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 does not contain $s$ nor $t$. The procedure terminates when the sets in $\mathcal{ K}$ separate $s$ and $t$. 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.
Categories: cs.CG cs.DS