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Steven Fortune - One of the best experts on this subject based on the ideXlab platform.

  • Vertex-Rounding a Three-Dimensional Polyhedral Subdivision
    Discrete & Computational Geometry, 1999
    Co-Authors: Steven Fortune
    Abstract:

    Let P be a polyhedral subdivision in R 3 with a total of n faces. We show that there is an embedding σ of the vertices, edges, and facets of P into a subdivision Q , where every Vertex Coordinate of Q is an integral multiple of $2^{- \lceil \log_2 n + 2 \rceil}$ . For each face f of P , the Hausdorff distance in the L ∈fty metric between f and σ(f) is at most 3/2 . The embedding σ preserves or collapses vertical order on faces of P . The subdivision Q has O(n 4 ) vertices in the worst case, and can be computed in the same time.

  • Symposium on Computational Geometry - Vertex-rounding a three-dimensional polyhedral subdivision
    Proceedings of the fourteenth annual symposium on Computational geometry - SCG '98, 1998
    Co-Authors: Steven Fortune
    Abstract:

    Let P be a polyhedral subdivision in R3 with a total of n faces. We show that there is an embedding σ of the vertices, edges, and facets of P into a subdivision Q , where every Vertex Coordinate of Q is an integral multiple of \(2^{- \lceil \log_2 n + 2 \rceil}\) . For each face f of P , the Hausdorff distance in the L ∈fty metric between f and σ(f) is at most 3/2 . The embedding σ preserves or collapses vertical order on faces of P . The subdivision Q has O(n 4 ) vertices in the worst case, and can be computed in the same time.

Sander Verdonschot - One of the best experts on this subject based on the ideXlab platform.

  • Optimal Local Routing on Delaunay Triangulations Defined by Empty Equilateral Triangles
    SIAM Journal on Computing, 2015
    Co-Authors: Prosenjit Bose, Rolf Fagerberg, André Van Renssen, Sander Verdonschot
    Abstract:

    We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-$\theta_6$-graph (the half-$\theta_6$-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most $5/\sqrt{3} \approx 2.887$ times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-$\theta_6$-graph is 2, meaning that even though there always exists a path whose length is at most twice the Euclidean distance, we cannot always find such a path when routing locally. Since every triangulation can be embedded in the plane as a half-$\theta_6$-graph using $O(\log n)$ bits per Vertex Coordinate via Schnyder's embedding scheme [W. Schnyder, Embedding planar graphs on the grid, in Proceedings of the 1st Annual ...

  • Optimal local routing on Delaunay triangulations defined by empty equilateral triangles
    arXiv: Computational Geometry, 2014
    Co-Authors: Prosenjit Bose, Rolf Fagerberg, André Van Renssen, Sander Verdonschot
    Abstract:

    We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-$\theta_6$-graph (the half-$\theta_6$-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most $5/\sqrt{3} \approx 2.887$ times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-$\theta_6$-graph is 2, meaning that even though there always exists a path whose lengths is at most twice the Euclidean distance, we cannot always find such a path when routing locally. Since every triangulation can be embedded in the plane as a half-$\theta_6$-graph using $O(\log n)$ bits per Vertex Coordinate via Schnyder's embedding scheme (SODA 1990), our result provides a competitive local routing algorithm for every such embedded triangulation. Finally, we show how our routing algorithm can be adapted to provide a routing ratio of $15/\sqrt{3} \approx 8.660$ on two bounded degree subgraphs of the half-$\theta_6$-graph.

  • SODA - Competitive routing in the half-θ 6 -graph
    2012
    Co-Authors: Prosenjit Bose, Rolf Fagerberg, André Van Renssen, Sander Verdonschot
    Abstract:

    We present a deterministic local routing scheme that is guaranteed to find a path between any pair of vertices in a half-θ6-graph whose length is at most 5/√3 = 2.886... times the Euclidean distance between the pair of vertices. The half-θ6-graph is identical to the Delaunay triangulation where the empty region is an equilateral triangle. Moreover, we show that no local routing scheme can achieve a better competitive spanning ratio thereby implying that our routing scheme is optimal. This is somewhat surprising because the spanning ratio of the half-θ6-graph is 2. Since every triangulation can be embedded in the plane as a half-θ6-graph using O(log n) bits per Vertex Coordinate via Schnyder's embedding scheme (SODA 1990), our result provides a competitive local routing scheme for every such embedded triangulation.

Antoine Vigneron - One of the best experts on this subject based on the ideXlab platform.

  • Navigating Weighted Regions with Scattered Skinny Tetrahedra
    International Journal of Computational Geometry & Applications, 2017
    Co-Authors: Siu-wing Cheng, Man-kwun Chiu, Jiongxin Jin, Antoine Vigneron
    Abstract:

    We propose an algorithm for finding a (1 + 𝜀)-approximate shortest path through a weighted 3D simplicial complex 𝒯. The weights are integers from the range [1,W] and the vertices have integral Coordinates. Let N be the largest Vertex Coordinate magnitude, and let n be the number of tetrahedra in 𝒯. Let ρ be some arbitrary constant. Let κ be the size of the largest connected component of tetrahedra whose aspect ratios exceed ρ. There exists a constant C dependent on ρ but independent of 𝒯 such that if κ ≤ 1 Cloglog n + O(1), the running time of our algorithm is polynomial in n, 1/𝜀 and log(NW). If κ = O(1), the running time reduces to O(n𝜀−O(1)(log(NW))O(1)).

  • ISAAC - Navigating Weighted Regions with Scattered Skinny Tetrahedra
    Algorithms and Computation, 2015
    Co-Authors: Siu-wing Cheng, Man-kwun Chiu, Jiongxin Jin, Antoine Vigneron
    Abstract:

    We propose an algorithm for finding a \((1+\varepsilon )\)-approximate shortest path through a weighted 3D simplicial complex \(\mathcal T\). The weights are integers from the range [1, W] and the vertices have integral Coordinates. Let N be the largest Vertex Coordinate magnitude, and let n be the number of tetrahedra in \(\mathcal T\). Let \(\rho \) be some arbitrary constant. Let \(\kappa \) be the size of the largest connected component of tetrahedra whose aspect ratios exceed \(\rho \). There exists a constant C dependent on \(\rho \) but independent of \(\mathcal T\) such that if \(\kappa \le \frac{1}{C}\log \log n + O(1)\), the running time of our algorithm is polynomial in n, \(1/\varepsilon \) and \(\log (NW)\). If \(\kappa = O(1)\), the running time reduces to \(O(n \varepsilon ^{-O(1)}(\log (NW))^{O(1)})\).

S. Dow - One of the best experts on this subject based on the ideXlab platform.

  • First-year experience with the BaBar silicon Vertex tracker
    Nuclear Instruments & Methods in Physics Research Section A-accelerators Spectrometers Detectors and Associated Equipment, 2001
    Co-Authors: C. Bozzi, V. Carassiti, A. Cotta Ramusino, S. Dittongo, M. Folegani, L. Piemontese, Brad Abbott, Breon, A. R. Clark, S. Dow
    Abstract:

    Abstract Within its first year of operation, the BaBar Silicon Vertex Tracker (SVT) has accomplished its primary design goal, measuring the z Vertex Coordinate with sufficient accuracy as to allow the measurement of the time-dependent CP asymmetry in the neutral B-meson system. The SVT consists of five layers of double-sided, AC-coupled silicon-strip detectors of 300 μm thickness with a readout strip pitch of 50–210 μm and a stereo angle of 90° between the strips on the two sides. Detector alignment and performance with respect to spatial resolution and efficiency in the reconstruction of single hits are discussed. In the day-to-day operation of the SVT, radiation damage and protection issues were of primary concern. The SVT is equipped with a dedicated system (SVTRAD) for radiation monitoring and protection, using reverse-biased photodiodes. The evolution of the SVTRAD thresholds on the tolerated radiation level is described. Results on the first-year radiation exposure as measured with the SVTRAD system and on the so far accumulated damage are presented. The implications of test-irradiation results and possible future PEP-II luminosity upgrades on the radiation limited lifetime of the SVT are discussed.

  • First-year experience with the BaBar silicon Vertex tracker
    Nuclear Instruments and Methods in Physics Research Section A: Accelerators Spectrometers Detectors and Associated Equipment, 2001
    Co-Authors: C. Bozzi, V. Carassiti, S. Dittongo, M. Folegani, L. Piemontese, A. R. Clark, A Cotta Ramusino, B.k Abbott, A.b Breon, S. Dow
    Abstract:

    Within its first year of operation, the BaBar Silicon Vertex Tracker (SVT) has accomplished its primary design goal, measuring the z Vertex Coordinate with sufficient accuracy as to allow the measurement of the time-dependent CP asymmetry in the neutral B-meson system. The SVT consists of five layers of double-sided. AC-coupled silicon-strip detectors of 300 mum thickness with a readout strip pitch of 50-210 mum and a stereo angle of 90 degrees between the strips on the two sides. Detector alignment and performance with respect to spatial resolution and efficiency in the reconstruction of single hits are discussed. In the day-to-day operation of the SVT., radiation damage and protection issues were of primary concern. The SVT is equipped with a dedicated system (SVTRAD) for radiation monitoring and protection, using reverse-biased photodiodes. The evolution of the SVTRAD thresholds on the tolerated radiation level is described. Results on the first-year radiation exposure as measured with the SVTRAD system and on the so far accumulated damage are presented. The implications of test-irradiation results and possible future PEP-II luminosity upgrades on the radiation limited lifetime of the SVT are discussed. (C) 2001 Elsevier Science B.V. All rights reserved

Prosenjit Bose - One of the best experts on this subject based on the ideXlab platform.

  • Optimal Local Routing on Delaunay Triangulations Defined by Empty Equilateral Triangles
    SIAM Journal on Computing, 2015
    Co-Authors: Prosenjit Bose, Rolf Fagerberg, André Van Renssen, Sander Verdonschot
    Abstract:

    We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-$\theta_6$-graph (the half-$\theta_6$-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most $5/\sqrt{3} \approx 2.887$ times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-$\theta_6$-graph is 2, meaning that even though there always exists a path whose length is at most twice the Euclidean distance, we cannot always find such a path when routing locally. Since every triangulation can be embedded in the plane as a half-$\theta_6$-graph using $O(\log n)$ bits per Vertex Coordinate via Schnyder's embedding scheme [W. Schnyder, Embedding planar graphs on the grid, in Proceedings of the 1st Annual ...

  • Optimal local routing on Delaunay triangulations defined by empty equilateral triangles
    arXiv: Computational Geometry, 2014
    Co-Authors: Prosenjit Bose, Rolf Fagerberg, André Van Renssen, Sander Verdonschot
    Abstract:

    We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-$\theta_6$-graph (the half-$\theta_6$-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most $5/\sqrt{3} \approx 2.887$ times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-$\theta_6$-graph is 2, meaning that even though there always exists a path whose lengths is at most twice the Euclidean distance, we cannot always find such a path when routing locally. Since every triangulation can be embedded in the plane as a half-$\theta_6$-graph using $O(\log n)$ bits per Vertex Coordinate via Schnyder's embedding scheme (SODA 1990), our result provides a competitive local routing algorithm for every such embedded triangulation. Finally, we show how our routing algorithm can be adapted to provide a routing ratio of $15/\sqrt{3} \approx 8.660$ on two bounded degree subgraphs of the half-$\theta_6$-graph.

  • SODA - Competitive routing in the half-θ 6 -graph
    2012
    Co-Authors: Prosenjit Bose, Rolf Fagerberg, André Van Renssen, Sander Verdonschot
    Abstract:

    We present a deterministic local routing scheme that is guaranteed to find a path between any pair of vertices in a half-θ6-graph whose length is at most 5/√3 = 2.886... times the Euclidean distance between the pair of vertices. The half-θ6-graph is identical to the Delaunay triangulation where the empty region is an equilateral triangle. Moreover, we show that no local routing scheme can achieve a better competitive spanning ratio thereby implying that our routing scheme is optimal. This is somewhat surprising because the spanning ratio of the half-θ6-graph is 2. Since every triangulation can be embedded in the plane as a half-θ6-graph using O(log n) bits per Vertex Coordinate via Schnyder's embedding scheme (SODA 1990), our result provides a competitive local routing scheme for every such embedded triangulation.