Abstract
Let \(\mathcal{O} = \{O_1, \ldots, O_m\}\) be a set of m convex polygons in ℝ2 with a total of n vertices, and let B be another convex k-gon. A placement of B, any congruent copy of B (without reflection), is called free if B does not intersect the interior of any polygon in \(\mathcal{O}\) at this placement. A placement z of B is called critical if B forms three “distinct” contacts with \(\mathcal{O}\) at z. Let \(\varphi(B, \mathcal{O})\) be the number of free critical placements. A set of placements of B is called a stabbing set of \(\mathcal{O}\) if each polygon in \(\mathcal{O}\) intersects at least one placement of B in this set.
We develop efficient Monte Carlo algorithms that compute a stabbing set of size h = O(h *logm), with high probability, where h * is the size of the optimal stabbing set of \(\mathcal{O}\). We also improve bounds on \(\varphi(B, \mathcal{O})\) for the following three cases, namely, (i) B is a line segment and the obstacles in \(\mathcal{O}\) are pairwise-disjoint, (ii) B is a line segment and the obstacles in \(\mathcal{O}\) may intersect (iii) B is a convex k-gon and the obstacles in \(\mathcal{O}\) are disjoint, and use these improved bounds to analyze the running time of our stabbing-set algorithm.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Work by P.A, S.G, and M.S, was supported by a grant from the U.S.-Israel Binational Science Foundation. Work by P.A. and S.G. was also supported by NSF under grants CNS-05-40347, CFF-06-35000, and DEB-04-25465, by ARO grants W911NF-04-1-0278 and W911NF-07-1-0376, and by an NIH grant 1P50-GM-08183-01 and by a DOE grant OEGP200A070505. Work by M.S. was partially supported by NSF Grant CCF-05-14079, by grant 155/05 from the Israel Science Fund, by a grant from the AFIRST joint French-Israeli program, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work of D.C. was supported in part by the NSF under Grant CCF-0515203.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Agarwal, P.K., Chen, D.Z., Ganjugunte, S.K., Misołek, E., Sharir, M., Tang, K.: Stabbing convex polygons with a segment or a polygon (2008), http://www.cs.duke.edu/~shashigk/sstab/shortstab.pdf
Agarwal, P.K., Sharir, M.: Arrangements and their applications. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 49–119. Elsevier, Amsterdam (2000)
Aronov, B., Har-Peled, S.: On approximating the depth and related problems. In: Proc. of the 16th Annu. ACM-SIAM Sympos. Discrete Algorithms, pp. 886–894 (2005)
Berman, P., DasGupta, B.: Complexities of efficient solutions of rectilinear polygon cover problems. Algorithmica 17, 331–356 (1997)
Clarkson, K.L., Varadarajan, K.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37, 43–58 (2007)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001)
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45, 634–652 (1998)
Halperin, D., Kavraki, L., Latombe, J.-C.: Robotics. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 755–778. CRC Press, Boca Raton (1997)
Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985)
Leven, D., Sharir, M.: An efficient and simple motion planning algorithm for a ladder moving in two-dimensional space amidst polygonal barriers. In: Proc. 1st Annu. Sympos. on Comput. Geom., pp. 221–227. ACM, New York (1985)
Leven, D., Sharir, M.: On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space. Discrete Comput. Geom. 2, 255–270 (1987)
Megiddo, N., Supowit, K.J.: On the complexity of some common geometric location problems. SIAM J. Comput. 13, 182–196 (1984)
Megiddo, N., Tamir, A.: On the complexity of locating linear facilities in the plane. Operations Research Letters 1, 194–197 (1982)
Sharir, M.: Algorithmic motion planning in robotics. IEEE Computer 22, 9–20 (1989)
Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and their Geometric Applications. Cambridge University Press, New York (1995)
Vazirani, V.: Approximation Algorithms. Springer, Heidelberg (2004)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Agarwal, P.K., Chen, D.Z., Ganjugunte, S.K., Misiołek, E., Sharir, M., Tang, K. (2008). Stabbing Convex Polygons with a Segment or a Polygon. In: Halperin, D., Mehlhorn, K. (eds) Algorithms - ESA 2008. ESA 2008. Lecture Notes in Computer Science, vol 5193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87744-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-87744-8_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87743-1
Online ISBN: 978-3-540-87744-8
eBook Packages: Computer ScienceComputer Science (R0)