License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2022.58
URN: urn:nbn:de:0030-drops-173431
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17343/
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Buchin, Kevin ; Custers, Bram ; van der Hoog, Ivor ; Löffler, Maarten ; Popov, Aleksandr ; Roeloffzen, Marcel ; Staals, Frank

Segment Visibility Counting Queries in Polygons

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LIPIcs-ISAAC-2022-58.pdf (1 MB)

Abstract

Let P be a simple polygon with n vertices, and let A be a set of m points or line segments inside P. We develop data structures that can efficiently count the objects from A that are visible to a query point or a query segment. Our main aim is to obtain fast, O(polylog nm), query times, while using as little space as possible.
In case the query is a single point, a simple visibility-polygon-based solution achieves O(log nm) query time using O(nm²) space. In case A also contains only points, we present a smaller, O(n + m^{2+ε} log n)-space, data structure based on a hierarchical decomposition of the polygon.
Building on these results, we tackle the case where the query is a line segment and A contains only points. The main complication here is that the segment may intersect multiple regions of the polygon decomposition, and that a point may see multiple such pieces. Despite these issues, we show how to achieve O(log n log nm) query time using only O(nm^{2+ε} + n²) space. Finally, we show that we can even handle the case where the objects in A are segments with the same bounds.

BibTeX - Entry

@InProceedings{buchin_et_al:LIPIcs.ISAAC.2022.58,
  author =	{Buchin, Kevin and Custers, Bram and van der Hoog, Ivor and L\"{o}ffler, Maarten and Popov, Aleksandr and Roeloffzen, Marcel and Staals, Frank},
  title =	{{Segment Visibility Counting Queries in Polygons}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{58:1--58:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/17343},
  URN =		{urn:nbn:de:0030-drops-173431},
  doi =		{10.4230/LIPIcs.ISAAC.2022.58},
  annote =	{Keywords: Visibility, Data Structure, Polygons, Complexity}
}

Keywords: Visibility, Data Structure, Polygons, Complexity
Collection: 33rd International Symposium on Algorithms and Computation (ISAAC 2022)
Issue Date: 2022
Date of publication: 14.12.2022

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