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Guest Editorsʼ foreword
Computational Geometry 46 (2013) 401
Contents lists available at SciVerse ScienceDirect
Computational Geometry: Theory and Applications www.elsevier.com/locate/comgeo
Editorial
Guest Editors’ foreword This special issue of Computational Geometry: Theory and Applications contains a selection of the best papers that were presented at the 27th Annual ACM Symposium on Computational Geometry, which was held in Paris, France, on June 13–15, 2012. The five papers in this issue were invited, submitted, and then reviewed according to the usual, high standards of the journal. The revised versions of these papers are what you can find in this issue. It is our pleasure to briefly introduce these papers. The paper by John Hershberger presents a snap rounding of line segments in the plane, improving on the standard method that may suffer from instability after iteration. The author shows how the existing algorithms for snap rounding can be modified to compute a stable snap rounding under the new model, with the same time complexity as the previous approach. The paper by Esther Ezra and Wolfgang Mulzer considers the problem of, given is a set of lines in the plane, to build a data structure such that the convex hull of any point set having one point on each line can be computed quickly. They show that such a structure of O (n2 ) space (and preprocessing time) exists, answering such queries in O (nα (n) log∗ n) time. They also extend their results to k-levels in line arrangements and show that for several related problems a similar result is not possible. The paper by Chao Chen and Michael Kerber shows how to compute the persistence diagram of a filtered simplicial complex in an output-sensitive manner. Only the homology classes with a persistence above a given threshold are returned, which is relevant because in many applications the classes with low persistence are considered noise. The paper by Dominique Attali, André Lieutier and David Salinas focuses on proving that Vietoris–Rips complexes provide topologically correct reconstructions of sampled shapes. They associate two real-valued functions to every compact set X (including point clouds P ) embedded in some Euclidean space. Using these functions, they provide conditions under which ˇ the Rips complex of a point cloud collapses to the Cech complex of a point cloud, and they also provide conditions under ˇ which the Rips (or the Cech) complex of a point cloud P sampled from a space X deformation retracts onto X . The paper by Vít Jelínek, Jan Kratochvíl, and Ignaz Rutter concerns a graph layout problem. Suppose a planar graph is given, and some parts of it are already embedded in the plane. The question is whether the graph can be embedded in a planar way by extending the given embedding. A characterization similar to the well-known Kuratowski theorem is given, using forbidden substructures. Acknowledgements We wish to thank all authors of invited papers to this special issue. Special thanks go to all reviewers, who ensured the quality of the papers published in this issue with their anonymous but much appreciated work.
Ferran Hurtado Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya (UPC), Barcelona, Spain E-mail address: [email protected] Marc van Kreveld Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands E-mail address: [email protected] Available online 26 October 2012
0925-7721/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comgeo.2012.10.007
Contents lists available at SciVerse ScienceDirect
Computational Geometry: Theory and Applications www.elsevier.com/locate/comgeo
Editorial
Guest Editors’ foreword This special issue of Computational Geometry: Theory and Applications contains a selection of the best papers that were presented at the 27th Annual ACM Symposium on Computational Geometry, which was held in Paris, France, on June 13–15, 2012. The five papers in this issue were invited, submitted, and then reviewed according to the usual, high standards of the journal. The revised versions of these papers are what you can find in this issue. It is our pleasure to briefly introduce these papers. The paper by John Hershberger presents a snap rounding of line segments in the plane, improving on the standard method that may suffer from instability after iteration. The author shows how the existing algorithms for snap rounding can be modified to compute a stable snap rounding under the new model, with the same time complexity as the previous approach. The paper by Esther Ezra and Wolfgang Mulzer considers the problem of, given is a set of lines in the plane, to build a data structure such that the convex hull of any point set having one point on each line can be computed quickly. They show that such a structure of O (n2 ) space (and preprocessing time) exists, answering such queries in O (nα (n) log∗ n) time. They also extend their results to k-levels in line arrangements and show that for several related problems a similar result is not possible. The paper by Chao Chen and Michael Kerber shows how to compute the persistence diagram of a filtered simplicial complex in an output-sensitive manner. Only the homology classes with a persistence above a given threshold are returned, which is relevant because in many applications the classes with low persistence are considered noise. The paper by Dominique Attali, André Lieutier and David Salinas focuses on proving that Vietoris–Rips complexes provide topologically correct reconstructions of sampled shapes. They associate two real-valued functions to every compact set X (including point clouds P ) embedded in some Euclidean space. Using these functions, they provide conditions under which ˇ the Rips complex of a point cloud collapses to the Cech complex of a point cloud, and they also provide conditions under ˇ which the Rips (or the Cech) complex of a point cloud P sampled from a space X deformation retracts onto X . The paper by Vít Jelínek, Jan Kratochvíl, and Ignaz Rutter concerns a graph layout problem. Suppose a planar graph is given, and some parts of it are already embedded in the plane. The question is whether the graph can be embedded in a planar way by extending the given embedding. A characterization similar to the well-known Kuratowski theorem is given, using forbidden substructures. Acknowledgements We wish to thank all authors of invited papers to this special issue. Special thanks go to all reviewers, who ensured the quality of the papers published in this issue with their anonymous but much appreciated work.
Ferran Hurtado Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya (UPC), Barcelona, Spain E-mail address: [email protected] Marc van Kreveld Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands E-mail address: [email protected] Available online 26 October 2012
0925-7721/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comgeo.2012.10.007