Detecting Weakly Simple Polygons
نویسندگان
چکیده
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 first efficient algorithm to determine whether an 10 arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n2 log n) time. We also 11 describe algorithms that detect weak simplicity in O(n log n) time for two interesting classes 12 of polygons. Finally, we discuss subtle errors in several previously published definitions of 13 weak simplicity. 14 ∗Work on this paper was partially supported by NSF grant CCF-0915519. See http://www.cs.illinois.edu/~jeffe/pubs/weak. html for the most recent version of this paper. Detecting Weakly Simple Polygons 1
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