Detecting Weakly Simple Polygons
Abstract
A closed curve in the plane is weakly simple if it is the limit (in the Fréchet 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 nvertex 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2014
 DOI:
 10.48550/arXiv.1407.3340
 arXiv:
 arXiv:1407.3340
 Bibcode:
 2014arXiv1407.3340C
 Keywords:

 Computer Science  Computational Geometry
 EPrint:
 25 pages and 13 figures, submitted to SODA 2015