1,700 research outputs found

    Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

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    We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

    TR-2012001: Algebraic Algorithms

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    An Open Access Database of Genome-wide Association Results

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    <p>Abstract</p> <p>Background</p> <p>The number of genome-wide association studies (GWAS) is growing rapidly leading to the discovery and replication of many new disease loci. Combining results from multiple GWAS datasets may potentially strengthen previous conclusions and suggest new disease loci, pathways or pleiotropic genes. However, no database or centralized resource currently exists that contains anywhere near the full scope of GWAS results.</p> <p>Methods</p> <p>We collected available results from 118 GWAS articles into a database of 56,411 significant SNP-phenotype associations and accompanying information, making this database freely available here. In doing so, we met and describe here a number of challenges to creating an open access database of GWAS results. Through preliminary analyses and characterization of available GWAS, we demonstrate the potential to gain new insights by querying a database across GWAS.</p> <p>Results</p> <p>Using a genomic bin-based density analysis to search for highly associated regions of the genome, positive control loci (e.g., MHC loci) were detected with high sensitivity. Likewise, an analysis of highly repeated SNPs across GWAS identified replicated loci (e.g., <it>APOE</it>, <it>LPL</it>). At the same time we identified novel, highly suggestive loci for a variety of traits that did not meet genome-wide significant thresholds in prior analyses, in some cases with strong support from the primary medical genetics literature (<it>SLC16A7, CSMD1, OAS1</it>), suggesting these genes merit further study. Additional adjustment for linkage disequilibrium within most regions with a high density of GWAS associations did not materially alter our findings. Having a centralized database with standardized gene annotation also allowed us to examine the representation of functional gene categories (gene ontologies) containing one or more associations among top GWAS results. Genes relating to cell adhesion functions were highly over-represented among significant associations (p < 4.6 × 10<sup>-14</sup>), a finding which was not perturbed by a sensitivity analysis.</p> <p>Conclusion</p> <p>We provide access to a full gene-annotated GWAS database which could be used for further querying, analyses or integration with other genomic information. We make a number of general observations. Of reported associated SNPs, 40% lie within the boundaries of a RefSeq gene and 68% are within 60 kb of one, indicating a bias toward gene-centricity in the findings. We found considerable heterogeneity in information available from GWAS suggesting the wider community could benefit from standardization and centralization of results reporting.</p
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