35 research outputs found

    Disjoint paths in geometric graphs

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    Construction of the shortest paths connecting two given nodes in a geometric graph is the quintessential problem in computational geometry. We consider several variations of this problem with applications that include robotics, Geographic Information Systems, and sensor networks. The first problem we address is the development of an efficient algorithm for constructing a pair of short node-disjoint paths connecting start and target nodes. The second problem investigated is the development of efficient algorithms for constructing narrow and in-range broadcast corridors in triangulated geometric graphs. Finally, we consider the development of an approximation algorithm for constructing reduced overlap trees in three-colored geometric graphs. Theoretical analysis and a detailed experimental investigation of the proposed algorithms are also presented

    Building Blocks for Mapping Services

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    Mapping services are ubiquitous on the Internet. These services enjoy a considerable user base. But it is often overlooked that providing a service on a global scale with virtually millions of users has been the playground of an oligopoly of a select few service providers are able to do so. Unfortunately, the literature on these solutions is more than scarce. This thesis adds a number of building blocks to the literature that explain how to design and implement a number of features

    Collection of abstracts of the 24th European Workshop on Computational Geometry

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    International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    In pursuit of linear complexity in discrete and computational geometry

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    Many computational problems arise naturally from geometric data. In this thesis, we consider three such problems: (i) distance optimization problems over point sets, (ii) computing contour trees over simplicial meshes, and (iii) bounding the expected complexity of weighted Voronoi diagrams. While these topics are broad, here the focus is on identifying structure which implies linear (or near linear) algorithmic and descriptive complexity. The first topic we consider is in geometric optimization. More specifically, we define a large class of distance problems, for which we provide linear time exact or approximate solutions. Roughly speaking, the class of problems facilitate either clustering together close points (i.e. netting) or throwing out outliers (i.e pruning), allowing for successively smaller summaries of the relevant information in the input. A surprising number of classical geometric optimization problems are unified under this framework, including finding the optimal k-center clustering, the kth ranked distance, the kth heaviest edge of the MST, the minimum radius ball enclosing k points, and many others. In several cases we get the first known linear time approximation algorithm for a given problem, where our approximation ratio matches that of previous work. The second topic we investigate is contour trees, a fundamental structure in computational topology. Contour trees give a compact summary of the evolution of level sets on a mesh, and are typically used on massive data sets. Previous algorithms for computing contour trees took Θ(n log n) time and were worst-case optimal. Here we provide an algorithm whose running time lies between Θ(nα(n)) and Θ(n log n), and varies depending on the shape of the tree, where α(n) is the inverse Ackermann function. In particular, this is the first algorithm with O(nα(n)) running time on instances with balanced contour trees. Our algorithmic results are complemented by lower bounds indicating that, up to a factor of α(n), on all instance types our algorithm performs optimally. For the final topic, we consider the descriptive complexity of weighted Voronoi diagrams. Such diagrams have quadratic (or higher) worst-case complexity, however, as was the case for contour trees, here we push beyond worst-case analysis. A new diagram, called the candidate diagram, is introduced, which allows us to bound the complexity of weighted Voronoi diagrams arising from a particular probabilistic input model. Specifically, we assume weights are randomly permuted among fixed Voronoi sites, an assumption which is weaker than the more typical sampled locations assumption. Under this assumption, the expected complexity is shown to be near linear

    Large bichromatic point sets admit empty monochromatic 4-gons

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    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

    Searching and mining in enriched geo-spatial data

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    The emergence of new data collection mechanisms in geo-spatial applications paired with a heightened tendency of users to volunteer information provides an ever-increasing flow of data of high volume, complex nature, and often associated with inherent uncertainty. Such mechanisms include crowdsourcing, automated knowledge inference, tracking, and social media data repositories. Such data bearing additional information from multiple sources like probability distributions, text or numerical attributes, social context, or multimedia content can be called multi-enriched. Searching and mining this abundance of information holds many challenges, if all of the data's potential is to be released. This thesis addresses several major issues arising in that field, namely path queries using multi-enriched data, trend mining in social media data, and handling uncertainty in geo-spatial data. In all cases, the developed methods have made significant contributions and have appeared in or were accepted into various renowned international peer-reviewed venues. A common use of geo-spatial data is path queries in road networks where traditional methods optimise results based on absolute and ofttimes singular metrics, i.e., finding the shortest paths based on distance or the best trade-off between distance and travel time. Integrating additional aspects like qualitative or social data by enriching the data model with knowledge derived from sources as mentioned above allows for queries that can be issued to fit a broader scope of needs or preferences. This thesis presents two implementations of incorporating multi-enriched data into road networks. In one case, a range of qualitative data sources is evaluated to gain knowledge about user preferences which is subsequently matched with locations represented in a road network and integrated into its components. Several methods are presented for highly customisable path queries that incorporate a wide spectrum of data. In a second case, a framework is described for resource distribution with reappearance in road networks to serve one or more clients, resulting in paths that provide maximum gain based on a probabilistic evaluation of available resources. Applications for this include finding parking spots. Social media trends are an emerging research area giving insight in user sentiment and important topics. Such trends consist of bursts of messages concerning a certain topic within a time frame, significantly deviating from the average appearance frequency of the same topic. By investigating the dissemination of such trends in space and time, this thesis presents methods to classify trend archetypes to predict future dissemination of a trend. Processing and querying uncertain data is particularly demanding given the additional knowledge required to yield results with probabilistic guarantees. Since such knowledge is not always available and queries are not easily scaled to larger datasets due to the #P-complete nature of the problem, many existing approaches reduce the data to a deterministic representation of its underlying model to eliminate uncertainty. However, data uncertainty can also provide valuable insight into the nature of the data that cannot be represented in a deterministic manner. This thesis presents techniques for clustering uncertain data as well as query processing, that take the additional information from uncertainty models into account while preserving scalability using a sampling-based approach, while previous approaches could only provide one of the two. The given solutions enable the application of various existing clustering techniques or query types to a framework that manages the uncertainty.Das Erscheinen neuer Methoden zur Datenerhebung in rĂ€umlichen Applikationen gepaart mit einer erhöhten Bereitschaft der Nutzer, Daten ĂŒber sich preiszugeben, generiert einen stetig steigenden Fluss von Daten in großer Menge, komplexer Natur, und oft gepaart mit inhĂ€renter Unsicherheit. Beispiele fĂŒr solche Mechanismen sind Crowdsourcing, automatisierte Wissensinferenz, Tracking, und Daten aus sozialen Medien. Derartige Daten, angereichert mit mit zusĂ€tzlichen Informationen aus verschiedenen Quellen wie Wahrscheinlichkeitsverteilungen, Text- oder numerische Attribute, sozialem Kontext, oder Multimediainhalten, werden als multi-enriched bezeichnet. Suche und Datamining in dieser weiten Datenmenge hĂ€lt viele Herausforderungen bereit, wenn das gesamte Potenzial der Daten genutzt werden soll. Diese Arbeit geht auf mehrere große Fragestellungen in diesem Feld ein, insbesondere Pfadanfragen in multi-enriched Daten, Trend-mining in Daten aus sozialen Netzwerken, und die Beherrschung von Unsicherheit in rĂ€umlichen Daten. In all diesen FĂ€llen haben die entwickelten Methoden signifikante ForschungsbeitrĂ€ge geleistet und wurden veröffentlicht oder angenommen zu diversen renommierten internationalen, von Experten begutachteten Konferenzen und Journals. Ein gĂ€ngiges Anwendungsgebiet rĂ€umlicher Daten sind Pfadanfragen in Straßennetzwerken, wo traditionelle Methoden die Resultate anhand absoluter und oft auch singulĂ€rer Maße optimieren, d.h., der kĂŒrzeste Pfad in Bezug auf die Distanz oder der beste Kompromiss zwischen Distanz und Reisezeit. Durch die Integration zusĂ€tzlicher Aspekte wie qualitativer Daten oder Daten aus sozialen Netzwerken als Anreicherung des Datenmodells mit aus diesen Quellen abgeleitetem Wissen werden Anfragen möglich, die ein breiteres Spektrum an Anforderungen oder PrĂ€ferenzen erfĂŒllen. Diese Arbeit prĂ€sentiert zwei AnsĂ€tze, solche multi-enriched Daten in Straßennetze einzufĂŒgen. Zum einen wird eine Reihe qualitativer Datenquellen ausgewertet, um Wissen ĂŒber NutzerprĂ€ferenzen zu generieren, welches darauf mit Örtlichkeiten im Straßennetz abgeglichen und in das Netz integriert wird. Diverse Methoden werden prĂ€sentiert, die stark personalisierbare Pfadanfragen ermöglichen, die ein weites Spektrum an Daten mit einbeziehen. Im zweiten Fall wird ein Framework prĂ€sentiert, das eine Ressourcenverteilung im Straßennetzwerk modelliert, bei der einmal verbrauchte Ressourcen erneut auftauchen können. Resultierende Pfade ergeben einen maximalen Ertrag basieren auf einer probabilistischen Evaluation der verfĂŒgbaren Ressourcen. Eine Anwendung ist die Suche nach ParkplĂ€tzen. Trends in sozialen Medien sind ein entstehendes Forscchungsgebiet, das Einblicke in Benutzerverhalten und wichtige Themen zulĂ€sst. Solche Trends bestehen aus großen Mengen an Nachrichten zu einem bestimmten Thema innerhalb eines Zeitfensters, so dass die Auftrittsfrequenz signifikant ĂŒber den durchschnittlichen Level liegt. Durch die Untersuchung der Fortpflanzung solcher Trends in Raum und Zeit prĂ€sentiert diese Arbeit Methoden, um Trends nach Archetypen zu klassifizieren und ihren zukĂŒnftigen Weg vorherzusagen. Die Anfragebearbeitung und Datamining in unsicheren Daten ist besonders herausfordernd, insbesondere im Hinblick auf das notwendige Zusatzwissen, um Resultate mit probabilistischen Garantien zu erzielen. Solches Wissen ist nicht immer verfĂŒgbar und Anfragen lassen sich aufgrund der \P-VollstĂ€ndigkeit des Problems nicht ohne Weiteres auf grĂ¶ĂŸere DatensĂ€tze skalieren. Dennoch kann Datenunsicherheit wertvollen Einblick in die Struktur der Daten liefern, der mit deterministischen Methoden nicht erreichbar wĂ€re. Diese Arbeit prĂ€sentiert Techniken zum Clustering unsicherer Daten sowie zur Anfragebearbeitung, die die Zusatzinformation aus dem Unsicherheitsmodell in Betracht ziehen, jedoch gleichzeitig die Skalierbarkeit des Ansatzes auf große Datenmengen sicherstellen

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum
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