Grid-Obstacle Representations with Connections to Staircase Guarding

In this paper, we study grid-obstacle representations of graphs where we assign grid-points to vertices and define obstacles such that an edge exists if and only if an $xy$-monotone grid path connects the two endpoints without hitting an obstacle or another vertex. It was previously argued that all planar graphs have a grid-obstacle representation in 2D, and all graphs have a grid-obstacle representation in 3D. In this paper, we show that such constructions are possible with significantly smaller grid-size than previously achieved. Then we study the variant where vertices are not blocking, and show that then grid-obstacle representations exist for bipartite graphs. The latter has applications in so-called staircase guarding of orthogonal polygons; using our grid-obstacle representations, we show that staircase guarding is \textsc{NP}-hard in 2D.Comment: To appear in the proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Eco-ISEA3H, a machine learning ready spatial database for ecometric and species distribution modeling

We present the Eco-ISEA3H database, a compilation of global spatial data characterizing climate, geology, land cover, physical and human geography, and the geographic ranges of nearly 900 large mammalian species. The data are tailored for machine learning (ML)-based ecological modeling, and are intended primarily for continental- to global-scale ecometric and species distribution modeling. Such models are trained on present-day data and applied to the geologic past, or to future scenarios of climatic and environmental change. Model training requires integrated global datasets, describing species' occurrence and environment via consistent observational units. The Eco-ISEA3H database incorporates data from 17 sources, and includes 3,033 variables. The database is built on the Icosahedral Snyder Equal Area (ISEA) aperture 3 hexagonal (3H) discrete global grid system (DGGS), which partitions the Earth's surface into equal-area hexagonal cells. Source data were incorporated at six nested ISEA3H resolutions, using scripts developed and made available here. We demonstrate the utility of the database in a case study analyzing the bioclimatic envelopes of ten large, widely distributed mammalian species.Peer reviewe

Steinitz Theorems for Orthogonal Polyhedra

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

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

An Exploratory Data Analysis Approach for Land Use-Transportation Interaction: The Design and Implementation of Transland Spatio-Temporal Data Model

Land use and transportation interaction is a complex and dynamic process. Many models have been used to study this interaction during the last several decades. Empirical studies suggest that land use and transportation patterns can be highly variable between geographic areas and at different spatial and temporal scales. Identifying these changes presents a major challenge. When we recognize that long-term changes could be affected by other factors such as population growth, economic development, and policy decisions, the challenge becomes even more overwhelming. Most existing land use and transportation interaction models are based on some prior theories and use mathematical or simulation approaches to study the problem. However, the literature also suggests that little consensus regarding the conclusions can be drawn from empirical studies that apply these models. There is a clear research need to develop alternative methods that will allow us to examine the land use and transportation patterns in more flexible ways and to help us identify potential improvements to the existing models. This dissertation presents a spatio-temporal data model that offers exploratory data analysis capabilities to interactively examine the land use and transportation interaction at use-specified spatial and temporal scales. The spatio-temporal patterns and the summary statistics derived from this interactive exploratory analysis process can be used to help us evaluate the hypotheses and modify the structures used in the existing models. The results also can suggest additional analyses for a better understanding of land use and transportation interaction. This dissertation first introduces a conceptual framework for the spatio-temporal data model. Then, based on a systematic method for explorations of various data sets relevant to land use and transportation interaction, this dissertation details procedures of designing and implementing the spatio-temporal data model. Finally, the dissertation describes procedures of creating tools for generating the proposed spatio-temporal data model from existing snapshot GIS data sets and illustrate its use by means of exploratory data analysis. Use of the spatio-temporal data model in this dissertation study makes it feasible to analyze spatio-temporal interaction patterns in a more effective and efficient way than the conventional snapshot GIS approach. Extending Sinton’s measurement framework into a spatio-temporal conceptual interaction framework, on the other hand, provides a systematic means of exploring land use and transportation interaction. Preliminary experiments of data collected for Dade County (Miami), Florida suggest that the spatio-temporal exploratory data analysis implemented for this dissertation can help transportation planners identify and visualize interaction patterns of land use and transportation by controlling the spatial, attribute, and temporal components. Although the identified interaction patterns do not necessarily lead to rules that can be applied to different areas, they do provide useful information for transportation modelers to re-evaluate the current model structure to validate the existing model parameter

Geometric optimization on visibility problems: metaheuristic and exact solutions

Doutoramento em MatemáticaOs problemas de visibilidade têm diversas aplicações a situações reais. Entre os mais conhecidos, e exaustivamente estudados, estão os que envolvem os conceitos de vigilância e ocultação em estruturas geométricas (problemas de vigilância e ocultação). Neste trabalho são estudados problemas de visibilidade em estruturas geométricas conhecidas como polígonos, uma vez que estes podem representar, de forma apropriada, muitos dos objectos reais e são de fácil manipulação computacional. O objectivo dos problemas de vigilância é a determinação do número mínimo de posições para a colocação de dispositivos num dado polígono, de modo a que estes dispositivos consigam “ver” a totalidade do polígono. Por outro lado, o objectivo dos problemas de ocultação é a determinação do número máximo de posições num dado polígono, de modo a que quaisquer duas posições não se consigam “ver”. Infelizmente, a maior parte dos problemas de visibilidade em polígonos são NP-difíceis, o que dá origem a duas linhas de investigação: o desenvolvimento de algoritmos que estabelecem soluções aproximadas e a determinação de soluções exactas para classes especiais de polígonos. Atendendo a estas duas linhas de investigação, o trabalho é dividido em duas partes. Na primeira parte são propostos algoritmos aproximados, baseados essencialmente em metaheurísticas e metaheurísticas híbridas, para resolver alguns problemas de visibilidade, tanto em polígonos arbitrários como ortogonais. Os problemas estudados são os seguintes: “Maximum Hidden Vertex Set problem”, “Minimum Vertex Guard Set problem”, “Minimum Vertex Floodlight Set problem” e “Minimum Vertex k-Modem Set problem”. São também desenvolvidos métodos que permitem determinar a razão de aproximação dos algoritmos propostos. Para cada problema são implementados os algoritmos apresentados e é realizado um estudo estatístico para estabelecer qual o algoritmo que obtém as melhores soluções num tempo razoável. Este estudo permite concluir que as metaheurísticas híbridas são, em geral, as melhores estratégias para resolver os problemas de visibilidade estudados. Na segunda parte desta dissertação são abordados os problemas “Minimum Vertex Guard Set”, “Maximum Hidden Set” e “Maximum Hidden Vertex Set”, onde são identificadas e estudadas algumas classes de polígonos para as quais são determinadas soluções exactas e/ou limites combinatórios.Visibility problems have several applications to real-life problems. Among the most distinguished and exhaustively studied visibility problems are the ones involving concepts of guarding and hiding on geometrical structures (guarding and hiding problems). This work deals with visibility problems on geometrical structures known as polygons, since polygons are appropriate representations of many real-world objects and are easily handled by computers. The objective of the guarding problems studied in this thesis is to find a minimum number of device positions on a given polygon such that these devices collectively ''see'' the whole polygon. On the other hand, the goal of the hiding problems is to find a maximum number of positions on a given polygon such that no two of these positions can “see" each other. Unfortunately, most of the visibility problems on polygons are NP-hard, which opens two lines of investigation: the development of algorithms that establish approximate solutions and the determination of exact solutions on special classes of polygons. Accordingly, this work is divided in two parts where these two lines of investigation are considered. The first part of this thesis proposes approximation algorithms, mainly based on metaheuristics and hybrid metaheuristics, to tackle some visibility problems on arbitrary and orthogonal polygons. The addressed problems are the Maximum Hidden Vertex Set problem, the Minimum Vertex Guard Set problem, the Minimum Vertex Floodlight Set problem and the Minimum Vertex k-Modem Set problem. Methods that allow the determination of the performance ratio of the developed algorithms are also proposed. For each problem, the proposed algorithms are implemented and a statistical study is performed to determine which of the developed methods obtains the best solution in a reasonable amount of time. This study allows to conclude that, in general, the hybrid metaheuristics are the best approach to solve the studied visibility problems. The second part of this dissertation addresses the Minimum Vertex Guard Set problem, the Maximum Hidden Set problem and the Maximum Hidden Vertex Set problem, where some classes of polygons are identified and studied and for which are determined exact solutions and/or combinatorial bounds

Contributions on secretary problems, independent sets of rectangles and related problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D

Topological Quantum Codes On Compact Surfaces With Genus G≥2

In this paper we propose a construction procedure of a class of topological quantum error-correcting codes on surfaces with genus g2. This generalizes the toric codes construction. We also tabulate all possible surface codes with genus 2-5. In particular, this construction reproduces the class of codes obtained when considering the embedding of complete graphs Ks, for s1 mod 4, on surfaces with appropriate genus. We also show a table comparing the rate of different codes when fixing the distance to 3-5. © 2009 American Institute of Physics.502Shor, P.W., , p. 124. , Proceedings of the 35th Annual Symposium on Foundations of Computer Science (unpublished),Shor, P.W., (1995) Phys. Rev. A, 52, p. 2493. , 1050-2947 10.1103/PhysRevA.52.R2493Calderbank, A.R., Shor, P.W., (1996) Phys. Rev. A, 54, p. 1098. , 1050-2947 10.1103/PhysRevA.54.1098Steane, A.M., (1996) Phys. Rev. Lett., 77, p. 793. , 0031-9007 10.1103/PhysRevLett.77.793Gottesman, D., (1996) Phys. Rev. A, 54, p. 1862. , 1050-2947 10.1103/PhysRevA.54.1862Kitaev, A.Yu., (2003) Ann. Phys., 303, p. 2. , 0003-3804 10.1016/S0003-4916(02)00018-0Freedman, M.H., Meyer, D.A., (1998), www.arXiv.org/quant-ph/9810055Dennis, E., Kitaev, A., Landahl, A., Preskill, J., (2002) J. Math. Phys., 43, p. 4452. , 0022-2488 10.1063/1.1499754Bombin, H., Martin-Delgado, M.A., (2007) J. Math. Phys., 48, p. 052105. , 0022-2488 10.1063/1.2731356Cavalcante, R.G., Lazari, H., Lima, J.D., Palazzo Jr., R., (2005) Discrete Mathematics and Theoretical Computer Science, 68, pp. 145-177. , in, DIMACS Series, edited by A. Ashikhimin and A. Barg (American Mathematical Society, Providence), VolSilva, E.B., Firer, M., Costa, S.R., Palazzo Jr., R., (2006) J. Franklin Inst., 343, p. 69. , 0016-0032 10.1016/j.jfranklin.2005.09.001De Albuquerque, C.D., Palazzo Jr., R., Da Silva, E.B., (2008), pp. 391-395. , Proceedings of the Information Theory Workshop, Porto, Portugal, May (unpublished)Gottesman, D., (1997), www.arxiv.org/abs/quant-ph/9705052Nilsen, M.A., Chuang, I.L., (2000) Quantum Computation and Quantum Information, , (Cambridge University Press, Cambridge)Katok, S., (1992) Fuchsian Groups, , (The University of Chicago Press, Chicago)Beardon, A., (1983) The Geometry of Discrete Groups, , (Springer-Verlag, New York)Stillwell, J., (2000) Geometry of Surfaces, , (Springer-Verlag, New York)Firby, P.A., Gardiner, C.F., (1991) Surface Topology, , Ellis Horwood Series in Mathematics and Its Applications (Halsted Press, New York)Edmonds, A.L., Ewing, J.H., Kulkarni, R.S., (1982) Ann. Math., 116, p. 113. , 0003-486X 10.2307/2007049Artin, E., Braun, H., (1969) Introduction to Algebraic Topology, , (Charles E. Merrill, Columbus, Ohio

Konvexgeometrie

[no abstract available