describing some kind of region of influence around each satellite. The influence exerted by non-cooperative GEO satellites over the satellite of interest, aid in estimating the threat perception. Once the threat is narrowed down, the costly and limited number of ground surveillance sensors like telescopes and radars will be optimally reoriented to closely monitor the non-cooperative satellites of interest.

A fundamental problem in computational geometry is determining whether a point is inside a B-rep solid. Methods currently used for such point classification are unreliable or inefficient or both. A new approach is illustrated by showing how a simple method for loops of planar curves represented by B-splines can be extended from two dimensions to three. The plan in two dimensions is to construct a polygon so that the point will be inside the loop if and only if it is inside the polygon. Once such a polygon is found, it is easy to compute its winding number with respect to the point. In three dimensions, an analogous (although more complicated) method is robust and efficient.

Answering range queries is a problem of fundamental importance in spatial information retrieval and computational geometry. The objective is to store a set of n points P in R d, each associated with a weight, so that it is possible to count, or more generally to compute some function of the weights of the points lying inside a given query range. Range searching is among the most heavily studied problems, and many search structures have been proposed and analyzed [1, 7]. There is a spectrum of space-time tradeoffs. The most relevant work to ours involves halfspace range counting queries, which Matouˇsek [6] has shown can be answered in n/m 1/d time from a data structure of space O(m). Nearly matching lower bounds were given by by Brönnimann, Chazelle and Pach [4] (or BCP). Given the relatively high complexity of range searching, it is natural to consider the problem in the context of approximation. We are given an approximation parameter ε> 0 and assume that ranges are bounded. Let η denote a range, and let diam(η) denote its diameter. All the points that lie in the range must be counted, and any of the points that lie within distance ε · diam(η) of the range’s boundary may be counted as well. Arya and Mount [3] showed that in any fixed dimension d with O(n log n) preprocessing time and O(n) space, ε-approximate range queries for any bounded convex range can be answered in time O(log n+1/ε d−1) [3]. Later, Chazelle, Liu, and Magen [5] considered approximate halfspace range and Euclidean ball searching in the high dimensional setting. Ignoring polylogarithmic factors, they showed that is possible to answer queries in O(d/ε 2) time with O(dn O(1/ε2) ) space.

A fundamental problem in computational geometry is determining whether it is possible to travel between two points in the plane while avoiding a set of obstacles. If the obstacles are known in advance, they can be preprocessed so that path-existence queries can be performed efficiently.

Maxima-finding problems are the fundamental problem in computational geometry with a great deal of application in many areas, and it has resurfaced with the advent of Skyline Queries for relational databases and data mining recently. The existed algorithms in the field of Maxima-finding problem (Skyline Queries) have been summarized in this paper. But for the massive data sets, there has no I/O linear algorithm yet. A new kind of External Memory Algorithm of Maxima-finding problem (EMMF) has been presented, the I/O complexity of algorithm is linear, the corresponding reliability has been validated from experiments, and the status of 2-dimensional space has been proved in theory too.

This paper presents a framework for multi-core implementations of divide and conquer algorithms and shows its efficiency and ease of use by applying it to some fundamental problems in computational geometry. The framework supports automatic parallelization of any D&C algorithm. It is only required that the algorithm is implemented by a C++ class implementing a so-called job-interface. We also report on experimental results and discuss some aspects of the automatic parallelization of randomized incremental algorithms. Some results of this paper have been presented in the 20th Annual Canadian Conference on Computational Geometry ([13]).KeywordsConvex HullComputational GeometryIncremental AlgorithmDivide OperationGeometric AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Chapter 7 A survey of computational geometry

Computational geometry is the study of algorithms for manipulating sets of points, lines, polygons, planes and other geometric objects. For many problems in this realm, the sets considered are static and the data structures representing them do not permit efficient insertion and deletion of objects (e.g. points). Dynamic problems, in which the set and the geometric data structure change over time, are also of interest. The topic of this thesis is the presentation of fast algorithms for fully dynamic maintenance of some common geometric structures. The following configurations are examined: planar nearest-point and farthest-point Voronoi diagrams, convex hulls (in two and three dimensions), common intersection of halfspaces (2-D and 3-D), and contour of maximal vectors (2-D and 3-D). The principal techniques exploited are fast merging of substructures, and the use of extra storage. Dynamic geometric search structures based upon the configurations are also presented.

This paper has two goals. First, we point out that most problems in computational geometry in fact have fast parallel algorithms (that is, in NC*) by reduction to the cell decomposition result of Kozen and Yap. We illustrate this using a new notion of generalized Voronoi diagrams that subsumes all known instances. While the existence of NC* algorithms for computational geometry is theoretically significant, it leaves much to be desired for specific problems. Therefore, the second part of the paper surveys some recent results in a fast growing list of parallel algorithms for computational geometry.

A classical problem in computational geometry is the planar point location problem. This problem calls for preprocessing a polygonal subdivision of the plane defined by n line segments so that, given a sequence of points, the polygon containing each point can be determined quickly on-line. Several ways of solving this problem in O(log n) query time and O(n) space are known, but they are all rather complicated. We propose a simple O(log n)-query-time, O(n)-space solution, using persistent search trees. A persistent search tree differs from an ordinary search tree in that after an insertion or deletion, the old version of the tree can still be accessed. We develop a persistent form of binary search tree that supports insertions and deletions in the present and queries in the past. The time per query or update is O(log m), where m is the total number of updates, and the space needed is O(1) per update. Our planar point location algorithm is an immediate application of this data structure. The structure also provides an alternative to Chazelle's hive graph structure, which has a variety of applications in geometric retrieval.