Exact VC-dimension for $L_1$-visibility of points in simple polygons
Elmar Langetepe, Simone Lehmann
TL;DR
This paper establishes that the VC-dimension for the $L_1$-visibility of points in simple polygons is exactly 5, providing both an upper bound proof and a matching example.
Contribution
It proves the exact VC-dimension for $L_1$-visibility in simple polygons, which was previously unknown, using a novel proof approach.
Findings
VC-dimension is at most 5 for $L_1$-visibility in simple polygons
An example exists where 5 points are shattered by $L_1$-visibility polygons
The VC-dimension is exactly 5
Abstract
The VC-dimension plays an important role for the algorithmic problem of guarding art galleries efficiently. We prove that inside a simple polygon at most points can be shattered by -visibility polygons and give an example where 5 points are shattered. The VC-dimension is exactly . The proof idea for the upper bound is different from previous approaches. Keywords: Art gallery, VC-dimension, -visibility, polygons
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Optimization and Search Problems · Complexity and Algorithms in Graphs