Semi-dynamic connectivity in the plane

TL;DR

This paper studies a semi-dynamic process for separating two points in the plane using convex sets, providing an efficient method to update the separating set with near-constant amortized time per addition.

Contribution

It introduces an efficient algorithm for dynamically adding convex sets to separate two points, with amortized time complexity involving the inverse Ackermann function.

Findings

Efficiently adds convex sets in O(1 + kα(n)) amortized time

Provides a method to maintain separation of points in the plane

Analyzes the complexity of the dynamic separation process

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 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.