LATIN 2010: LATIN 2010: Theoretical Informatics pp 456-467

Matching Points with Things

• Greg Aloupis
• Jean Cardinal
• Sébastien Collette
• Erik D. Demaine
• Martin L. Demaine
• Muriel Dulieu
• Ruy Fabila-Monroy
• Vi Hart
• Stefan Langerman
• Maria Saumell
• Carlos Seara
• Perouz Taslakian
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6034)

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.

Keywords

Line Segment Computational Geometry Convex Polygon Strong Connected Component Matching Edge
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.

References

1. 1.
Agarwal, P., Aronov, B., Sharir, M., Suri, S.: Selecting distances in the plane. Algorithmica 9(5), 495–514 (1993)
2. 2.
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)
3. 3.
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)Google Scholar
4. 4.
Alt, H., Guibas, L.: Discrete geometric shapes: Matching, interpolation, and approximation. In: Handbook of computational geometry, pp. 121–154 (1999)Google Scholar
5. 5.
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)Google Scholar
6. 6.
Bentley, J.L., Ottmann, T.A.: Algorithms for reporting and counting geometric intersections. IEEE Transactions on Computers 28(9), 643–647 (1979)
7. 7.
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)
8. 8.
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)Google Scholar
9. 9.
Chazelle, B.: Triangulating a simple polygon in linear time. Discrete and Computational Geometry 6(1), 485–542 (1991)
10. 10.
Chew, L., Dor, D., Efrat, A., Kedem, K.: Geometric pattern matching in d-dimensional space. Discrete and Computational Geometry 21(2), 257–274 (1999)
11. 11.
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)
12. 12.
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)Google Scholar
13. 13.
Cohen, S.: Finding color and shape patterns in images. PhD thesis, Stanford University, Department of Computer Science (1999)Google Scholar
14. 14.
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)
15. 15.
Efrat, A., Itai, A., Katz, M.: Geometry helps in bottleneck matching and related problems. Algorithmica 31(1), 1–28 (2001)
16. 16.
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)Google Scholar
17. 17.
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)
18. 18.
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)Google Scholar
19. 19.
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)Google Scholar
20. 20.
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)Google Scholar
21. 21.
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)Google Scholar
22. 22.
Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane—a survey. Discrete & Computational Geometry 25, 551–570 (2003)
23. 23.
Kirkpatrick, D.: Optimal search in planar subdivisions. SIAM Journal on Computing 12(1), 28–35 (1983)
24. 24.
Lovász, L., Plummer, M.D.: Matching theory. Elsevier Science Ltd., Amsterdam (1986)
25. 25.
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)Google Scholar
26. 26.
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)Google Scholar
27. 27.
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)

Authors and Affiliations

• Greg Aloupis
• 1
• Jean Cardinal
• 1
• Sébastien Collette
• 1
• Erik D. Demaine
• 2
• Martin L. Demaine
• 2
• Muriel Dulieu
• 3
• Ruy Fabila-Monroy
• 4
• Vi Hart
• 5
• 6
• Stefan Langerman
• 1
• Maria Saumell
• 6
• Carlos Seara
• 6
• Perouz Taslakian
• 1
1. 1.Université Libre de BruxellesBrusselsBelgium
2. 2.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA
3. 3.Polytechnic Institute of NYUUSA
4. 4.Instituto de MatemáticasUniversidad Nacional Autónoma de México
5. 5.Stony Brook UniversityStony BrookUSA
6. 6.Universitat Politècnica de CatalunyaBarcelonaSpain