Sweeping Arrangements of Non-Piercing Regions in the 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 \(\alpha , \beta \in \Gamma \), both \(\tilde{\alpha } \,\setminus \, \tilde{\beta }\) and \(\tilde{\beta }\,\setminus \,\tilde{\beta }\) 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 sweep curve \(\gamma \in {\Gamma }\), then the arrangement can be 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 \(\gamma \) outwards so that P lies outside \(\tilde{\gamma }\), and the arrangement remains non-piercing. We give several applications of our results to combinatorial and algorithmic questions including to the multi-hitting set problem involving points and non-piercing regions.