Sweeping Arrangements of Non-Piercing Curves in Plane

Abstract

Let $\Gamma$ be an arrangement of Jordan curves in the plane, i.e., simple closed curves in the plane. For any curve $\gamma \in \Gamma$, we denote the bounded region enclosed by $\gamma$ as $\tilde{\gamma}$. We say that $\Gamma$ is non-piercing if for any two curves $\alpha , \beta \in \Gamma$, $\tilde{\alpha} \,\setminus\, \tilde{\beta}$ is connected. A non-piercing arrangement of curves generalizes a set of $2$-intersecting curves in which each pair of curves intersect in at most two points. Snoeyink and Hershberger (``Sweeping Arrangements of Curves'', SoCG '89) proved that if we are given an arrangement $\Gamma$ of $2$-intersecting curves and a {\em sweep} curve $\gamma\in{\Gamma}$, then the arrangement can be \emph{swept} by $\gamma$ while always maintaining the $2$-intersecting property of the curves in $\Gamma$. We generalize the result of Snoeyink and Hershberger to the setting of non-piercing arrangements. Given an arrangement $\Gamma$ of non-piercing curves, a sweep curve $\gamma\in \Gamma$, and a point $P$ in $\tilde{\gamma}$, we show that we can continuously shrink $\gamma$ to $P$ so that throughout the process, the arrangement remains non-piercing (except at a finite set of points in time where $\gamma$ crosses other curves), and $P$ lies in $\tilde{\gamma}$. We show that our arguments can be modified if $P$ lies outside $\tilde{\gamma}$, and we want to sweep $\gamma$ \emph{outwards} so that $P$ lies outside $\tilde{\gamma}$, and the arrangement remains non-piercing. As a second contribution, we give an alternate proof of the result of Snoeyink and Hershberger, and give several applications of our results to combinatorial and algorithmic questions including to the \emph{multi-hitting set} problem involving points and non-piercing regions.