A Simple Sampling Lemma: Analysis and Applications in Geometric Optimization

We show that a Simple Stochastic Game (SSG) can be formulated as an LP-type problem. Using this formulation, and the known algorithm of Sharir and Welzl [SW] for LP-type problems, we obtain the first strongly subexponential solution for SSGs (a strongly subexponential algorithm has only been known for binary SSGs [L]). Using known reductions between various games, we achieve the first strongly subexponential solutions for Discounted and Mean Payoff Games. We also give alternative simple proofs for the best known upper bounds for Parity Games and binary SSGs. To the best of our knowledge, the LP-type framework has been used so far only in order to yield linear or close to linear time algorithms for various problems in computational geometry and location theory. Our approach demonstrates the applicability of the LP-type framework in other fields, and for achieving subexponential algorithms.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Polygon Placement Revisited: (Degree of Freedom + 1)-SUM Hardness and an Improvement via Offline Dynamic Rectangle UnionMarvin Künnemann and André NusserMarvin Künnemann and André Nusserpp.3181 - 3201Chapter DOI:https://doi.org/10.1137/1.9781611977073.124PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We revisit a classical problem in computational geometry: Determine the largest copy of a simple polygon P that can be placed into the simple polygon Q. Despite significant effort studying a number of settings, known algorithms require high polynomial running times, even for the interesting case when either P or Q have constant size. (Barequet and Har-Peled, 2001) give a conditional lower bound of n2–o(1) under the 3SUM conjecture when P and Q are (convex) polygons with Θ(n) vertices each. This leaves open whether we can establish (1) hardness beyond quadratic time and (2) any superlinear bound for constant-sized P or Q. In this paper, we affirmatively answer these questions under the higher-order kSUM conjecture, proving natural hardness results that increase with each degree of freedom (scaling, x-translation, y-translation, rotation): (scaling, x-translation:) Finding the largest copy of P that can be x-translated into Q requires time n2–o(1) under the 3SUM conjecture, even for orthogonal (rectilinear) polygons P, Q with O(1) and n vertices, respectively. (scaling, x-translation, y-translation:) Finding the largest copy of P that can be arbitrarily translatedinto Q requires time n2–o(1) under the 4SUM conjecture, even for orthogonal polygons P, Q with O(1) and n vertices, respectively. This establishes the same lower bound under the assumption that Subset Sum cannot be solved in time O((2–∊)n/2) for any ∊ > 0. The above lower bounds are almost tight when one of the polygons is of constant size: Using an offline dynamic algorithm for maintaining the area of a union of rectangles due to Overmars and Yap, we obtain an -time algorithm for orthogonal polygons P, Q with p and q vertices, respectively. This matches the lower bounds up to an n1/2+o(1)-factor when P, Q have O(1) and n vertices. (scaling, x-translation, y-translation, rotation:) Finally, finding the largest copy of P that can be arbitrarily rotated and translated into Q requires time n3–o(1) under the 5SUM conjecture. As in our reductions, each degree of freedom determines one summand in a kSUM instance, these lower bounds appear likely to be best possible under kSUM. We are not aware of any other such natural (degree of freedom + 1)-SUM hardness for a geometric optimization problem. Finally, we prove an additional tight OV hardness of the translations-only case. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771

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.

Chapter 7 A survey of computational geometry

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.

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.

Certified computation of the sign of the determinant of a matrix is a central problem in computational geometry. Certification by known methods is practically difficult because the magnitude of the determinant of an integer input matrix A may vary dramatically, from 1 to || A || n , and the round-off error bound of the determinant computation varies proportionally. Because of such a variation, high precision computation of \det A is required to ensure that the error bound is smaller than the magnitude of the determinant. We observe, however, that our certification task of determining only a single bit of \det A , that is, the bit carrying the sign, does not require us to estimate the latter round-off error. Instead, we solve a much simpler task of computing numerically the factorization of a matrix by Gaussian elimination with pivoting (which is a subroutine of LAPACK and LINPACK) and of estimating the minimum distance 1/||A -1 || from A to a singular matrix. Such an estimate gives us a desired range for the round-off error of the factorization such that the invariance of the sign of det A is ensured as long as the error varies in this range. Based on these simple but novel observations, we devise new effective arithmetic filters for the certification of the sign, compare them with the known filters, and confirm the efficiency of our techniques by some numerical tests.

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.

We present an O(n) time algorithm for computing row-wise maxima or minima of an implicit, totally monotone $n \times n$ matrix whose entries represent shortest-path distances between pairs of vertices in a simple polygon. We apply this result to derive improved algorithms for several well-known problems in computational geometry. Most prominently, we obtain linear-time algorithms for computing the geodesic diameter, all farthest neighbors, and external farthest neighbors of a simple polygon, improving the previous best result by a factor of O(log n) in each case.

We propose the first graphics processing unit (GPU) solution to compute the 2D constrained Delaunay triangulation (CDT) of a planar straight line graph (PSLG) consisting of points and edges. There are many existing CPU algorithms to solve the CDT problem in computational geometry, yet there has been no prior approach to solve this problem efficiently using the parallel computing power of the GPU. For the special case of the CDT problem where the PSLG consists of just points, which is simply the normal Delaunay triangulation (DT) problem, a hybrid approach using the GPU together with the CPU to partially speed up the computation has already been presented in the literature. Our work, on the other hand, accelerates the entire computation on the GPU. Our implementation using the CUDA programming model on NVIDIA GPUs is numerically robust, and runs up to an order of magnitude faster than the best sequential implementations on the CPU. This result is reflected in our experiment with both randomly generated PSLGs and real-world GIS data having millions of points and edges.

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.

Given a polygonal domain (or polygon with holes) in the plane, we study the problem of computing the visibility polygon of any query point. As a special case of visibility problems, we also study the ray-shooting problem of finding the first point on the polygon boundaries that is hit by any query ray. These are fundamental problems in computational geometry and have been studied extensively. We present new algorithms and data structures that improve the previous results.

Constructing a two dimensional shape from a given set of point sites is a well-known problem in computational geometry. We present a critical review of the existing algorithms for constructing polygonal shapes from given point sites. We present a new approach called 'inward denting' for constructing polygonal shapes. We then extend the proposed approach for modeling polygons with holes. We also present experimental investigations of the quality of the solutions generated by the proposed algorithm. For obtaining experimental results we implemented the proposed algorithm in Java programming language. The experimental results show that the perimeter of the generated shape is at most 33% more than the length of the minimum spanning tree.

In this paper we propose a generalized Roman domination problem called connected strong k-Roman dominating set problem. It is NP-hard even in a unit ball graph. Unit ball graphs are the intersection graphs of equal sized balls in the three-dimensional space, they are widely used as a mathematical model for wireless sensor networks and some problems in computational geometry. This paper presents the first constant approximation algorithm with a guaranteed performance ratio at most $$6(k+2)$$ in unit ball graphs, where k is a positive integer.