Semi-dynamic Connectivity in the Plane

Abstract

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 $\mathcal {K}=\emptyset$. At each step we add to $\mathcal {K}$ a compact convex set that is disjoint from s and t. We must recognize when the union of the sets in $\mathcal {K}$ separates s and t, at which point the procedure terminates. We show how to add one set to $\mathcal {K}$ in $O(1+k\alpha (n))$ amortized time plus the time needed to find all sets of $\mathcal {K}$ intersecting the newly added set, where n is the cardinality of $\mathcal {K}$, k is the number of sets in $\mathcal {K}$ intersecting the newly added set, and $\alpha (\cdot )$ is the inverse of the Ackermann function.