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DOI: 10.4230/LIPIcs.ESA.2019.22

Randomized Incremental Construction of Delaunay Triangulations of Nice Point Sets

Authors: Jean-Daniel Boissonnat, Olivier Devillers, Kunal Dutta, and Marc Glisse

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract

Randomized incremental construction (RIC) is one of the most important paradigms for building geometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms that are both simple and efficient in theory and in practice. Randomized incremental constructions are most of the time space and time optimal in the worst-case, as exemplified by the construction of convex hulls, Delaunay triangulations and arrangements of line segments. However, the worst-case scenario occurs rarely in practice and we would like to understand how RIC behaves when the input is nice in the sense that the associated output is significantly smaller than in the worst-case. For example, it is known that the Delaunay triangulations of nicely distributed points on polyhedral surfaces in E^3 has linear complexity, as opposed to a worst-case quadratic complexity. The standard analysis does not provide accurate bounds on the complexity of such cases and we aim at establishing such bounds in this paper. More precisely, we will show that, in the case of nicely distributed points on polyhedral surfaces, the complexity of the usual RIC is O(n log n), which is optimal. In other words, without any modification, RIC nicely adapts to good cases of practical value. Our proofs also work for some other notions of nicely distributed point sets, such as (epsilon, kappa)-samples. Along the way, we prove a probabilistic lemma for sampling without replacement, which may be of independent interest.

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Jean-Daniel Boissonnat, Olivier Devillers, Kunal Dutta, and Marc Glisse. Randomized Incremental Construction of Delaunay Triangulations of Nice Point Sets. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 22:1-22:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{boissonnat_et_al:LIPIcs.ESA.2019.22,
  author =	{Boissonnat, Jean-Daniel and Devillers, Olivier and Dutta, Kunal and Glisse, Marc},
  title =	{{Randomized Incremental Construction of Delaunay Triangulations of Nice Point Sets}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{22:1--22:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.22},
  URN =		{urn:nbn:de:0030-drops-111437},
  doi =		{10.4230/LIPIcs.ESA.2019.22},
  annote =	{Keywords: Randomized incremental construction, Delaunay triangulations, Voronoi diagrams, polyhedral surfaces, probabilistic analysis}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2018.30

3D Snap Rounding

Authors: Olivier Devillers, Sylvain Lazard, and William J. Lenhart

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract

Let P be a set of n polygons in R^3, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps P to a simplicial complex Q whose vertices have integer coordinates. Every face of P is mapped to a set of faces (or edges or vertices) of Q and the mapping from P to Q can be done through a continuous motion of the faces such that (i) the L_infty Hausdorff distance between a face and its image during the motion is at most 3/2 and (ii) if two points become equal during the motion, they remain equal through the rest of the motion. In the worst case, the size of Q is O(n^{15}) and the time complexity of the algorithm is O(n^{19}) but, under reasonable hypotheses, these complexities decrease to O(n^{5}) and O(n^{6}sqrt{n}).

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Olivier Devillers, Sylvain Lazard, and William J. Lenhart. 3D Snap Rounding. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{devillers_et_al:LIPIcs.SoCG.2018.30,
  author =	{Devillers, Olivier and Lazard, Sylvain and Lenhart, William J.},
  title =	{{3D Snap Rounding}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{30:1--30:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.30},
  URN =		{urn:nbn:de:0030-drops-87438},
  doi =		{10.4230/LIPIcs.SoCG.2018.30},
  annote =	{Keywords: Geometric algorithms, Robustness, Fixed-precision computations}
}

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DOI: 10.4230/LIPIcs.SoCG.2016.33

Qualitative Symbolic Perturbation

Authors: Olivier Devillers, Menelaos Karavelas, and Monique Teillaud

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract

In a classical Symbolic Perturbation scheme, degeneracies are handled by substituting some polynomials in epsilon for the inputs of a predicate. Instead of a single perturbation, we propose to use a sequence of (simpler) perturbations. Moreover, we look at their effects geometrically instead of algebraically; this allows us to tackle cases that were not tractable with the classical algebraic approach.

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Olivier Devillers, Menelaos Karavelas, and Monique Teillaud. Qualitative Symbolic Perturbation. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{devillers_et_al:LIPIcs.SoCG.2016.33,
  author =	{Devillers, Olivier and Karavelas, Menelaos and Teillaud, Monique},
  title =	{{Qualitative Symbolic Perturbation}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{33:1--33:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.33},
  URN =		{urn:nbn:de:0030-drops-59259},
  doi =		{10.4230/LIPIcs.SoCG.2016.33},
  annote =	{Keywords: Robustness issues, Symbolic perturbations, Apollonius diagram}
}

Document

DOI: 10.4230/LIPIcs.SOCG.2015.224

On the Smoothed Complexity of Convex Hulls

Authors: Olivier Devillers, Marc Glisse, Xavier Goaoc, and Rémy Thomasse

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract

We establish an upper bound on the smoothed complexity of convex hulls in R^d under uniform Euclidean (L^2) noise. Specifically, let {p_1^*, p_2^*, ..., p_n^*} be an arbitrary set of n points in the unit ball in R^d and let p_i = p_i^* + x_i, where x_1, x_2, ..., x_n are chosen independently from the unit ball of radius r. We show that the expected complexity, measured as the number of faces of all dimensions, of the convex hull of {p_1, p_2, ..., p_n} is O(n^{2-4/(d+1)} (1+1/r)^{d-1}); the magnitude r of the noise may vary with n. For d=2 this bound improves to O(n^{2/3} (1+r^{-2/3})). We also analyze the expected complexity of the convex hull of L^2 and Gaussian perturbations of a nice sample of a sphere, giving a lower-bound for the smoothed complexity. We identify the different regimes in terms of the scale, as a function of n, and show that as the magnitude of the noise increases, that complexity varies monotonically for Gaussian noise but non-monotonically for L^2 noise.

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Olivier Devillers, Marc Glisse, Xavier Goaoc, and Rémy Thomasse. On the Smoothed Complexity of Convex Hulls. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 224-239, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{devillers_et_al:LIPIcs.SOCG.2015.224,
  author =	{Devillers, Olivier and Glisse, Marc and Goaoc, Xavier and Thomasse, R\'{e}my},
  title =	{{On the Smoothed Complexity of Convex Hulls}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{224--239},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.224},
  URN =		{urn:nbn:de:0030-drops-51451},
  doi =		{10.4230/LIPIcs.SOCG.2015.224},
  annote =	{Keywords: Probabilistic analysis, Worst-case analysis, Gaussian noise}
}

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Randomized Incremental Construction of Delaunay Triangulations of Nice Point Sets

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3D Snap Rounding

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Qualitative Symbolic Perturbation

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On the Smoothed Complexity of Convex Hulls

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