Helly-type theorems for approximate covering

• Authors: Julien Demouth;Olivier Devillers;Marc Glisse;Xavier Goaoc
• Venue: Proceedings of the twenty-fourth annual symposium on Computational geometry
• Year: 2008

Let F ∪ {U} be a collection of convex sets in Rd such that F covers U. We show that if the elements of F and U have comparable size, in the sense that each contains a ball of radius r and is contained in a ball of radius R for some fixed r and R, then for any ε 0 there exists Hε ⊂ F, whose size |Hε| is polynomial in 1/ε and independent of |F|, that covers U except for a volume of at most ε. The size of the smallest such subset depends on the geometry of the elements of F; specifically, we prove that it is O(1/ε) when F consists of axis-parallel unit squares in the plane and Õ(ε1--d/2) when F consists of unit balls in Rd (here, Õ(n) means O(n log n) for some constant), and that these bounds are, in the worst-case, tight up to the logarithmic factors. We extend these results to surface-to-surface visibility in 3 dimensions: if a collection F of disjoint unit balls occludes visibility between two balls then a subset of F of size Õ(ε--7/2) blocks visibility along all but a set of lines of measure ε. Finally, for each of the above situations we give an algorithm that takes F and U as input and outputs in time O(|F|*|Hε|) either a point in U not covered by F or a subset Hε covering U up to a measure ε, with |Hε| satisfying the above bounds.