Abstract
We consider a natural variation of the concept of stabbing a segment by a simple polygon: a segment is stabbed by a simple polygon \(\mathcal{P}\) if at least one of its two endpoints is contained in \(\mathcal{P}\). A segment set S is stabbed by \(\mathcal{P}\) if every segment of S is stabbed by \(\mathcal{P}\). We show that if S is a set of pairwise disjoint segments, the problem of computing the minimum perimeter polygon stabbing S can be solved in polynomial time. We also prove that for general segments the problem is NP-hard. Further, an adaptation of our polynomial-time algorithm solves an open problem posed by Löffler and van Kreveld [Algorithmica 56(2), 236–269 (2010)] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments.
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
Arkin, E.M., Díaz-Báñez, J.M., Hurtado, F., Kumar, P., Mitchell, J.S.B., Palop, B., Pérez-Lantero, P., Saumell, M., Silveira, R.I.: Bichromatic 2-center of pairs of points. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 25–36. Springer, Heidelberg (2012)
Arkin, E.M., Dieckmann, C., Knauer, C., Mitchell, J.S., Polishchuk, V., Schlipf, L., Yang, S.: Convex transversals. Comput. Geom. (2012) (article in press)
Blömer, J.: Computing sums of radicals in polynomial time. In: FOCS 1991, pp. 670–677. IEEE Computer Society (1991)
Daescu, O., Ju, W., Luo, J.: NP-completeness of spreading colored points. In: Wu, W., Daescu, O. (eds.) COCOA 2010, Part I. LNCS, vol. 6508, pp. 41–50. Springer, Heidelberg (2010)
Díaz-Báñez, J.M., Korman, M., Pérez-Lantero, P., Pilz, A., Seara, C., Silveira, R.I.: New results on stabbing segments with a polygon. CoRR abs/1211.1490 (2012)
Dumitrescu, A., Jiang, M.: Minimum-perimeter intersecting polygons. Algorithmica 63(3), 602–615 (2012)
Goodrich, M.T., Snoeyink, J.: Stabbing parallel segments with a convex polygon. Computer Vision, Graphics, and Image Processing 49(2), 152–170 (1990)
Hassanzadeh, F., Rappaport, D.: Approximation algorithms for finding a minimum perimeter polygon intersecting a set of line segments. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 363–374. Springer, Heidelberg (2009)
Klincsek, G.T.: Minimal triangulations of polygonal domains. Ann. Discrete Math. 9, 121–123 (1980)
Löffler, M., van Kreveld, M.J.: Largest and smallest convex hulls for imprecise points. Algorithmica 56(2), 235–269 (2010)
Löffler, M., van Kreveld, M.J.: Largest bounding box, smallest diameter, and related problems on imprecise points. Comput. Geom. 43(4), 419–433 (2010)
Meijer, H., Rappaport, D.: Minimum polygon covers of parallel line segments. In: CCCG 1990, pp. 324–327 (1990)
Mukhopadhyay, A., Kumar, C., Greene, E., Bhattacharya, B.K.: On intersecting a set of parallel line segments with a convex polygon of minimum area. Inf. Proc. Lett. 105(2), 58–64 (2008)
Rappaport, D.: Minimum polygon transversals of line segments. Int. J. Comput. Geometry Appl. 5(3), 243–256 (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Díaz-Báñez, J.M., Korman, M., Pérez-Lantero, P., Pilz, A., Seara, C., Silveira, R.I. (2013). New Results on Stabbing Segments with a Polygon. In: Spirakis, P.G., Serna, M. (eds) Algorithms and Complexity. CIAC 2013. Lecture Notes in Computer Science, vol 7878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38233-8_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-38233-8_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38232-1
Online ISBN: 978-3-642-38233-8
eBook Packages: Computer ScienceComputer Science (R0)