Stabbing Convex Polygons with a Segment or a Polygon

Abstract

Let $\mathcal{O} = \{O_1, \ldots, O_m\}$ be a set of m convex polygons in ℝ² with a total of n vertices, and let B be another convex k-gon. A placement of B, any congruent copy of B (without reflection), is called free if B does not intersect the interior of any polygon in $\mathcal{O}$ at this placement. A placement z of B is called critical if B forms three “distinct” contacts with $\mathcal{O}$ at z. Let $\varphi(B, \mathcal{O})$ be the number of free critical placements. A set of placements of B is called a stabbing set of $\mathcal{O}$ if each polygon in $\mathcal{O}$ intersects at least one placement of B in this set.

We develop efficient Monte Carlo algorithms that compute a stabbing set of size h = O(h ^*logm), with high probability, where h ^* is the size of the optimal stabbing set of $\mathcal{O}$. We also improve bounds on $\varphi(B, \mathcal{O})$ for the following three cases, namely, (i) B is a line segment and the obstacles in $\mathcal{O}$ are pairwise-disjoint, (ii) B is a line segment and the obstacles in $\mathcal{O}$ may intersect (iii) B is a convex k-gon and the obstacles in $\mathcal{O}$ are disjoint, and use these improved bounds to analyze the running time of our stabbing-set algorithm.

Work by P.A, S.G, and M.S, was supported by a grant from the U.S.-Israel Binational Science Foundation. Work by P.A. and S.G. was also supported by NSF under grants CNS-05-40347, CFF-06-35000, and DEB-04-25465, by ARO grants W911NF-04-1-0278 and W911NF-07-1-0376, and by an NIH grant 1P50-GM-08183-01 and by a DOE grant OEGP200A070505. Work by M.S. was partially supported by NSF Grant CCF-05-14079, by grant 155/05 from the Israel Science Fund, by a grant from the AFIRST joint French-Israeli program, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work of D.C. was supported in part by the NSF under Grant CCF-0515203.