Abstract
Let T be a triangulation of a simple polygon. A flip in T is the operation of replacing one diagonal of T by a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study. We show that computing the flip distance between two triangulations of a simple polygon is NP-hard. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.
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Notes
The flip-kernel of D might not be completely inside the polygon D. This is in contrast to the “visibility kernel” of a polygon.
The exact time bounds shown in the proof are irrelevant for the NP-hardness reduction (which even requires a different model of computation). We mention them only as they may be of general interest.
Actually, any dual transform will do. When thinking of the rounding process as a continuous transformation, a change of the order type would involve a collapsing triangular cell of the dual arrangement, indicating a “close” point triple.
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Acknowledgments
O. Aichholzer and A. Pilz were supported by the ESF EUROCORES programme EuroGIGA - ComPoSe, Austrian Science Fund (FWF): I 648-N18. W. Mulzer was supported in part by DFG project MU/3501/1. Part of this work was done while A. Pilz was recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology, Austria. Preliminary versions have appeared as O. Aichholzer, W. Mulzer, and A. Pilz, Flip Distance Between Triangulations of a Simple Polygon is NP-Complete in Proc. 29th EuroCG, pp. 115–118, 2013, and in Proc. 21st ESA, pp. 13–24, 2013 [2, 3].
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Aichholzer, O., Mulzer, W. & Pilz, A. Flip Distance Between Triangulations of a Simple Polygon is NP-Complete. Discrete Comput Geom 54, 368–389 (2015). https://doi.org/10.1007/s00454-015-9709-7
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DOI: https://doi.org/10.1007/s00454-015-9709-7