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Time-Space Trade-Off for Finding the k-Visibility Region of a Point in a Polygon

  • Yeganeh Bahoo
  • Bahareh Banyassady
  • Prosenjit Bose
  • Stephane Durocher
  • Wolfgang Mulzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10167)

Abstract

We study the problem of computing the k-visibility region in the memory-constrained model. In this model, the input resides in a randomly accessible read-only memory of O(n) words, with \(O(\log {n})\) bits each. An algorithm can read and write O(s) additional words of workspace during its execution, and it writes its output to write-only memory. In a given polygon P and for a given point \(q \in P\), we say that a point p is inside the k-visibility region of q, if and only if the line segment pq intersects the boundary of P at most k times. Given a simple n-vertex polygon P stored in a read-only input array and a point \(q \in P\), we give a time-space trade-off algorithm which reports the k-visibility region of q in P in \(O(cn/s+n\log {s}+ \min \{{\lceil k/s \rceil n,n \log {\log _s{n}}}\})\) expected time using O(s) words of workspace. Here \(c\le n\) is the number of critical vertices for q, i.e., the vertices of P where the visibility region may change. We also show how to generalize this result for polygons with holes and for sets of non-crossing line segments.

Keywords

Memory-constrained model k-visibility region Time-space trade-off 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Yeganeh Bahoo
    • 1
  • Bahareh Banyassady
    • 2
  • Prosenjit Bose
    • 3
  • Stephane Durocher
    • 1
  • Wolfgang Mulzer
    • 2
  1. 1.Department of Computer ScienceUniversity of ManitobaWinnipegCanada
  2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany
  3. 3.School of Computer ScienceCarleton UniversityOttawaCanada

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