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DOI: 10.4230/LIPIcs.SoCG.2016.63

Approximating Convex Shapes With Respect to Symmetric Difference Under Homotheties

Authors: Juyoung Yon, Sang Won Bae, Siu-Wing Cheng, Otfried Cheong, and Bryan T. Wilkinson

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

Abstract

The symmetric difference is a robust operator for measuring the error of approximating one shape by another. Given two convex shapes P and C, we study the problem of minimizing the volume of their symmetric difference under all possible scalings and translations of C. We prove that the problem can be solved by convex programming. We also present a combinatorial algorithm for convex polygons in the plane that runs in O((m+n) log^3(m+n)) expected time, where n and m denote the number of vertices of P and C, respectively.

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Juyoung Yon, Sang Won Bae, Siu-Wing Cheng, Otfried Cheong, and Bryan T. Wilkinson. Approximating Convex Shapes With Respect to Symmetric Difference Under Homotheties. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{yon_et_al:LIPIcs.SoCG.2016.63,
  author =	{Yon, Juyoung and Bae, Sang Won and Cheng, Siu-Wing and Cheong, Otfried and Wilkinson, Bryan T.},
  title =	{{Approximating Convex Shapes With Respect to Symmetric Difference Under Homotheties}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{63:1--63:15},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.63},
  URN =		{urn:nbn:de:0030-drops-59551},
  doi =		{10.4230/LIPIcs.SoCG.2016.63},
  annote =	{Keywords: shape matching, convexity, symmetric difference, homotheties}
}

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DOI: 10.4230/LIPIcs.STACS.2012.290

Linear-Space Data Structures for Range Mode Query in Arrays

Authors: Timothy M. Chan, Stephane Durocher, Kasper Green Larsen, Jason Morrison, and Bryan T. Wilkinson

Published in: LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Abstract

A mode of a multiset S is an element a in S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1:n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i, j) for which a mode of A[i:j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (ISAAC 2003), requires O(sqrt(n) loglog n) query time. We improve their result and present an O(n)-space data structure that supports range mode queries in O(sqrt(n / log n)) worst-case time. Furthermore, we present strong evidence that a query time significantly below sqrt(n) cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two sqrt(n) by sqrt(n) matrices reduces to n range mode queries in an array of size O(n). Additionally, we give linear-space data structures for orthogonal range mode in higher dimensions (queries in near O(n^(1-1/2d)) time) and for halfspace range mode in higher dimensions (queries in O(n^(1-1/d^2)) time).

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Timothy M. Chan, Stephane Durocher, Kasper Green Larsen, Jason Morrison, and Bryan T. Wilkinson. Linear-Space Data Structures for Range Mode Query in Arrays. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 290-301, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{chan_et_al:LIPIcs.STACS.2012.290,
  author =	{Chan, Timothy M. and Durocher, Stephane and Larsen, Kasper Green and Morrison, Jason and Wilkinson, Bryan T.},
  title =	{{Linear-Space Data Structures for Range Mode Query in Arrays}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{290--301},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{D\"{u}rr, Christoph and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.290},
  URN =		{urn:nbn:de:0030-drops-34254},
  doi =		{10.4230/LIPIcs.STACS.2012.290},
  annote =	{Keywords: mode, range query, data structure, linear space, array}
}

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Documents authored by Wilkinson, Bryan T.

Approximating Convex Shapes With Respect to Symmetric Difference Under Homotheties

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Linear-Space Data Structures for Range Mode Query in Arrays

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