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Segment Visibility Counting Queries in Polygons

Authors Kevin Buchin , Bram Custers , Ivor van der Hoog, Maarten Löffler, Aleksandr Popov , Marcel Roeloffzen , Frank Staals



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Author Details

Kevin Buchin
  • Department of Computer Science, TU Dortmund, Germany
Bram Custers
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Ivor van der Hoog
  • Department of Applied Mathematics and Computer Science, DTU, Copenhagen, Denmark
Maarten Löffler
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Aleksandr Popov
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Marcel Roeloffzen
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Frank Staals
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

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Kevin Buchin, Bram Custers, Ivor van der Hoog, Maarten Löffler, Aleksandr Popov, Marcel Roeloffzen, and Frank Staals. Segment Visibility Counting Queries in Polygons. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 58:1-58:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ISAAC.2022.58

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Visibility
  • Data Structure
  • Polygons
  • Complexity

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