A plane-sweep algorithm for finding a closest pair among convex planar objects

Abstract

Given a set of geometric objects a closest pair is a pair of objects whose mutual distance is smallest. We present a plane-sweep algorithm which finds a closest pair with respect to any LP-metric, 1≤p≤∞, for planar configurations consisting of n (possibly intersecting) compact convex objects such as line segments, circular discs and convex polygons. For configurations of line segments or discs the algorithm runs in asymptotically optimal time O(n log n). For a configuration of n convex m-gons given in a suitable representation it finds a closest pair with respect to the Euclidean metric L2 in time O(n log(n·m)).