Detecting Weakly Simple Polygons

TL;DR

This paper presents efficient algorithms for detecting weakly simple polygons and closed curves, improving previous methods and analyzing the concept's subtle nuances in geometric graph theory.

Contribution

It introduces the first efficient algorithms for weakly simple polygon detection, reducing complexity from cubic to near-linear time for specific cases.

Findings

Algorithm for weakly simple closed walk in O(n log n) time

Efficient detection of weakly simple polygons in O(n^2 log n) time

Discussion of errors in prior definitions of weak simplicity

Abstract

A closed curve in the plane is weakly simple if it is the limit (in the Fr\'echet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We also describe algorithms that detect weak simplicity in O(n log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.