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Documents authored by Perković, Ljubomir

Found 2 Possible Name Variants:

Perkovic, Ljubomir

Document
DOI: 10.4230/LIPIcs.SoCG.2020.34
The Stretch Factor of Hexagon-Delaunay Triangulations

Authors: Michael Dennis, Ljubomir Perković, and Duru Türkoğlu

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation is one of the longstanding open problems in computational geometry. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop techniques for computing the exact stretch factor of "easier" types of Delaunay triangulations, in particular those defined using regular-polygons instead of a circle. Tight bounds exist for Delaunay triangulations defined using an equilateral triangle and a square. In this paper, we determine the exact stretch factor of Delaunay triangulations defined using a regular hexagon: It is 2. We think that the main contribution of this paper are the two techniques we have developed to compute tight upper bounds for the stretch factor of Hexagon-Delaunay triangulations.
Cite as

Michael Dennis, Ljubomir Perković, and Duru Türkoğlu. The Stretch Factor of Hexagon-Delaunay Triangulations. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dennis_et_al:LIPIcs.SoCG.2020.34,
  author =	{Dennis, Michael and Perkovi\'{c}, Ljubomir and T\"{u}rko\u{g}lu, Duru},
  title =	{{The Stretch Factor of Hexagon-Delaunay Triangulations}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{34:1--34:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.34},
  URN =		{urn:nbn:de:0030-drops-121920},
  doi =		{10.4230/LIPIcs.SoCG.2020.34},
  annote =	{Keywords: Delaunay triangulation, geometric spanner, plane spanner, stretch factor, spanning ratio}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.45
Degree Four Plane Spanners: Simpler and Better

Authors: Iyad Kanj, Ljubomir Perkovic, and Duru Türkoglu

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

Abstract
Let P be a set of n points embedded in the plane, and let C be the complete Euclidean graph whose point-set is P. Each edge in C between two points p, q is realized as the line segment [pq], and is assigned a weight equal to the Euclidean distance |pq|. In this paper, we show how to construct in O(nlg{n}) time a plane spanner of C of maximum degree at most 4 and of stretch factor at most 20. This improves a long sequence of results on the construction of bounded degree plane spanners of C. Our result matches the smallest known upper bound of 4 by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from 156.82 to 20. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner, and reveals useful structural properties of the Delaunay triangulations defined with respect to the equilateral-triangle distance.
Cite as

Iyad Kanj, Ljubomir Perkovic, and Duru Türkoglu. Degree Four Plane Spanners: Simpler and Better. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kanj_et_al:LIPIcs.SoCG.2016.45,
  author =	{Kanj, Iyad and Perkovic, Ljubomir and T\"{u}rkoglu, Duru},
  title =	{{Degree Four Plane Spanners: Simpler and Better}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{45:1--45:15},
  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.2016.45},
  URN =		{urn:nbn:de:0030-drops-59376},
  doi =		{10.4230/LIPIcs.SoCG.2016.45},
  annote =	{Keywords: Nonnumerical Algorithms and Problems,Computational Geometry and Object Modeling}
}
Document
DOI: 10.4230/LIPIcs.STACS.2008.1320
On Geometric Spanners of Euclidean and Unit Disk Graphs

Authors: Ljubomir Perkovic and Iyad A. Kanj

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

Abstract
We consider the problem of constructing bounded-degree planar geometric spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor $C_{delapprox 2.42$; however, its degree may not be bounded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor $ ho =1+2pi(kcos{frac{pi{k)^{-1$ and degree bounded by $k$, for any integer parameter $kgeq 14$. This result immediately implies an algorithm for constructing a planar geometric spanner of a Euclidean graph with stretch factor $ ho cdot C_{del$ and degree bounded by $k$, for any integer parameter $kgeq 14$. Moreover, the resulting spanner contains a Euclidean Minimum Spanning Tree (EMST) as a subgraph. Our second contribution lies in developing the structural results necessary to transfer our analysis and algorithm from Euclidean graphs to unit disk graphs, the usual model for wireless ad-hoc networks. We obtain a very simple distributed, {em strictly-localized algorithm that, given a unit disk graph embedded in the plane, constructs a geometric spanner with the above stretch factor and degree bound, and also containing an EMST as a subgraph. The obtained results dramatically improve the previous results in all aspects, as shown in the paper.
Cite as

Ljubomir Perkovic and Iyad A. Kanj. On Geometric Spanners of Euclidean and Unit Disk Graphs. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 409-420, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{perkovic_et_al:LIPIcs.STACS.2008.1320,
  author =	{Perkovic, Ljubomir and Kanj, Iyad A.},
  title =	{{On Geometric Spanners of Euclidean and Unit Disk Graphs}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{409--420},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1320},
  URN =		{urn:nbn:de:0030-drops-13200},
  doi =		{10.4230/LIPIcs.STACS.2008.1320},
  annote =	{Keywords: Geometric spanner, euclidean graph, unit disk graph, wireless ad-hoc networks}
}

Perković, Ljubomir

Document
DOI: 10.4230/LIPIcs.SoCG.2020.34
The Stretch Factor of Hexagon-Delaunay Triangulations

Authors: Michael Dennis, Ljubomir Perković, and Duru Türkoğlu

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation is one of the longstanding open problems in computational geometry. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop techniques for computing the exact stretch factor of "easier" types of Delaunay triangulations, in particular those defined using regular-polygons instead of a circle. Tight bounds exist for Delaunay triangulations defined using an equilateral triangle and a square. In this paper, we determine the exact stretch factor of Delaunay triangulations defined using a regular hexagon: It is 2. We think that the main contribution of this paper are the two techniques we have developed to compute tight upper bounds for the stretch factor of Hexagon-Delaunay triangulations.
Cite as

Michael Dennis, Ljubomir Perković, and Duru Türkoğlu. The Stretch Factor of Hexagon-Delaunay Triangulations. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{dennis_et_al:LIPIcs.SoCG.2020.34,
  author =	{Dennis, Michael and Perkovi\'{c}, Ljubomir and T\"{u}rko\u{g}lu, Duru},
  title =	{{The Stretch Factor of Hexagon-Delaunay Triangulations}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{34:1--34:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.34},
  URN =		{urn:nbn:de:0030-drops-121920},
  doi =		{10.4230/LIPIcs.SoCG.2020.34},
  annote =	{Keywords: Delaunay triangulation, geometric spanner, plane spanner, stretch factor, spanning ratio}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.45
Degree Four Plane Spanners: Simpler and Better

Authors: Iyad Kanj, Ljubomir Perkovic, and Duru Türkoglu

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

Abstract
Let P be a set of n points embedded in the plane, and let C be the complete Euclidean graph whose point-set is P. Each edge in C between two points p, q is realized as the line segment [pq], and is assigned a weight equal to the Euclidean distance |pq|. In this paper, we show how to construct in O(nlg{n}) time a plane spanner of C of maximum degree at most 4 and of stretch factor at most 20. This improves a long sequence of results on the construction of bounded degree plane spanners of C. Our result matches the smallest known upper bound of 4 by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from 156.82 to 20. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner, and reveals useful structural properties of the Delaunay triangulations defined with respect to the equilateral-triangle distance.
Cite as

Iyad Kanj, Ljubomir Perkovic, and Duru Türkoglu. Degree Four Plane Spanners: Simpler and Better. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

Copy BibTex To Clipboard

@InProceedings{kanj_et_al:LIPIcs.SoCG.2016.45,
  author =	{Kanj, Iyad and Perkovic, Ljubomir and T\"{u}rkoglu, Duru},
  title =	{{Degree Four Plane Spanners: Simpler and Better}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{45:1--45:15},
  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.2016.45},
  URN =		{urn:nbn:de:0030-drops-59376},
  doi =		{10.4230/LIPIcs.SoCG.2016.45},
  annote =	{Keywords: Nonnumerical Algorithms and Problems,Computational Geometry and Object Modeling}
}
Document
DOI: 10.4230/LIPIcs.STACS.2008.1320
On Geometric Spanners of Euclidean and Unit Disk Graphs

Authors: Ljubomir Perkovic and Iyad A. Kanj

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

Abstract
We consider the problem of constructing bounded-degree planar geometric spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor $C_{delapprox 2.42$; however, its degree may not be bounded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor $ ho =1+2pi(kcos{frac{pi{k)^{-1$ and degree bounded by $k$, for any integer parameter $kgeq 14$. This result immediately implies an algorithm for constructing a planar geometric spanner of a Euclidean graph with stretch factor $ ho cdot C_{del$ and degree bounded by $k$, for any integer parameter $kgeq 14$. Moreover, the resulting spanner contains a Euclidean Minimum Spanning Tree (EMST) as a subgraph. Our second contribution lies in developing the structural results necessary to transfer our analysis and algorithm from Euclidean graphs to unit disk graphs, the usual model for wireless ad-hoc networks. We obtain a very simple distributed, {em strictly-localized algorithm that, given a unit disk graph embedded in the plane, constructs a geometric spanner with the above stretch factor and degree bound, and also containing an EMST as a subgraph. The obtained results dramatically improve the previous results in all aspects, as shown in the paper.
Cite as

Ljubomir Perkovic and Iyad A. Kanj. On Geometric Spanners of Euclidean and Unit Disk Graphs. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 409-420, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

Copy BibTex To Clipboard

@InProceedings{perkovic_et_al:LIPIcs.STACS.2008.1320,
  author =	{Perkovic, Ljubomir and Kanj, Iyad A.},
  title =	{{On Geometric Spanners of Euclidean and Unit Disk Graphs}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{409--420},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1320},
  URN =		{urn:nbn:de:0030-drops-13200},
  doi =		{10.4230/LIPIcs.STACS.2008.1320},
  annote =	{Keywords: Geometric spanner, euclidean graph, unit disk graph, wireless ad-hoc networks}
}
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