Abstract
We present a new algorithm for computing the straight skeleton of a polygon. For a polygon with n vertices, among which r are reflex vertices, we give a deterministic algorithm that reduces the straight skeleton computation to a motorcycle graph computation in O(n (logn)logr) time. It improves on the previously best known algorithm for this reduction, which is randomized, and runs in expected \(O(n \sqrt{h+1}\log^2 n)\) time for a polygon with h holes. Using known motorcycle graph algorithms, our result yields improved time bounds for computing straight skeletons. In particular, we can compute the straight skeleton of a non-degenerate polygon in O(n (logn) logr + r 4/3 + ε) time for any ε > 0. On degenerate input, our time bound increases to O(n (logn) logr + r 17/11 + ε).
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References
Aichholzer, O., Alberts, D., Aurenhammer, F., Gärtner, B.: A novel type of skeleton for polygons. Journal of Universal Computer Science 1(12), 752–761 (1995)
Barequet, G., Goodrich, M., Levi-Steiner, A., Steiner, D.: Straight-skeleton based contour interpolation. In: Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 119–127 (2003)
Berg, M.D., Cheong, O., Kreveld, M.V., Overmars, M.: Computational Geometry: Algorithms and Applications. Springer (2008)
Bowers, J.: Computing the straight skeleton of a simple polygon from its motorcycle graph in deterministic O(n log n) time. CoRR abs/1405.6260 (2014)
Cheng, S.W., Mencel, L., Vigneron, A.: A faster algorithm for computing straight skeletons. CoRR abs/1405.4691 (2014)
Cheng, S.W., Vigneron, A.: Motorcycle graphs and straight skeletons. Algorithmica 47(2), 159–182 (2007)
Chin, F., Snoeyink, J., Wang, C.A.: Finding the medial axis of a simple polygon in linear time. Discrete and Computational Geometry 21(3), 405–420 (1999)
Cloppet, F., Oliva, J., Stamon, G.: Angular bisector network, a simplified generalized voronoi diagram: Application to processing complex intersections in biomedical images. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(1), 120–128 (2000)
Coquillart, S., Oliva, J., Perrin, M.: 3d reconstruction of complex polyhedral shapes from contours using a simplified generalized voronoi diagram. Computer Graphics Forum 15(3), 397–408 (1996)
Day, A., Laycock, R.: Automatically generating large urban environments based on the footprint data of buildings. In: Proceedings of the 8th ACM Symposium on Solid Modeling and Applications, pp. 346–351 (2003)
Demaine, E.D., Demaine, M.L., Lubiw, A.: Folding and cutting paper. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 104–118. Springer, Heidelberg (2000)
Demaine, E.D., Demaine, M.L., Lubiw, A.: Folding and one straight cut suffice. In: Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 891–892 (1999)
Demaine, E.D., Demaine, M.L., Mitchell, J.S.B.: Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami. In: Proceedings of the 15th Annual ACM Symposium on Computational Geometry, pp. 105–114 (1999)
Eppstein, D., Erickson, J.: Raising roofs, crashing cycles, and playing pool: Applications of a data structure for finding pairwise interactions. Discrete and Computational Geometry 22(4), 569–592 (1999)
Felkel, P., Obdržálek, Š.: Straight skeleton implementation. In: Proceedings of the 14th Spring Conference on Computer Graphics, pp. 210–218 (1998)
Held, M., Huber, S.: Theoretical and practical results on straight skeletons of planar straight-line graphs. In: Proceedings of the 27th Symposium on Computational Geometry, pp. 171–178 (2011)
Held, M., Huber, S.: A fast straight-skeleton algorithm based on generalized motorcycle graphs. International Journal of Computational Geometry and Applications 22(5), 471–498 (2012)
Hershberger, J.: Finding the upper envelope of n line segments in O(n log n) time. Information Processing Letters 33(4), 169–174 (1989)
Kelly, T., Wonka, P.: Interactive architectural modeling with procedural extrusions. ACM Transactions on Graphics 30(2), 14:1–14:15 (2011)
von Peschka, G.: Kotirte Ebenen: Kotirte Projektionen und deren Anwendung. Buschak and Irrgang, Vorträge (1877)
Vigneron, A., Yan, L.: A faster algorithm for computing motorcycle graphs. In: Proceedings of the 29th Symposium on Computational Geometry, pp. 17–26 (2013)
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Cheng, SW., Mencel, L., Vigneron, A. (2014). A Faster Algorithm for Computing Straight Skeletons. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_23
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DOI: https://doi.org/10.1007/978-3-662-44777-2_23
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