Dynamic Streaming Algorithms for Epsilon-Kernels

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.

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.

The landing site decision has direct relation to the safety of a spacecraft. This paper presents the use of computational geometry (CG) and genetic algorithm (GA) in the landing site selection. In order to describe the obstacles, we use computational geometry to compute the minimum enclosing circles and ellipses. Taking the obstacles as discrete circles and ellipses, we use the genetic algorithm to simplify and to solve the difficult problem of finding the largest empty circle (LEC) in circles and ellipses for the spacecraft landing on the surface. The results of tests on the solution procedure are presented.

We consider a collectionH ofn hyperplanes in Ed (where the dimensiond is fixed). An ?-cutting forH is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all Ed and such that the interior of any simplex is intersected by at most?n hyperplanes ofH. We give a deterministic algorithm for finding a (1/r)-cutting withO(rd) simplices (which is asymptotically optimal). Forr≤n1??, where ?>0 is arbitrary but fixed, the running time of this algorithm isO(n(logn)O(1)rd?1). In the plane we achieve a time boundO(nr) forr≤n1??, which is optimal if we also want to compute the collection of lines intersecting each simplex of the cutting. This improves a result of Agarwal, and gives a conceptually simpler algorithm. For ann point setX⊆Ed and a parameterr, we can deterministically compute a (1/r)-net of sizeO(rlogr) for the range space (X, {X ? R; R is a simplex}), In timeO(n(logn)O(1)rd?1+rO(1)). The size of the (1/r)-net matches the best known existence result. By a simple transformation, this allows us to find ?-nets for other range spaces usually encountered in computational geometry. These results have numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly.

Solid modelling and computational geometry are based on classical topology and geometry in which the basic predicates and operations, such as membership, subset inclusion, union and intersection, are not continuous and therefore not computable. But a sound computational framework for solids and geometry can only be built in a framework with computable predicates and operations. In practice, correctness of algorithms in computational geometry is usually proved using the unrealistic Real RAM machine model of computation, which allows comparison of real numbers, with the undesirable result that correct algorithms, when implemented, turn into unreliable programs. Here, we use a domain-theoretic approach to recursive analysis to develop the basis of an e3ective and realistic framework for solid modelling. This framework is equipped with a well de5ned and realistic notion of computability which re6ects the observable properties of real solids. The basic predicates and operations on solids are computable in this model which admits regular and non-regular sets and supports a design methodology for actual robust algorithms. Moreover, the model is able to capture the uncertainties of input data in actual CAD situations. c 2002 Elsevier Science B.V. All rights reserved.

Algorithms in computational geometry often use the real-RAM model of computation. This model assumes that exact real numbers can be stored in memory and retreived in constant time, and that field operations (+, –, *, /) and certain other operations, square root, sine, and cosine for instance, are exact, and can be applied in constant time. These assumptions are often difficult to discharge at implementation time. Even well-understood algorithms, like line-sweep for polygon union [PS85], present much trouble. Why? Such algorithms obtain good combinatorial complexity bounds by exploiting geometric orders on the input. These relations are often implicit, so an algorithm can only probe them pointwise with a collection of predicate functions answering, e.g., “Is point p left of edge e?”. In implementations, the reply must depend on arithmetic with finitely represented numbers. The trouble originates here.

In school development problem, determining location is one of important things to consider. In this research, the purpose is to determine the location of SMAN 9 Denpasar if it will be built. One of algorithms in computational geometry that can be used to find solution of facility location problem is multiplicatively weighted Voronoi diagram in two dimensions. The result of weighted Voronoi diagram shows the influence of each site to the surrounding area. Then, the location of SMAN 9 Denpasar is obtained by determining the center of the largest empty circle of the weighted Voronoi diagram.

Algorithms in computational geometry are of increasing importance in computer-aided design, for example, in the layout of integrated circuits. The efficient computation of the intersection of several superimposed figures is a basic problem. Plane figures defined by points connected by straight line segments are considered, for example, polygons (not necessarily simple) and maps (embedded planar graphs). The regions into which the plane is partitioned by these intersecting figures are to be processed in various ways such as listing the boundary of each region in cyclic order or sweeping the interior of each region. Let m be the total number of points of all the figures involved and s be the total number of intersections of all line segments. A two plane-sweep algorithm that solves the problems above is presented; in the general case (non convexity) in time O((n+s)log-n) and space O(n+s); when the regions of each given figure are convex, the same can be achieved in time O(n log n +s) and space O(n)

Enhancements to array processors in the form of broadcast buses have been proposed for improving the speeds of algorithms in linear algebra, image processing and computational geometry. In this paper, we consider certain practical issues in such arrays which include fewer processors connected to the broadcast buses and finite time, namely log N time for broadcasting data to N processors. We propose a modified broadcast bus VLSI architecture which consists of a √N × √N mesh connected structure and a hierarchy of broadcast buses in each row and in each column such that every bus has k PE's. That is, in any row or column, in the first level there are \(\frac{{\sqrt N }}{k}\)buses with groups of k PE's connected to each bus. One PE from each of this group is connected to second level of buses in a similar manner. This is recursively done until there are only k PE's left which are connected by a broadcast bus. With this architecture, maximum, minimum, or sum of N values can be found in O(log N), median row of a binary picture in O(log N), shortest distance between two points in O(log N). This architecture is well suited for parallel processing of applications in Linear Algebra, Image Processing, Computational Geometry and Numerical Computations. These restricted connections to buses reduces the I/O ports significantly, and therefore, is more suitable for VLSI implementation.

This work presents an adapted version of the Computational Geometry Algorithm (CGA) used for the development of audio-based applications and services. The CGA algorithm analyses an audio stream and produces a unique set of points that can be considered to be the audio data “fingerprint”. It is shown that this fingerprint is coding-independent, a fact that can render the proposed algorithm suitable for multiple purposes, including the categorisation of content identity and the identification of audio clips, hence providing support for the realisation of audio sorting/searching tasks and services. Additionally, based on specific novel applications and services, the overall algorithmic performance and efficiency characteristics of the CGA algorithm are discussed and analysed.

COMBINATORIAL AND DISCRETE GEOMETRY Finite Point Configurations, J. Pach Packing and Covering, G. Fejes Toth Tilings, D. Schattschneider and M. Senechal Helly-Type Theorems and Geometric Transversals, R. Wenger Pseudoline Arrangements, J.E. Goodman Oriented Matroids, J. Richter-Gebert and G.M. Ziegler Lattice Points and Lattice Polytopes, A. Barvinok New! Low-Distortion Embeddings of Finite Metric Spaces, P. Indyk and J. Matousek New! Geometry and Topology of Polygonal Linkages, R. Connelly and E.D. Demaine New! Geometric Graph Theory, J. Pach Euclidean Ramsey Theory, R.L. Graham Discrete Aspects of Stochastic Geometry, R. Schneider Geometric Discrepancy Theory and Uniform Distribution, J.R. Alexander, J. Beck, and W.W.L. Chen Topological Methods, R.T. Zivaljevic Polyominoes, S.W. Golomb and D.A. Klarner POLYTOPES AND POLYHEDRA Basic Properties of Convex Polytopes, M. Henk, J. Richter-Gebert, and G.M. Ziegler Subdivisions and Triangulations of Polytopes, C.W. Lee Face Numbers of Polytopes and Complexes, L.J. Billera and A. Bjoerner Symmetry of Polytopes and Polyhedra, E. Schulte Polytope Skeletons and Paths, G. Kalai Polyhedral Maps, U. Brehm and E. Schulte ALGORITHMS AND COMPLEXITY OF FUNDAMENTAL GEOMETRIC OBJECTS Convex Hull Computations, R. Seidel Voronoi Diagrams and Delaunay Triangulations, S. Fortune Arrangements, D. Halperin Triangulations and Mesh Generation, M. Bern Polygons, J. O'Rourke and S. Suri Shortest Paths and Networks, J.S.B. Mitchell Visibility, J. O'Rourke Geometric Reconstruction Problems, S.S. Skiena New! Curve and Surface Reconstruction, T.K. Dey Computational Convexity, P. Gritzmann and V. Klee Computational Topology, G. Vegter Computational Real Algebraic Geometry, B. Mishra GEOMETRIC DATA STRUCTURES AND SEARCHING Point Location, J. Snoeyink New! Collision and Proximity Queries, M.C. Lin and D. Manocha Range Searching, P.K. Agarwal Ray Shooting and Lines in Space, M. Pellegrini Geometric Intersection, D.M. Mount New! Nearest Neighbors in High-Dimensional Spaces, P. Indyk COMPUTATIONAL TECHNIQUES Randomization and Derandomization, O. Cheong, K. Mulmuley, and E. Ramos Robust Geometric Computation, C.K. Yap Parallel Algorithms in Geometry, M.T. Goodrich Parametric Search, J.S. Salowe New! The Discrepancy Method in Computational Geometry, B. Chazelle APPLICATIONS OF DISCRETE AND COMPUTATIONAL GEOMETRY Linear Programming, M. Dyer, N. Megiddo, and E. Welzl Mathematical Programming, M.H. Todd Algorithmic Motion Planning, M. Sharir Robotics, D. Halperin, L.E. Kavraki, and J.-C. Latombe Computer Graphics, D. Dobkin and S. Teller New! Modeling Motion, L.J. Guibas Pattern Recognition, J. O'Rourke and G.T. Toussaint Graph Drawing, R. Tamassia and G. Liotta Splines and Geometric Modeling, C.L. Bajaj New! Surface Simplification and 3D Geometry Compression, J. Rossignac Manufacturing Processes, R. Janardan and T.C. Woo Solid Modeling, C.M. Hoffmann New! Computation of Robust Statistics: Depth, Median, and Related Measures, P.J. Rousseeuw and A. Struyf New! Geographic Information Systems, M. van Kreveld Geometric Application of the Grassmann-Cayley Algebra, N.L. White Rigidity and Scene Analysis, W. Whiteley Sphere Packing and Coding Theory, G.A. Kabatiansky and J.A. Rush Crystals and Quasicrystals, M. Senechal New! Biological Applications of Computational Topology, H. Edelsbrunner New! GEOMETRIC SOFTWARE Software, J. Joswig Two Computation Geometry Libraries: LEDA and CGAL, L. Kettner and S. Naher Index of Defined Terms New! Index of Cited Authors

Traditional algorithms in computational geometry assume that the input points are given precisely. In practice, data is usually imprecise, but information about the imprecision is often available. In this context, we investigate what the value of this information is. We show here how to preprocess a set of disjoint regions in the plane of total complexity n in $O(n\log n)$ time so that if one point per set is specified with precise coordinates, a triangulation of the points can be computed in linear time. In our solution, we solve another problem which we believe to be of independent interest. Given a triangulation with red and blue vertices, we show how to compute a triangulation of only the blue vertices in linear time.

A fundamental question is whether one can maintain a maximum independent set in polylogarithmic update time for a dynamic collection of geometric objects in Euclidean space. Already, for a set of intervals, it is known that no dynamic algorithm can maintain an exact maximum independent set in sublinear update time. Therefore, the typical objective is to explore the trade-off between update time and solution size. Substantial efforts have been made in recent years to understand this question for various families of geometric objects, such as intervals, hypercubes, hyperrectangles, and fat objects. We present the first fully dynamic approximation algorithm for disks of arbitrary radii in the plane that maintains a constant-factor approximate maximum independent set in polylogarithmic expected amortized update time. Moreover, for a fully dynamic set of n disks of unit radius in the plane, we show that a 12-approximate maximum independent set can be maintained with worst-case update time $$O(\log n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and optimal output-sensitive reporting. This result generalizes to fat objects of comparable sizes in any fixed dimension d , where the approximation ratio depends on the dimension and the fatness parameter. Further, we note that, even for a dynamic set of disks of unit radius in the plane, it is impossible to maintain $$O(1+\varepsilon )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -approximate maximum independent set in truly sublinear update time, under standard complexity assumptions. Our results build on two recent technical tools: (i) The MIX algorithm by Cardinal et al. (2021) that can smoothly transition from one independent set to another; hence it suffices to maintain a family of independent sets where the largest one is a constant-factor approximation of a maximum independent set. (ii) A dynamic nearest/farthest neighbor data structure for disks by Kaplan et al. (2020) and Liu (2022), which generalizes the dynamic convex hull data structure by Chan (2010), and allows us to quickly find a “replacement” disk (if any) when a disk in one of our independent sets is deleted.

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.

The workshop Algorithm Engineering , organized by Giuseppe Italiano (Roma), Petra Mutzel (Dortmund), Peter Sanders (Karlsruhe), and Martin Skutella (Dortmund) was held May 6th – May 12th, 2007. The goal of the workshop was to bring together researchers with different background, e.g., from combinatorial optimization, algorithmic theory, and algorithm engineering, in order to strengthen and foster collaborations in the area of algorithm engineering and to identify key research directions for the future. There were 47 international participants out of which 23 mainly work in the area of algorithm engineering, 11 in algorithm theory, and 13 in combinatorial optimization. A considerable number of participants visited the Oberwolfach Institute (MFO) for the first time. In five survey talks given by renowned specialists in the field, the state of the art in selected areas of high interest was demonstrated. The introductory talk was presented by Giuseppe Italiano (Roma) who gave “a personal and historical perspective” of algorithm engineering. Bill Cook (Atlanta) presented the newest results as well as engineering efforts for solving huge TSP instances. Friedhelm Meyer auf der Heide (Paderborn) suggested new algorithmic challenges in the area of Computer Graphics. Ian Munro (Waterloo) introduced the audience into the area of succinct data structures which is getting increasingly important with huge data sets. In this context, also external memory and cache-oblivious algorithms are getting growing interest. The survey talk by Gerth Brodal (Aarhus) presented theory and practice for these type of algorithms. In addition, we organized a series of sessions containing shorter contributions of roughly 30 minutes. Here, we particularly encouraged young researchers to present their recent results. The focus of the program lay on recent developments in various areas that are relevant for the broad topic of algorithm engineering. Examples are mathematical programming, external memory algorithms, succinct data structures, resilient data structures, dynamic data structures, online algorithms, graph algorithms, geometric computation, analysis of local search algorithms, and algorithmic game-theory. In total, we had 32 talks of which 15 can be assigned to the area of algorithm engineering, 8 to the area of algorithmic theory, and 7 to the area of combinatorial optimization, which we think was an ideal mixture for the purpose of this workshop. We found it encouraging that after the talks a lot of discussion took place between the participants. We also like to mention that Peter Sanders (Karlsruhe) introduced the new DFG Priority Program 1307 Algorithm Engineering , which will start in fall 2007. As the coordinator of this program he pointed out the specific goals of the program and the chances for the area of algorithm engineering. All of the participants had the chance to discuss different aspects of algorithm engineering in the open problem session on Thursday evening. In our view, and as expressed by many of the participants, the workshop was a great success. We (including many participants) have learned a lot during this week, also from the three different research communities present at the workshop, and got many new ideas for exciting research projects. The program of the workshop was well-received by the participants as the good quality of presentations and the enthusiasm of the speakers and the audience gave raise to joint research among the participants. In particular exchange between young researchers and experienced scientists was promoted. We wish to thank Oberwolfach for this unique opportunity to bring together the three different research groups at this great place in order to strengthen the area of algorithm engineering. The following collection of abstracts in chronological order gives a more detailed overview of the program.