13 research outputs found

    Towards Characterizing Graphs with a Sliceable Rectangular Dual

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    \u3cp\u3eLet G be a plane triangulated graph. A rectangular dual of G is a partition of a rectangle R into a set R of interior-disjoint rectangles, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge. A rectangular dual is sliceable if it can be recursively subdivided along horizontal or vertical lines. A graph is rectangular if it has a rectangular dual and sliceable if it has a sliceable rectangular dual. There is a clear characterization of rectangular graphs. However, a full characterization of sliceable graphs is still lacking. The currently best result (Yeap and Sarrafzadeh, 1995) proves that all rectangular graphs without a separating 4-cycle are slice- able. In this paper we introduce a recursively defined class of graphs and prove that these graphs are precisely the nonsliceable graphs with exactly one separating 4-cycle.\u3c/p\u3

    Drawing graphs for cartographic applications

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    Graph Drawing is a relatively young area that combines elements of graph theory, algorithms, (computational) geometry and (computational) topology. Research in this field concentrates on developing algorithms for drawing graphs while satisfying certain aesthetic criteria. These criteria are often expressed in properties like edge complexity, number of edge crossings, angular resolutions, shapes of faces or graph symmetries and in general aim at creating a drawing of a graph that conveys the information to the reader in the best possible way. Graph drawing has applications in a wide variety of areas which include cartography, VLSI design and information visualization. In this thesis we consider several graph drawing problems. The first problem we address is rectilinear cartogram construction. A cartogram, also known as value-by-area map, is a technique used by cartographers to visualize statistical data over a set of geographical regions like countries, states or counties. The regions of a cartogram are deformed such that the area of a region corresponds to a particular geographic variable. The shapes of the regions depend on the type of cartogram. We consider rectilinear cartograms of constant complexity, that is cartograms where each region is a rectilinear polygon with a constant number of vertices. Whether a cartogram is good is determined by how closely the cartogram resembles the original map and how precisely the area of its regions describe the associated values. The cartographic error is defined for each region as jAc¡Asj=As, where Ac is the area of the region in the cartogram and As is the specified area of that region, given by the geographic variable to be shown. In this thesis we consider the construction of rectilinear cartograms that have correct adjacencies of the regions and zero cartographic error. We show that any plane triangulated graph admits a rectilinear cartogram where every region has at most 40 vertices which can be constructed in O(nlogn) time. We also present experimental results that show that in practice the algorithm works significantly better than suggested by the complexity bounds. In our experiments on real-world data we were always able to construct a cartogram where the average number of vertices per region does not exceed five. Since a rectangle has four vertices, this means that most of the regions of our rectilinear car tograms are in fact rectangles. Moreover, the maximum number vertices of each region in these cartograms never exceeded ten. The second problem we address in this thesis concerns cased drawings of graphs. The vertices of a drawing are commonly marked with a disk, but differentiating between vertices and edge crossings in a dense graph can still be difficult. Edge casing is a wellknown method—used, for example, in electrical drawings, when depicting knots, and, more generally, in information visualization—to alleviate this problem and to improve the readability of a drawing. A cased drawing orders the edges of each crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. One can also envision that every edge is encased in a strip of the background color and that the casing of the upper edge covers the lower edge at the crossing. If there are no application-specific restrictions that dictate the order of the edges at each crossing, then we can in principle choose freely how to arrange them. However, certain orders will lead to a more readable drawing than others. In this thesis we formulate aesthetic criteria for a cased drawing as optimization problems and solve these problems. For most of the problems we present either a polynomial time algorithm or demonstrate that the problem is NP-hard. Finally we consider a combinatorial question in computational topology concerning three types of objects: closed curves in the plane, surfaces immersed in the plane, and surfaces embedded in space. In particular, we study casings of closed curves in the plane to decide whether these curves can be embedded as the boundaries of certain special surfaces. We show that it is NP-complete to determine whether an immersed disk is the projection of a surface embedded in space, or whether a curve is the boundary of an immersed surface in the plane that is not constrained to be a disk. However, when a casing is supplied with a self-intersecting curve, describing which component of the curve lies above and which below at each crossing, we can determine in time linear in the number of crossings whether the cased curve forms the projected boundary of a surface in space. As a related result, we show that an immersed surface with a single boundary curve that crosses itself n times has at most 2n=2 combinatorially distinct spatial embeddings and we discuss the existence of fixed-parameter tractable algorithms for related problems

    Steinitz Theorems for Orthogonal Polyhedra

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    We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

    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

    Visualization of dynamic multidimensional and hierarchical datasets

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    When it comes to tools and techniques designed to help understanding complex abstract data, visualization methods play a prominent role. They enable human operators to lever age their pattern finding, outlier detection, and questioning abilities to visually reason about a given dataset. Many methods exist that create suitable and useful visual represen tations of static abstract, non-spatial, data. However, for temporal abstract, non-spatial, datasets, in which the data changes and evolves through time, far fewer visualization tech niques exist. This thesis focuses on the particular cases of temporal hierarchical data representation via dynamic treemaps, and temporal high-dimensional data visualization via dynamic projec tions. We tackle the joint question of how to extend projections and treemaps to stably, accurately, and scalably handle temporal multivariate and hierarchical data. The literature for static visualization techniques is rich and the state-of-the-art methods have proven to be valuable tools in data analysis. Their temporal/dynamic counterparts, however, are not as well studied, and, until recently, there were few hierarchical and high-dimensional methods that explicitly took into consideration the temporal aspect of the data. In addi tion, there are few or no metrics to assess the quality of these temporal mappings, and even fewer comprehensive benchmarks to compare these methods. This thesis addresses the abovementioned shortcomings. For both dynamic treemaps and dynamic projections, we propose ways to accurately measure temporal stability; we eval uate existing methods considering the tradeoff between stability and visual quality; and we propose new methods that strike a better balance between stability and visual quality than existing state-of-the-art techniques. We demonstrate our methods with a wide range of real-world data, including an application of our new dynamic projection methods to support the analysis and classification of hyperkinetic movement disorder data.Quando se trata de ferramentas e técnicas projetadas para ajudar na compreensão dados abstratos complexos, métodos de visualização desempenham um papel proeminente. Eles permitem que os operadores humanos alavanquem suas habilidades de descoberta de padrões, detecção de valores discrepantes, e questionamento visual para a raciocinar sobre um determinado conjunto de dados. Existem muitos métodos que criam representações visuais adequadas e úteis de para dados estáticos, abstratos, e não-espaciais. No entanto, para dados temporais, abstratos, e não-espaciais, isto é, dados que mudam e evoluem no tempo, existem poucas técnicas apropriadas. Esta tese concentra-se nos casos específicos de representação temporal de dados hierárquicos por meio de treemaps dinâmicos, e visualização temporal de dados de alta dimen sionalidade via projeções dinâmicas. Nós abordar a questão conjunta de como estender projeções e treemaps de forma estável, precisa e escalável para lidar com conjuntos de dados hierárquico-temporais e multivariado-temporais. Em ambos os casos, a literatura para técnicas estáticas é rica e os métodos estado da arte provam ser ferramentas valiosas em análise de dados. Suas contrapartes temporais/dinâmicas, no entanto, não são tão bem estudadas e, até recentemente, existiam poucos métodos hierárquicos e de alta dimensão que explicitamente levavam em consideração o aspecto temporal dos dados. Além disso, existiam poucas métricas para avaliar a qualidade desses mapeamentos visuais temporais, e ainda menos benchmarks abrangentes para comparação esses métodos. Esta tese aborda as deficiências acima mencionadas para treemaps dinâmicos e projeções dinâmicas. Propomos maneiras de medir com precisão a estabilidade temporal; avalia mos os métodos existentes, considerando o compromisso entre estabilidade e qualidade visual; e propomos novos métodos que atingem um melhor equilíbrio entre estabilidade e a qualidade visual do que as técnicas estado da arte atuais. Demonstramos nossos mé todos com uma ampla gama de dados do mundo real, incluindo uma aplicação de nossos novos métodos de projeção dinâmica para apoiar a análise e classificação dos dados de transtorno de movimentos

    Machine Learning for Multi-Layer Open and Disaggregated Optical Networks

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    L'abstract è presente nell'allegato / the abstract is in the attachmen

    27th Annual European Symposium on Algorithms: ESA 2019, September 9-11, 2019, Munich/Garching, Germany

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    Enabling Technology in Optical Fiber Communications: From Device, System to Networking

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    This book explores the enabling technology in optical fiber communications. It focuses on the state-of-the-art advances from fundamental theories, devices, and subsystems to networking applications as well as future perspectives of optical fiber communications. The topics cover include integrated photonics, fiber optics, fiber and free-space optical communications, and optical networking

    Tools and Algorithms for the Construction and Analysis of Systems

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    This open access two-volume set constitutes the proceedings of the 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2021, which was held during March 27 – April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The total of 41 full papers presented in the proceedings was carefully reviewed and selected from 141 submissions. The volume also contains 7 tool papers; 6 Tool Demo papers, 9 SV-Comp Competition Papers. The papers are organized in topical sections as follows: Part I: Game Theory; SMT Verification; Probabilities; Timed Systems; Neural Networks; Analysis of Network Communication. Part II: Verification Techniques (not SMT); Case Studies; Proof Generation/Validation; Tool Papers; Tool Demo Papers; SV-Comp Tool Competition Papers

    Development and encoding of visual statistics in the primary visual cortex

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    How do circuits in the mammalian cerebral cortex encode properties of the sensory environment in a way that can drive adaptive behavior? This question is fundamental to neuroscience, but it has been very difficult to approach directly. Various computational and theoretical models can explain a wide range of phenomena observed in the primary visual cortex (V1), including the anatomical organization of its circuits, the development of functional properties like orientation tuning, and behavioral effects like surround modulation. However, so far no model has been able to bridge these levels of description to explain how the machinery that develops directly affects behavior. Bridging these levels is important, because phenomena at any one specific level can have many possible explanations, but there are far fewer possibilities to consider once all of the available evidence is taken into account. In this thesis we integrate the information gleaned about cortical development, circuit and cell-type specific interactions, and anatomical, behavioral and electrophysiological measurements, to develop a computational model of V1 that is constrained enough to make predictions across multiple levels of description. Through a series of models incorporating increasing levels of biophysical detail and becoming increasingly better constrained, we are able to make detailed predictions for the types of mechanistic interactions required for robust development of cortical maps that have a realistic anatomical organization, and thereby gain insight into the computations performed by the primary visual cortex. The initial models focus on how existing anatomical and electrophysiological knowledge can be integrated into previously abstract models to give a well-grounded and highly constrained account of the emergence of pattern-specific tuning in the primary visual cortex. More detailed models then address the interactions between specific excitatory and inhibitory cell classes in V1, and what role each cell type may play during development and function. Finally, we demonstrate how these cell classes come together to form a circuit that gives rise not only to robust development but also the development of realistic lateral connectivity patterns. Crucially, these patterns reflect the statistics of the visual environment to which the model was exposed during development. This property allows us to explore how the model is able to capture higher-order information about the environment and use that information to optimize neural coding and aid the processing of complex visual tasks. Using this model we can make a number of very specific predictions about the mechanistic workings of the brain. Specifically, the model predicts a crucial role of parvalbumin-expressing interneurons in robust development and divisive normalization, while it implicates somatostatin immunoreactive neurons in mediating longer range and feature-selective suppression. The model also makes predictions about the role of these cell classes in efficient neural coding and under what conditions the model fails to organize. In particular, we show that a tight coupling of activity between the principal excitatory population and the parvalbumin population is central to robust and stable responses and organization, which may have implications for a variety of diseases where parvalbumin interneuron function is impaired, such as schizophrenia and autism. Further the model explains the switch from facilitatory to suppressive surround modulation effects as a simple by-product of the facilitating response function of long-range excitatory connections targeting a specialized class of inhibitory interneurons. Finally, the model allows us to make predictions about the statistics that are encoded in the extensive network of long-range intra-areal connectivity in V1, suggesting that even V1 can capture high-level statistical dependencies in the visual environment. The final model represents a comprehensive and well constrained model of the primary visual cortex, which for the first time can relate the physiological properties of individual cell classes to their role in development, learning and function. While the model is specifically tuned for V1, all mechanisms introduced are completely general, and can be used as a general cortical model, useful for studying phenomena across the visual cortex and even the cortex as a whole. This work is also highly relevant for clinical neuroscience, as the cell types studied here have been implicated in neurological disorders as wide ranging as autism, schizophrenia and Parkinson’s disease
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