Elsevier

Computational Geometry

Volume 21, Issue 3, March 2002, Pages 193-204
Computational Geometry

Illumination in the presence of opaque line segments in the plane

https://doi.org/10.1016/S0925-7721(01)00057-8Get rights and content
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Abstract

What is the minimal number of light sources which is always sufficient to illuminate the plane in the presence of n disjoint opaque line segments? For n⩾5, O'Rourke proved that ⌊2n/3⌋ light sources are always sufficient and sometimes necessary, if light sources can be placed on the line segments and thus they can illuminate both sides of a segment.

We prove that ⌊2(n+1)/3⌋ light sources are always sufficient and sometimes necessary, if light sources cannot be placed on the line segments. An O(nlogn) time algorithm is presented which allocates at most ⌊2(n+1)/3⌋ light sources collectively illuminating the plane.

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The author acknowledges support from the joint Berlin–Zürich graduate program “Combinatorics, Geometry, and Computation”, financed by the German Science Foundation (DFG) and ETH Zürich.