Stabbing Convex Polygons with a Segment or a Polygon

Let O = {O1, . . . , Om} be a set of m convex polygons in R 2 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 φ(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∗ logm), with high probability, where h∗ is the size of the optimal stabbing set of O. We also improve bounds on φ(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, and use these improved bounds to analyze the running time of our stabbing-set algorithm.