A new upper bound for the VC-dimension of visibility regions

  • Authors:
  • Alexander Gilbers;Rolf Klein

  • Affiliations:
  • -;-

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2014

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Abstract

In this paper we are proving the following fact. Let P be an arbitrary simple polygon, and let S be an arbitrary set of 15 points inside P. Then there exists a subset T of S that is not ''visually discernible'', that is, Tvis(v)@?S holds for the visibility regions vis(v) of all points v in P. In other words, the VC-dimension d of visibility regions in a simple polygon cannot exceed 14. Since Valtr [15] proved in 1998 that d@?[6,23] holds, no progress has been made on this bound. By @e-net theorems our reduction immediately implies a smaller upper bound to the number of guards needed to cover P.