Stabbing Convex Polygons with a Segment or a Polygon

Let O = {O1, . . . , Om} 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 O at this placement. A placement z of B is called critical if B forms three "distinct" contacts with O at z. Let (B, O) be the number of free critical placements. A set of placements of B is called a stabbing set of O if each polygon in 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 log m), with high probability, where h is the size of the optimal stabbing set of O. We also improve bounds on (B, O) for the following three cases, namely, (i) B is a line segment and the obstacles in O are pairwise-disjoint, (ii) B is a line segment and the obstacles in O may intersect (iii) B is a convex k-gon and the obstacles in O are disjoint, a...

• Type: Conference
• Year: 2008
• Where: ESA
• Authors: Pankaj K. Agarwal, Danny Z. Chen, Shashidhara K. Ganjugunte, Ewa Misiolek, Micha Sharir, Kai Tang
• Keywords: Algorithms, Convex K-gon, ESA 2008, Polygon, Stabbing Set