Abstract
Let C be a polygonal cycle on n vertices in the plane. A randomized algorithm is presented which computes in O(n log3 n) expected time, the edge of C whose removal results in a polygonal path of smallest possible dilation. It is also shown that the edge whose removal gives a polygonal path of largest possible dilation can be computed in O(n logn) time. If C is a convex polygon, the running time for the latter problem becomes O(n). Finally, it is shown that for each edge e of C, a (1 − ε)-approximation to the dilation of the path C ∖ {e} can be computed in O(n logn) total time.
Part of this work was done during the Korean Workshop on Computational Geometry at Schloß Dagstuhl in 2006.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Agarwal, P.K., Klein, R., Knauer, C., Sharir, M.: Computing the detour of polygonal curves. Technical Report B 02-03, Fachbereich Mathematik und Informatik, Freie Universität Berlin (2002)
Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. Journal of the ACM 42, 67–90 (1995)
Clarkson, K.L., Shor, P.W.: Algorithms for diametral pairs and convex hulls that are optimal, randomized, and incremental. In: SCG 1988: Proceedings of the fourth annual symposium on Computational geometry, pp. 12–17. ACM Press, New York (1988)
Fortune, S.J.: A sweepline algorithm for Voronoi diagrams. Algorithmica 2, 153–174 (1987)
Fredman, M.L., Komlos, J., Szemeredi, E.: Storing a sparse table with O(1) worst case access time. Journal of the ACM 31, 538–544 (1984)
Kirkpatrick, D.G.: Optimal search in planar subdivisions. SIAM Journal on Computing 12, 28–35 (1983)
Langerman, S., Morin, P., Soss, M.: Computing the maximum detour and spanning ratio of planar paths, trees and cycles. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 250–261. Springer, Heidelberg (2002)
Lee, D.T., Preparata, F.P.: The all nearest-neighbor problem for convex polygons. Information Processing Letters 7, 189–192 (1978)
Lenhof, H.P., Smid, M.: Sequential and parallel algorithms for the k closest pairs problem. International Journal of Computational Geometry & Applications 5, 273–288 (1995)
Narasimhan, G., Smid, M.: Approximating the stretch factor of Euclidean graphs. SIAM Journal on Computing 30, 978–989 (2000)
Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)
Smid, M.: Closest-point problems in computational geometry. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 877–935. Elsevier Science, Amsterdam (2000)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ahn, HK., Farshi, M., Knauer, C., Smid, M., Wang, Y. (2007). Dilation-Optimal Edge Deletion in Polygonal Cycles. In: Tokuyama, T. (eds) Algorithms and Computation. ISAAC 2007. Lecture Notes in Computer Science, vol 4835. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77120-3_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-77120-3_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77118-0
Online ISBN: 978-3-540-77120-3
eBook Packages: Computer ScienceComputer Science (R0)