Abstract
We introduce a new measure for planar point sets S. Intuitively, it describes the combinatorial distance from a convex set: The reflexivity ρ(S) of S is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study naturally takes us into the examination of some closely related quantities, such as the convex cover number κ 1 (S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex partition number κ 2 (S), which is given by the smallest number of disjoint convex chains that cover S. We prove that it is NP-complete to determine the convex cover or the convex partition number, and we give logarithmic-approximation algorithms for determining each.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
P. K. Agarwal. Ray shooting and other applications of spanning trees with low stabbing number. SIAM J. Comput., 21, 540–570, 1992.
A. Aggarwal, D. Coppersmith, S. Khanna, R. Motwani, and B. Schieber. The angular-metric traveling salesman problem. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 221–229, Jan. 1997.
N. Amenta, M. Bern, and D. Eppstein. The crust and the β-skeleton: Combinatorial curve reconstruction. Graphical Models and Image Processing, 60, 125–135, 1998.
B. Chazelle. Computational geometry and convexity. Ph.D. thesis, Dept. Comput. Sci., Yale Univ., New Haven, CT, 1979. Carnegie-Mellon Univ. Report CS-80-150.
T. K. Dey and P. Kumar. A simple provable algorithm for curve reconstruction. In Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, pages 893–894, Jan. 1999.
T. K. Dey, K. Mehlhorn, and E. A. Ramos. Curve reconstruction: Connecting dots with good reason. In Proc. 15th Annu. ACM Sympos. Comput. Geom., pages 197–206, 1999.
D. P. Dobkin, H. Edelsbrunner, and M. H. Overmars. Searching for empty convex polygons. Algorithmica, 5, 561–571, 1990.
P. Erdős and G. Szekeres. A combinatorial problem in geometry. Compositio Math., 2, 463–470, 1935.
P. Erdős and G. Szekeres. On some extremum problem in geometry. Ann. Univ. Sci. Budapest, 3–4, 53–62, 1960.
S. P. Fekete and G. J. Woeginger. Angle-restricted tours in the plane. Comp. Geom. Theory Appl., 8, 195–218, 1997.
A. Garcýa, M. Noy, and J. Tejel. Lower bounds for the number of crossing-free subgraphs of K n. In Proc. 7th Canad. Conf. Comput. Geom., pages 97–102, 1995.
J. Hershberger and S. Suri. A pedestrian approach to ray shooting: Shoot a ray, take a walk. J. Algorithms, 18, 403–431, 1995.
S. Hertel and K. Mehlhorn. Fast triangulation of the plane with respect to simple polygons. Inf. Control, 64, 52–76, 1985.
J. Hershberger and S. Suri. Applications of a semi-dynamic convex hull algorithm. BIT, 32, 249–267, 1992.
J. Horton. Sets with no empty convex 7-gons. Canad. Math. Bull., 26, 482–484, 1983.
F. Hurtado and M. Noy. Triangulations, visibility graph and reflex vertices of a simple polygon. Comput. Geom. Theory Appl., 6, 355–369, 1996.
J. M. Keil. Polygon decomposition. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 491–518. Elsevier Science Publishers B.V. North-Holland, Amsterdam, 2000.
D. Lichtenstein. Planar formulae and their uses. SIAM J. Comput., 11, 329–343, 1982.
J. S. B. Mitchell. Approximation algorithms for geometric separation problems. Technical report, Department of Applied Mathematics, SUNY Stony Brook, NY, July 1993.
J. S. B. Mitchell, G. Rote, G. Sundaram, and G. Woeginger. Counting convex polygons in planar point sets. Inform. Process. Lett., 56, 191–194, 1995.
J. Pach (ed.). Discrete and Computational Geometry, 19, Special issue dedicated to Paul Erdös, 1998.
M. Urabe. On a partition into convex polygons. Discrete Appl. Math., 64, 179–191, 1996.
M. Urabe. On a partition of point sets into convex polygons. In Proc. 9th Canad. Conf. Comp. Geom., pages 21–24, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Arkin, E.M. et al. (2001). On the Reflexivity of Point Sets. In: Dehne, F., Sack, JR., Tamassia, R. (eds) Algorithms and Data Structures. WADS 2001. Lecture Notes in Computer Science, vol 2125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44634-6_18
Download citation
DOI: https://doi.org/10.1007/3-540-44634-6_18
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42423-9
Online ISBN: 978-3-540-44634-7
eBook Packages: Springer Book Archive