Semi-dynamic Connectivity in the Plane

@inproceedings{Cabello2015SemidynamicCI,
  title={Semi-dynamic Connectivity in the Plane},
  author={Sergio Cabello and Michael Kerber},
  booktitle={WADS},
  year={2015}
}
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… 

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