Semi-dynamic Connectivity in the Plane

Motivated by a path planning problem we consider the following procedure. Assume that we have two points s and t in the plane and take K = ∅. At each step we add to K a compact convex set that does not contain s nor t. The procedure terminates when the sets in K separate s and t. We show how to add one set to K in O(1 + kα(n)) amortized time plus the time needed to find all sets of K intersecting the newly added set, where n is the cardinality of K, k is the number of sets in K intersecting the newly added set, and α(·) is the inverse of the Ackermann function.