New Results on Abstract Voronoi Diagrams

Voronoi diagrams are a fundamental structure used in many areas of science. For a given set of objects, called sites, the Voronoi diagram separates the plane into regions, such that points belonging to the same region have got the same nearest site. This definition clearly depends on the type of given objects, they may be points, line segments, polygons, etc. and the distance measure used. To free oneself from these geometric notions, Klein introduced abstract Voronoi diagrams as a general construct covering many concrete Voronoi diagrams. Abstract Voronoi diagrams are based on a system of bisecting curves, one for each pair of abstract sites, separating the plane into two dominance regions, belonging to one site each. The intersection of all dominance regions belonging to one site p defines its Voronoi region. The system of bisecting curves is required to fulfill only some simple combinatorial properties, like Voronoi regions to be connected, the union of their closures cover the whole plane, and the bisecting curves are unbounded. These assumptions are enough to show that an abstract Voronoi diagram of n sites is a planar graph of complexity O(n) and can be computed in expected time O(n log n) by a randomized incremental construction. In this thesis we widen the notion of abstract Voronoi diagrams in several senses. One step is to allow disconnected Voronoi regions. We assume that in a diagram of a subset of three sites each Voronoi region may consist of at most s connected components, for a constant s, and show that the diagram can be constructed in expected time O(s2 n ∑3 ≤ j ≤ n mj / j), where mj is the expected number of connected components of a Voronoi region over all diagrams of a subset of j sites. The case that all Voronoi regions are connected is a subcase, where this algorithm performs in optimal O(n log n) time, because here s = mj =1. The next step is to additionally allow bisecting curves to be closed. We present an algorithm constructing such diagrams which runs in expected time O(s2 n log(max{s,n}) ∑2 ≤ j≤ n mj / j). This algorithm is slower by a log n-factor compared to the one for disconnected regions and unbounded bisectors. The extra time is necessary to be able to handle special phenomenons like islands, where a Voronoi region is completely surrounded by another region, something that can occur only when bisectors are closed. However, this algorithm solves many open problems and improves the running time of some existing algorithms, for example for the farthest Voronoi diagram of n simple polygons of constant complexity. Another challenge was to study higher order abstract Voronoi diagrams. In the concrete sense of an order-k Voronoi diagram points are collected in the same Voronoi region, if they have the same k nearest sites. By suitably intersecting the dominance regions this can be defined also for abstract Voronoi diagrams. The question arising is about the complexity of an order-k Voronoi diagram. There are many subsets of size k but fortunately many of them have an empty order-k region. For point sites it has already been shown that there can be at most O(k (n-k)) many regions and even though order-k regions may be disconnected when considering line segments, still the complexity of the order-k diagram remains O(k(n-k)). The proofs used to show this strongly depended on the geometry of the sites and the distance measure, and were thus not applicable for our abstract higher order Voronoi diagrams. The proofs used to show this strongly depended on the geometry of the sites and the distance measure, and were thus not applicable for our abstract higher order Voronoi diagrams. Nevertheless, we were able to come up with proofs of purely topological and combinatorial nature of Jordan curves and certain permutation sequences, and hence we could show that also the order-k abstract Voronoi diagram has complexity O(k (n-k)), assuming that bisectors are unbounded, and the order-1 regions are connected. Finally, we discuss Voronoi diagrams having the shape of a tree or forest. Aggarwal et. al. showed that if points are in convex position, then given their ordering along the convex hull, their Voronoi diagram, which is a tree, can be computed in linear time. Klein and Lingas have generalized this idea to Hamiltonian abstract Voronoi diagrams, where a curve is given, intersecting each Voronoi region with respect to any subset of sites exactly once. If the ordering of the regions along the curve is known in advance, all Voronoi regions are connected, and all bisectors are unbounded, then the abstract Voronoi diagram can be computed in linear time. This algorithm also applies to diagrams which are trees for all subsets of sites and the ordering of the unbounded regions around the diagram is known. In this thesis we go one step further and allow the diagram to be a forest for subsets of sites as long as the complete diagram is a tree. We show that also these diagrams can be computed in linear time

Approximation algorithms for multi-facility location

This thesis deals with the development and implementation of efficient algorithms to obtain acceptable solutions for the location of several facilities to serve customer sites. The general version of facility location problem is known to be NP-hard; For locating multiple facilities we use Voronoi diagram of initial facility locations to partition the customer sites into k clusters. On each Voronoi region, solutions for single facility problem is obtained by using both Weizfield\u27s algorithm and Center of Gravity. The customer space is again partitioned by using the newly computed locations. This iteration is continued to obtain a better solution for multi-facility location problem. We call the resulting algorithm: Voronoi driven k-median algorithm ; We report experimental results on several test data that include randomly distributed customers and distinctly clustered customers. The observed results show that the proposed approximation algorithm produces good results

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

A GRASP-Tabu Heuristic Approach to Territory Design for Pickup and Delivery Operations for Large-Scale Instances

We address a logistics districting problem faced by a parcel company whose operations consist of picking up and delivering packages over a service region. The districting process aims to find a partition of the service region into delivery and collection zones that may be served by a single vehicle that departs from a central depot. Criteria to be optimized are to balance workload content among the districts and to create districts of compact shape. A solution approach based on a hybrid procedure that combines elements of GRASP and Tabu Search (TS) is proposed to solve large-scale instances. Numerical experimentation is performed considering different instance sizes and types. Results show that the proposed solution approach is able to solve large-scale instances in reasonable computational times with good quality of the solutions obtained. To determine the quality of the solutions, results are compared with CPLEX solutions and with the current real solution to highlight the benefits of the proposed approach. Conclusions and recommendations for further research are provided

Man-made Surface Structures from Triangulated Point Clouds

Photogrammetry aims at reconstructing shape and dimensions of objects captured with cameras, 3D laser scanners or other spatial acquisition systems. While many acquisition techniques deliver triangulated point clouds with millions of vertices within seconds, the interpretation is usually left to the user. Especially when reconstructing man-made objects, one is interested in the underlying surface structure, which is not inherently present in the data. This includes the geometric shape of the object, e.g. cubical or cylindrical, as well as corresponding surface parameters, e.g. width, height and radius. Applications are manifold and range from industrial production control to architectural on-site measurements to large-scale city models. The goal of this thesis is to automatically derive such surface structures from triangulated 3D point clouds of man-made objects. They are defined as a compound of planar or curved geometric primitives. Model knowledge about typical primitives and relations between adjacent pairs of them should affect the reconstruction positively. After formulating a parametrized model for man-made surface structures, we develop a reconstruction framework with three processing steps: During a fast pre-segmentation exploiting local surface properties we divide the given surface mesh into planar regions. Making use of a model selection scheme based on minimizing the description length, this surface segmentation is free of control parameters and automatically yields an optimal number of segments. A subsequent refinement introduces a set of planar or curved geometric primitives and hierarchically merges adjacent regions based on their joint description length. A global classification and constraint parameter estimation combines the data-driven segmentation with high-level model knowledge. Therefore, we represent the surface structure with a graphical model and formulate factors based on likelihood as well as prior knowledge about parameter distributions and class probabilities. We infer the most probable setting of surface and relation classes with belief propagation and estimate an optimal surface parametrization with constraints induced by inter-regional relations. The process is specifically designed to work on noisy data with outliers and a few exceptional freeform regions not describable with geometric primitives. It yields full 3D surface structures with watertightly connected surface primitives of different types. The performance of the proposed framework is experimentally evaluated on various data sets. On small synthetically generated meshes we analyze the accuracy of the estimated surface parameters, the sensitivity w.r.t. various properties of the input data and w.r.t. model assumptions as well as the computational complexity. Additionally we demonstrate the flexibility w.r.t. different acquisition techniques on real data sets. The proposed method turns out to be accurate, reasonably fast and little sensitive to defects in the data or imprecise model assumptions.Künstliche Oberflächenstrukturen aus triangulierten Punktwolken Ein Ziel der Photogrammetrie ist die Rekonstruktion der Form und Größe von Objekten, die mit Kameras, 3D-Laserscannern und anderern räumlichen Erfassungssystemen aufgenommen wurden. Während viele Aufnahmetechniken innerhalb von Sekunden triangulierte Punktwolken mit Millionen von Punkten liefern, ist deren Interpretation gewöhnlicherweise dem Nutzer überlassen. Besonders bei der Rekonstruktion künstlicher Objekte (i.S.v. engl. man-made = „von Menschenhand gemacht“ ist man an der zugrunde liegenden Oberflächenstruktur interessiert, welche nicht inhärent in den Daten enthalten ist. Diese umfasst die geometrische Form des Objekts, z.B. quaderförmig oder zylindrisch, als auch die zugehörigen Oberflächenparameter, z.B. Breite, Höhe oder Radius. Die Anwendungen sind vielfältig und reichen von industriellen Fertigungskontrollen über architektonische Raumaufmaße bis hin zu großmaßstäbigen Stadtmodellen. Das Ziel dieser Arbeit ist es, solche Oberflächenstrukturen automatisch aus triangulierten Punktwolken von künstlichen Objekten abzuleiten. Sie sind definiert als ein Verbund ebener und gekrümmter geometrischer Primitive. Modellwissen über typische Primitive und Relationen zwischen Paaren von ihnen soll die Rekonstruktion positiv beeinflussen. Nachdem wir ein parametrisiertes Modell für künstliche Oberflächenstrukturen formuliert haben, entwickeln wir ein Rekonstruktionsverfahren mit drei Verarbeitungsschritten: Im Rahmen einer schnellen Vorsegmentierung, die lokale Oberflächeneigenschaften berücksichtigt, teilen wir die gegebene vermaschte Oberfläche in ebene Regionen. Unter Verwendung eines Schemas zur Modellauswahl, das auf der Minimierung der Beschreibungslänge beruht, ist diese Oberflächensegmentierung unabhängig von Kontrollparametern und liefert automatisch eine optimale Anzahl an Regionen. Eine anschließende Verbesserung führt eine Menge von ebenen und gekrümmten geometrischen Primitiven ein und fusioniert benachbarte Regionen hierarchisch basierend auf ihrer gemeinsamen Beschreibungslänge. Eine globale Klassifikation und bedingte Parameterschätzung verbindet die datengetriebene Segmentierung mit hochrangigem Modellwissen. Dazu stellen wir die Oberflächenstruktur in Form eines graphischen Modells dar und formulieren Faktoren basierend auf der Likelihood sowie auf apriori Wissen über die Parameterverteilungen und Klassenwahrscheinlichkeiten. Wir leiten die wahrscheinlichste Konfiguration von Flächen- und Relationsklassen mit Hilfe von Belief-Propagation ab und schätzen eine optimale Oberflächenparametrisierung mit Bedingungen, die durch die Relationen zwischen benachbarten Primitiven induziert werden. Der Prozess ist eigens für verrauschte Daten mit Ausreißern und wenigen Ausnahmeregionen konzipiert, die nicht durch geometrische Primitive beschreibbar sind. Er liefert wasserdichte 3D-Oberflächenstrukturen mit Oberflächenprimitiven verschiedener Art. Die Leistungsfähigkeit des vorgestellten Verfahrens wird an verschiedenen Datensätzen experimentell evaluiert. Auf kleinen, synthetisch generierten Oberflächen untersuchen wir die Genauigkeit der geschätzten Oberflächenparameter, die Sensitivität bzgl. verschiedener Eigenschaften der Eingangsdaten und bzgl. Modellannahmen sowie die Rechenkomplexität. Außerdem demonstrieren wir die Flexibilität bzgl. verschiedener Aufnahmetechniken anhand realer Datensätze. Das vorgestellte Rekonstruktionsverfahren erweist sich als genau, hinreichend schnell und wenig anfällig für Defekte in den Daten oder falsche Modellannahmen

Machining accuracy enhancement using machine tool error compensation and metrology

This dissertation aims to enhance machining accuracy by machine tool error reduction and workpiece metrology. The error characteristics are studied by building a quasi-static error model. Perturbed forward kinematic model is used for modeling a 5-axis Computer Numerical Control (CNC) machine with one redundant linear axis. It is found that the 1st order volumetric error model of the 5-axis machine is attributed to 32 error parameter groups. To identify the model by estimating these parameter groups using the least-squares fitting, errors at 290 quasi-randomly generated measurement points over the machine’s workspace are measured using a laser tracker. The identified error model explains 90% of the mean error of the training data sets. However, the measurements using the laser tracker take about 90 minutes, which may cause the identified error parameters to be inaccurate due to the slow varying and transient natures of thermal errors. To shorten the measurement time, an experimental design approach, which suggests the optimal observation locations such that the corresponding robustness of identification is maximized, is applied to design the optimal error observers. Since the observers must be uniformly distributed over the workspace for gaining redundancy, the constrained K-optimal designs are used to select 80 K-optimal observers for the 5-axis machine. Six measurement cycles using 80 observers are done at machine’s different thermal states within a 400-minute experiment. Six error models are trained with consistent performances and are found to be comparable to the one trained by 290 quasi-random observations. This shows the feasibility of using smaller but more strategical-chosen point-set in data-driven error models. More importantly, the growth on mean nominal (119.1 to 181.9 microns) and modeled error (26.3 to 33.9 microns) suggest the necessity of thermal error tracking for enhancing the machining accuracy. A point-set based metrology is also developed to compensate the inaccuracies introduced by workpiece and fixtures and enhance machining accuracy. The machinability of all planar features is examined by virtually comparing the scanned data with the nominal machining planes, which are also known as virtual gages. The virtual gaging problem is modeled as a constrained linear program. The optimal solution to the problem can compensate the displacement introduced by workpiece and fixtures and hence guarantee a conforming finished part. To transfer point-set data into mathematical constraints, algorithms that align, segment, downsize and filter the point-set data are exploited. The concept of virtual gage analysis is demonstrated using experimental data for a simple raw casting. However, for the case where the casting is defective, and some virtual gages are not feasible, the corresponding linear program was found to have no solution. By introducing slack variables to the original linear programming problem, the extended problem has been solved. The extended model is validated for the data obtained for another casting. Further, the analysis predicts the machining allowances on all functional features. Cylindrical surface and its tolerance verification play important role in machining process. Although there exist many approaches that can fit the maximum, minimum and minimum zone cylinders, the cylinder fitting problems can be even simplified. The proposed methodology seeks to reduce the number of parameters used in cylinder fitting model by using the projection model that considers the degenerated tolerance specifications of the projected 2-D point-set. Also, to avoid the problem of local optimum by introducing the optimal direction of projection such that the 2-D point projected onto this direction has optimal tolerance specifications (maximum, minimum and minimum zone circles), global optimum solver such as Particle Swarm Optimization (PSO) is used. The proposed simplified method shows consistent results compared with the results from literature

Spatial Optimization Methods And System For Redistricting Problems

Redistricting is the process of dividing space into districts or zones while optimizing a set of spatial criteria under certain constraints. Example applications of redistricting include political redistricting, school redistricting, business service planning, and city management, among many others. Redistricting is a mission-critical component in operating governments and businesses alike. In research fields, redistricting (or region building) are also widely used, such as climate zoning, traffic zone analysis, and complex network analysis. However, as a combinatorial optimization problem, redistricting optimization remains one of the most difficult research challenges. There are currently few automated redistricting methods that have the optimization capability to produce solutions that meet practical needs. The absence of effective and efficient computational approaches for redistricting makes it extremely time-consuming and difficult for an individual person to consider multiple criteria/constraints and manually create solutions using a trial-and-error approach. To address both the scientific and practical challenges in solving real-world redistricting problems, this research advances the methodology and application of redistricting by developing a new computational spatial optimization method and a system platform that can address a wide range of redistricting problems, in an automated and computation-assisted manner. The research has three main contributions. First, an efficient and effective spatial optimization method is developed for redistricting. The new method is based on a spatially constrained and Tabu-based heuristics, which can optimize multiple criteria under multiple constraints to construct high-quality optimization solutions. The new approach is evaluated with real-world redistricting applications and compared with existing methods. Evaluation results show that the new optimization algorithm is more efficient (being able to allow real-time user interaction), more flexible (considering multiple user-expressed criteria and constraints), and more powerful (in terms of optimization quality) than existing methods. As such, it has the potential to enable general users to perform complex redistricting tasks. Second, a redistricting system, iRedistrict, is developed based on the newly developed spatial optimization method to provide user-friendly visual interface for defining redistricting problems, incorporating domain knowledge, configuring optimization criteria and methodology parameters, and ultimately meeting the needs of real-world applications for tackling complex redistricting tasks. It is particularly useful for users of different skill levels, including researchers, practitioners, and the general public, and thus enables public participation in challenging redistricting tasks that are of immense public interest. Performance evaluations with real-world case studies are carried out. Further computational strategies are developed and implemented to handle large datasets. Third, the newly developed spatial optimization method is extended to solve a different spatial optimization problem, i.e., spatial community structure detection in complex networks, which is to partition networks to discover spatial communities by optimizing an objective function. Moreover, a series of new evaluations are carried out with synthetic datasets. This set of evaluations is different from the previous evaluations with case studies in that, the optimal solution is known with synthetic data and therefore it is possible to evaluate (1) whether the optimization method can discover the true pattern (global optima), and (2) how different data characteristics may affect the performance of the method. Evaluation results reveal that existing non-spatial methods are not robust in detecting spatial community structure, which may produce dramatically different outcomes for the same data with different characteristics, such as different spatial aggregations, sampling rates, or noise levels. The new optimization method with spatial constraints is significantly more stable and consistent. In addition to evaluations with synthetic datasets, a case study is also carried out to detect urban community structure with human movements, to demonstrate the application and effectiveness of the approach

Geospatial Analysis and Modeling of Textual Descriptions of Pre-modern Geography

Textual descriptions of pre-modern geography offer a different view of classical geography. The descriptions have been produced when none of the modern geographical concepts and tools were available. In this dissertation, we study pre-modern geography by primarily finding the existing structures of the descriptions and different cases of geographical data. We first explain four major geographical cases in pre-modern Arabic sources: gazetteer, administrative hierarchies, routes, and toponyms associated with people. Focusing on hierarchical divisions and routes, we offer approaches for manual annotation of administrative hierarchies and route sections as well as a semi-automated toponyms annotation. The latter starts with a fuzzy search of toponyms from an authority list and applies two different extrapolation models to infer true or false values, based on the context, for disambiguating the automatically annotated toponyms. Having the annotated data, we introduce mathematical models to shape and visualize regions based on the description of administrative hierarchies. Moreover, we offer models for comparing hierarchical divisions and route networks from different sources. We also suggest approaches to approximate geographical coordinates for places that do not have geographical coordinates - we call them unknown places - which is a major issue in visualization of pre-modern places on map. The final chapter of the dissertation introduces the new version of al-Ṯurayyā, a gazetteer and a spatial model of the classical Islamic world using georeferenced data of a pre-modern atlas with more than 2, 000 toponyms and routes. It offers search, path finding, and flood network functionalities as well as visualizations of regions using one of the models that we describe for regions. However the gazetteer is designed using the classical Islamic world data, the spatial model and features can be used for similarly prepared datasets.:1 Introduction 1 2 Related Work 8 2.1 GIS 8 2.2 NLP, Georeferencing, Geoparsing, Annotation 10 2.3 Gazetteer 15 2.4 Modeling 17 3 Classical Geographical Cases 20 3.1 Gazetteer 21 3.2 Routes and Travelogues 22 3.3 Administrative Hierarchy 24 3.4 Geographical Aspects of Biographical Data 25 4 Annotation and Extraction 27 4.1 Annotation 29 4.1.1 Manual Annotation of Geographical Texts 29 4.1.1.1 Administrative Hierarchy 30 4.1.1.2 Routes and Travelogues 32 4.1.2 Semi-Automatic Toponym Annotation 34 4.1.2.1 The Annotation Process 35 4.1.2.2 Extrapolation Models 37 4.1.2.2.1 Frequency of Toponymic N-grams 37 4.1.2.2.2 Co-occurrence Frequencies 38 4.1.2.2.3 A Supervised ML Approach 40 4.1.2.3 Summary 45 4.2 Data Extraction and Structures 45 4.2.1 Administrative Hierarchy 45 4.2.2 Routes and Distances 49 5 Modeling Geographical Data 51 5.1 Mathematical Models for Administrative Hierarchies 52 5.1.1 Sample Data 53 5.1.2 Quadtree 56 5.1.3 Voronoi Diagram 58 5.1.4 Voronoi Clippings 62 5.1.4.1 Convex Hull 62 5.1.4.2 Concave Hull 63 5.1.5 Convex Hulls 65 5.1.6 Concave Hulls 67 5.1.7 Route Network 69 5.1.8 Summary of Models for Administrative Hierarchy 69 5.2 Comparison Models 71 5.2.1 Hierarchical Data 71 5.2.1.1 Test Data 73 5.2.2 Route Networks 76 5.2.2.1 Post-processing 81 5.2.2.2 Applications 82 5.3 Unknown Places 84 6 Al-Ṯurayyā 89 6.1 Introducing al-Ṯurayyā 90 6.2 Gazetteer 90 6.3 Spatial Model 91 6.3.1 Provinces and Administrative Divisions 93 6.3.2 Pathfinding and Itineraries 93 6.3.3 Flood Network 96 6.3.4 Path Alignment Tool 97 6.3.5 Data Structure 99 6.3.5.1 Places 100 6.3.5.2 Routes and Distances 100 7 Conclusions and Further Work 10

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version