Abstract
We revisit some maximization problems for geometric networks design under the non-crossing constraint, first studied by Alon, Rajagopalan and Suri (ACM Symposium on Computational Geometry, 1995). Given a set of n points in the plane in general position (no three points collinear), compute a longest non-crossing configuration composed of straight line segments that is: (a) a matching, (b) a Hamiltonian path, and (c) a spanning tree. We obtain some new results for (b) and (c), as well as for the Hamiltonian cycle problem.
(i) For the longest non-crossing Hamiltonian path problem, we give an approximation algorithm with ratio \(\frac{2}{\pi+1}\approx0.4829\). The previous best ratio, due to Alon et al., was \(\frac{1}{\pi }\approx0.3183\). The ratio of our algorithm is close to \(\frac{2}{\pi}\approx0.6366\) on a relatively broad class of instances: for point sets whose perimeter (or diameter) is much shorter than the maximum length matching. For instance “random” point sets meet the condition with high probability. The algorithm runs in O(n 7/3log n) time.
(ii) For the longest non-crossing spanning tree problem, we give an approximation algorithm with ratio 0.502 which runs in O(nlog n) time. The previous ratio, 1/2, due to Alon et al., was achieved by a quadratic time algorithm. Along the way, we first re-derive the result of Alon et al. with a faster algorithm and a very simple analysis.
(iii) For the longest non-crossing Hamiltonian cycle problem, we give an approximation algorithm whose ratio is close to 2/π on a relatively broad class of instances: for point sets where the product 〈diameter×convex hull size〉 is much smaller than the maximum length matching. Again “random” point sets meet the condition with high probability. However, this algorithm does not come with a constant approximation guarantee for all instances. The algorithm runs in O(n 7/3log n) time. No previous approximation results were known for this problem.
Article PDF
Similar content being viewed by others
References
Abellanas, M., Garcia, J., Hernández, G., Noy, M., Ramos, P.: Bipartite embeddings of trees in the plane. Discrete Appl. Math. 93, 141–148 (1999)
Aichholzer, O., Cabello, S., Fabila-Monroy, R., Flores-Peñaloza, D., Hackl, T., Huemer, C., Hurtado, F., Wood, D.R.: Edge-removal and non-crossing configurations in geometric graphs. Discrete Math. Theor. Comput. Sci. 12(1), 75–86 (2010)
Alon, N., Rajagopalan, S., Suri, S.: Long non-crossing configurations in the plane. Fund. Inf. 22, 385–394 (1995). Also in Proc. of the 9-th Annual Symposium on Computational Geometry, pp. 257–263. ACM Press (1993)
Asano, T., Ghosh, S., Shermer, T.: Visibility in the plane. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 829–876. Elsevier, Amsterdam (2000)
Bern, M., Eppstein, D.: Approximation algorithms for geometric problems. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-hard Problems, pp. 296–345. PWS, Boston (1997)
Černý, J., Dvoŕák, Z., Jelínek, V., Kára, J.: Noncrossing Hamiltonian paths in geometric graphs. Discrete Appl. Math. 155, 1096–1105 (2007)
Dey, T.K.: Improved bounds on planar k-sets and related problems. Discrete Comput. Geom. 19, 373–382 (1998)
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Heidelberg (1987)
Edelsbrunner, H., Welzl, E.: On the number of separations of a finite set in the plane. J. Comb. Theory Ser. A 38, 15–29 (1985)
Eppstein, D.: Spanning trees and spanners. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 425–461. Elsevier, Amsterdam (2000)
Erdős, P., Lovász, L., Simmons, A., Straus, E.: Dissection graphs of planar point sets. In: Srivastava, J.N. (ed.) A Survey of Combinatorial Theory, pp. 139–154. North-Holland, Amsterdam (1973)
Fekete, S.P.: Simplicity and hardness of the maximum traveling salesman problem under geometric distances. In: Proceedings of the 10-th ACM-SIAM Symposium on Discrete Algorithms, January 1999 (SODA ’99), pp. 337–345
Hurtado, F., Kano, M., Rappaport, D., Tóth, Cs.D.: Encompassing colored planar straight line graphs. Comput. Geom., Theory Appl. 39(1), 14–23 (2008)
Károlyi, G., Pach, J., Tóth, G.: Ramsey-type results for geometric graphs. I. Discrete Comput. Geom. 18, 247–255 (1997)
Károlyi, G., Pach, J., Tóth, G., Valtr, P.: Ramsey-type results for geometric graphs. II. Discrete Comput. Geom. 20, 375–388 (1998)
Lovász, L.: On the number of halving lines. Ann. Univ. Sci. Budapest, Eötvös, Sec. Math. 14, 107–108 (1971)
Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 633–701. Elsevier, Amsterdam (2000)
O’Rourke, J.: Visibility. In: Goodman, J., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., pp. 643–663. Chapman & Hall, London (2004)
Preparata, F., Shamos, M.: Computational Geometry: An Introduction. Springer, New York (1985)
Tóth, G.: Point sets with many k-sets. Discrete Comput. Geom. 26, 187–194 (2001)
Yaglom, I.M., Boltyanskiĭ, V.G.: Convex Figures. Holt, Rinehart and Winston, New York (1961)
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of this paper appeared in the Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010), Nancy, France, March 2010, pp. 299–310.
A. Dumitrescu was supported in part by NSF CAREER grant CCF-0444188. Part of the research by this author was done at Ecole Polytechnique Fédérale de Lausanne.
C.D. Tóth was supported in part by NSERC grant RGPIN 35586. Part of the research by this author was done at Tufts University.
Rights and permissions
About this article
Cite this article
Dumitrescu, A., Tóth, C.D. Long Non-crossing Configurations in the Plane. Discrete Comput Geom 44, 727–752 (2010). https://doi.org/10.1007/s00454-010-9277-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-010-9277-9