Nesting of Touching Polygons

TL;DR

This paper extends a sweep line algorithm to efficiently handle a set of polygons that may touch at boundaries, achieving optimal running time for convex polygons and similar cases.

Contribution

It adapts the sweep line algorithm to handle touching polygons and proves asymptotic optimality under certain conditions.

Findings

Algorithm runs in O(n+N log N) time, optimal for bounded maximal segments.

Successfully handles polygons sharing boundary points.

Applicable to convex polygons with bounded maximal segments.

Abstract

Polygons are cycles embedded into the plane; their vertices are associated with $x$- and $y$-coordinates and the edges are straight lines. Here, we consider a set of polygons with pairwise non-overlapping interior that may touch along their boundaries. Ideas of the sweep line algorithm by Bajaj and Dey for non-touching polygons are adapted to accommodate polygons that share boundary points. The algorithms established here achieves a running time of $\mathcal{O}(n+N\log N)$, where $n$ is the total number of vertices and $N<n$ is the total number of "maximal outstretched segments" of all polygons. It is asymptotically optimal if the number of maximal outstretched segments per polygon is bounded. In particular, this is the case for convex polygons.