Computational geometry: an introduction
Computational geometry: an introduction
The shortest curve that meets all the lines that meet a convex body
American Mathematical Monthly
An algorithm for determining an opaque minimal forest of a convex polygon
Information Processing Letters
An O(n3) algorithm for finding the minimal opaque forest of a convex polygon
Information Processing Letters
On a problem about covering lines by squares
Discrete & Computational Geometry
A counterexample to the algorithms for determining opaque minimal forests
Information Processing Letters
Minimum-perimeter intersecting polygons
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
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The problem of finding "small" sets that meet every straightline which intersects a given convex region was initiated by Mazurkiewicz in 1916. We call such a set an opaque set or a barrier for that region. We consider the problem of computing the shortest barrier for a given convex polygon with n vertices. No exact algorithm is currently known even for the simplest instances such as a square or an equilateral triangle. For general barriers, we present a O(n) time approximation algorithm with ratio 1/2 + 2+√2/π = 1.5867...For connected barriers, we can achieve the approximation ratio π+5/π+2 = 1.5834 ... again in O(n) time. We also show that if the barrier is restricted to the interior and the boundary of the input polygon, then the problem admits a fully polynomial-time approximation scheme for the connected case and a quadratic-time exact algorithm for the single-arc case. These are the first approximation algorithms obtained for this problem.