E cient Algorithms for Counting and Reporting Pairwise Intersections between Convex Polygons

Let S be a set of convex polygons in the plane with a total of n vertices, where a polygon consists of the boundary as well as the interior. E cient algorithms are presented for the problem of reporting output-sensitively (resp. counting) the I pairs of polygons that intersect. The algorithm for the reporting (resp. counting) problem runs in time O(n4=3+ +I) (resp. O(n4=3+ )), where > 0 is an arbitrarily small constant. This result is based on an interesting characterization of the intersection of two convex polygons in terms of the intersection of certain trapezoids from their trapezoidal decomposition. Also given is an alternative solution to the reporting problem, which runs in O(n4=3 logn + I) time, and is based on characterizing the intersection of two convex polygons via the intersection of their upper and lower chains and their leftmost vertices. The problems are interesting and challenging because the output size, I , can be much smaller than the total number of intersections between the boundaries of the polygons and because the number of polygons and their sizes can depend on n.