Recognizing Weakly Simple Polygons

Recognizing Weakly Simple Polygons

We present an O(n log n)-time algorithm that determines whether a given planar n-gon is weakly simple. This improves upon an O(n log n)-time algorithm by Chang, Erickson, and Xu [5]. Weakly simple polygons are required as input for several geometric algorithms. As such, how to recognize simple or weakly simple polygons is a fundamental question.

Detecting Weakly Simple Polygons

5 A closed curve in the plane is weakly simple if it is the limit (in the Fréchet metric) 6 of a sequence of simple closed curves. We describe an algorithm to determine whether 7 a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, 8 improving an earlier O(n3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As 9 an immediate corollary, we obtain the fir...

On Recognizing and Characterizing Visibility Graphs of Simple Polygons

Reconstruction of Weakly Simple Polygons from their Edges

Given n line segments in the plane, do they form the edge set of a weakly simple polygon; that is, can the segment endpoints be perturbed by at most ε, for any ε > 0, to obtain a simple polygon? While the analogous question for simple polygons can easily be answered in O(n logn) time, we show that it is NP-complete for weakly simple polygons. We give O(n)-time algorithms in two special cases: w...

Recognizing S-Star Polygons

We consider the problem of recognizing star-polygons under staircase visibility (s-visibility). We show that the s-visibility polygon from a point inside a simple orthogonal polygon of n sides can be computed in O(n) time. When the polygon contains holes the algorithm runs in O(n log n) time, which we prove to be optimal by linear time reduction from sorting. We present an algorithm for computi...