Abstract
Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a non-crossing matching between point-object pairs. We show that when the objects we match the points to are finite point sets, the problem is NP-complete in general, and polynomial when the objects are on a line or when their number is at most 2. When the objects are line segments, we show that the problem is NP-complete in general, and polynomial when the segments form a convex polygon or are all on a line. Finally, for objects that are straight lines, we show that the problem of finding a min-max non-crossing matching is NP-complete.
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References
Agarwal, P., Aronov, B., Sharir, M., Suri, S.: Selecting distances in the plane. Algorithmica 9(5), 495–514 (1993)
Aichholzer, O., Bereg, S., Dumitrescu, A., García, A., Huemer, C., Hurtado, F., Kano, M., Márquez, A., Rappaport, D., Smorodinsky, S., Souvaine, D., Urrutia, J., Wood, D.R.: Compatible geometric matchings. Computational Geometry: Theory and Applications 42, 617–626 (2009)
Aichholzer, O., Cabello, S., Fabila-Monroy, R., Flores-Penaloza, D., Hackl, T., Huemer, C., Hurtado, F., Wood, D.R.: Edge-Removal and Non-Crossing Configurations in Geometric Graphs. In: Proceedings of 24th European Conference on Computational Geometry, pp. 119–122 (2008)
Alt, H., Guibas, L.: Discrete geometric shapes: Matching, interpolation, and approximation. In: Handbook of computational geometry, pp. 121–154 (1999)
Arkin, E., Kedem, K., Mitchell, J., Sprinzak, J., Werman, M.: Matching points into noise regions: combinatorial bounds and algorithms. In: Proceedings of the second annual ACM-SIAM symposium on Discrete algorithms, pp. 42–51 (1991)
Bentley, J.L., Ottmann, T.A.: Algorithms for reporting and counting geometric intersections. IEEE Transactions on Computers 28(9), 643–647 (1979)
Cabello, S., Giannopoulos, P., Knauer, C., Rote, G.: Matching point sets with respect to the Earth Mover’s Distance. Computational Geometry: Theory and Applications 39(2), 118–133 (2008)
Cardoze, D., Schulman, L.: Pattern matching for spatial point sets. In: Proceedings. 39th Annual Symposium on Foundations of Computer Science (FOCS), 1998, pp. 156–165 (1998)
Chazelle, B.: Triangulating a simple polygon in linear time. Discrete and Computational Geometry 6(1), 485–542 (1991)
Chew, L., Dor, D., Efrat, A., Kedem, K.: Geometric pattern matching in d-dimensional space. Discrete and Computational Geometry 21(2), 257–274 (1999)
Chew, L., Goodrich, M., Huttenlocher, D., Kedem, K., Kleinberg, J., Kravets, D.: Geometric pattern matching under Euclidean motion. Computational Geometry: Theory and Applications 7(1-2), 113–124 (1997)
Chew, L., Kedem, K.: Improvements on geometric pattern matching problems. In: Nurmi, O., Ukkonen, E. (eds.) SWAT 1992. LNCS, vol. 621, pp. 318–325. Springer, Heidelberg (1992)
Cohen, S.: Finding color and shape patterns in images. PhD thesis, Stanford University, Department of Computer Science (1999)
Colannino, J., Damian, M., Hurtado, F., Iacono, J., Meijer, H., Ramaswami, S., Toussaint, G.: An O(n logn)-time algorithm for the restriction scaffold assignment problem. Journal of Computational Biology 13(4), 979–989 (2006)
Efrat, A., Itai, A., Katz, M.: Geometry helps in bottleneck matching and related problems. Algorithmica 31(1), 1–28 (2001)
Formella, A.: Approximate point set match for partial protein structure alignment. In: Proceedings of Bioinformatics: Knowledge Discovery in Biology (BKDB 2005), Facultade Ciencias Lisboa da Universidade de Lisboa, pp. 53–57 (2005)
Giannopoulos, P., Veltkamp, R.: A pseudo-metric for weighted point sets. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 715–730. Springer, Heidelberg (2002)
Grauman, K., Darrell, T.: Fast contour matching using approximate earth mover’s distance. In: Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 220–227 (2004)
Heffernan, P.: Generalized approximate algorithms for point set congruence. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1993. LNCS, vol. 709, pp. 373–373. Springer, Heidelberg (1993)
Heffernan, P., Schirra, S.: Approximate decision algorithms for point set congruence. In: Proceedings of the eighth annual Symposium on Computational geometry, pp. 93–101 (1992)
Huttenlocher, D., Kedem, K.: Efficiently computing the Hausdorff distance for point sets under translation. In: Proceedings of the Sixth ACM Symposium on Computational Geometry, pp. 340–349 (1990)
Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane—a survey. Discrete & Computational Geometry 25, 551–570 (2003)
Kirkpatrick, D.: Optimal search in planar subdivisions. SIAM Journal on Computing 12(1), 28–35 (1983)
Lovász, L., Plummer, M.D.: Matching theory. Elsevier Science Ltd., Amsterdam (1986)
Rappaport, D.: Tight bounds for visibility matching of f-equal width objects. In: Akiyama, J., Kano, M. (eds.) JCDCG 2002. LNCS, vol. 2866, pp. 246–250. Springer, Heidelberg (2002)
Typke, R., Giannopoulos, P., Veltkamp, R., Wiering, F., Van Oostrum, R.: Using transportation distances for measuring melodic similarity. In: Proceedings of the 4th International Conference on Music Information Retrieval (ISMIR 2003), pp. 107–114 (2003)
Vaidya, P.: Geometry helps in matching. In: STOC 1988: Proceedings of the twentieth annual ACM symposium on Theory of computing, pp. 422–425. ACM, New York (1988)
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Aloupis, G. et al. (2010). Matching Points with Things. In: López-Ortiz, A. (eds) LATIN 2010: Theoretical Informatics. LATIN 2010. Lecture Notes in Computer Science, vol 6034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12200-2_40
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DOI: https://doi.org/10.1007/978-3-642-12200-2_40
Publisher Name: Springer, Berlin, Heidelberg
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