License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SOCG.2015.186
URN: urn:nbn:de:0030-drops-51378
URL: https://drops.dagstuhl.de/opus/volltexte/2015/5137/
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### Geometric Spanners for Points Inside a Polygonal Domain

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### Abstract

Let P be a set of n points inside a polygonal domain D. A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set P with respect to the geodesic distance function d where for any two points p and q, d(p,q) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., h=0), we construct a (sqrt(10)+eps)-spanner that has O(n log^2 n) edges where eps is the a given positive real number. For a case where there are h holes, our construction gives a (5+eps)-spanner with the size of O(sqrt(h) n log^2 n).

Moreover, we study t-spanners for the visibility graph of P (VG(P), for short) with respect to a hole-free polygonal domain D. The graph VG(P) is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a (3+eps)-spanner of size almost O(n^{4/3}). In addition, we show that there is a set P of n points such that any (3-eps)-spanner of VG(P) must contain almost n^2 edges.

### BibTeX - Entry

```@InProceedings{abam_et_al:LIPIcs:2015:5137,
author =	{Mohammad Ali Abam and Marjan Adeli and Hamid Homapour and Pooya Zafar Asadollahpoor},
title =	{{Geometric Spanners for Points Inside a Polygonal Domain}},
booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
pages =	{186--197},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-83-5},
ISSN =	{1868-8969},
year =	{2015},
volume =	{34},
editor =	{Lars Arge and J{\'a}nos Pach},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5137},
URN =		{urn:nbn:de:0030-drops-51378},
doi =		{10.4230/LIPIcs.SOCG.2015.186},
annote =	{Keywords: Geometric Spanners, Polygonal Domain, Visibility Graph}
}
```

 Keywords: Geometric Spanners, Polygonal Domain, Visibility Graph Collection: 31st International Symposium on Computational Geometry (SoCG 2015) Issue Date: 2015 Date of publication: 12.06.2015

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