# Computing Largest Common Point Sets under Approximate Congruence

- First Online:

## Abstract

The problem of computing a *largest common point set* (LCP) between two point sets under ε-congruence with the bottleneck matching metric has recently been a subject of extensive study. Although polynomial time solutions are known for the planar case and for restricted sets of transformations and metrics (like translations and the Hausdorff-metric under L_{∞}-norm), no complexity results are formally known for the general problem. In this paper we give polynomial time algorithms for this problem under different classes of transformations and metrics for any fixed dimension, and establish NP-hardness for unbounded dimensions. Any solution to this (or related) problem, especially in higher dimensions, is generally believed to involve implementation difficulties because they rely on the computation of intersections between algebraic surfaces. We show that (contrary to intuitive expectations) this problem can be solved under a rational arithmetic model in a straightforward manner if the set of transformations is *extended* to general affine transformations under the *L*_{∞}-norm (difficulty of this problem is generally expected to be in the order: translations < rotation < isometry < more general). To the best of our knowledge this is also the first paper which deals with the LCP-problem under such a general class of transformations.

## Preview

Unable to display preview. Download preview PDF.

### References

- 1.T. Akutsu. On determining the congruence of point sets in
*d*dimensions.*Computational Geometry: Theory and Applications*, 9:247–256, 1998.MATHMathSciNetGoogle Scholar - 2.T. Akutsu and M.M. Halldórsson. On approximation of largest common subtrees and largest common point sets.
*Theoretical Computer Science*, 233(1–2):33–50, 2000.MATHCrossRefMathSciNetGoogle Scholar - 3.T. Akutsu, H. Tamaki, and T. Tokuyama. Distribution of distances and triangles in a point set and algorithms for computing the largest common point sets.
*Discrete and Computational Geometry*, 20:307–331, 1998.MATHCrossRefMathSciNetGoogle Scholar - 4.H. Alt and L. Guibas. Discrete geometric shapes: Matching, interpolation, and approximation. In J.-R. Sack and J. Urrutia, editors,
*Handbook of Computational Geometry*, pages 121–153. Elsevier Science Publishers B.V. North-Holland, 1999.Google Scholar - 5.H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl. Congruence, similarity, and symmetries of geometric objects.
*Discrete and Computational Geometry*, 3:237–256, 1988.MATHCrossRefMathSciNetGoogle Scholar - 6.D. Avis and K. Fukuda. Reverse search for enumeration.
*Discrete and Applied Mathematics*, 65:21–46, 1996.MATHCrossRefMathSciNetGoogle Scholar - 7.S. Basu, R. Pollack, and M.-F. Roy. A new algorithm to find a point in every cell defined by a family of polynomials. In B.F. Caviness and J. Johnson, editors,
*Proc. Symp. on Quantifier Elimination and Cylindrical Algebraic Decomposition*. Springer Verlag, 1995.Google Scholar - 8.S. Basu, R. Pollack, and M.-F. Roy. On computing a set of points meeting every semi-algebraically connected component of a family of polynomials on a variety.
*Journal of Complexity*, 13:28–37, 1997.MATHCrossRefMathSciNetGoogle Scholar - 9.D.E. Cardoze and L.J. Schulman. Pattern matching for spatial point sets. In
*Proc. 39th Annual Symposium on Foundations of Computer Science*, pages 156–165, 1998.Google Scholar - 10.S. Chakraborty and S. Biswas. Approximation algorithms for 3-D common substructure identification in drug and protein molecules. In
*Proc. 6th. International Workshop on Algorithms and Data Structures*, LNCS 1663, pages 253–264, 1999.CrossRefGoogle Scholar - 11.L.P. Chew, D. Dor, A. Efrat, and K. Kedem. Geometric pattern matching in
*d*-dimensional space.*Discrete and Computational Geometry*, 21:257–274, 1999.MATHCrossRefMathSciNetGoogle Scholar - 12.P. Chew, M. Goodrich, D. Huttenlocher, K. Kedem, J. Kleinberg, and D. Kravets. Geometric pattern matching under eucledian motion.
*Computational Geometry: Theory and Applications*, 7:113–124, 1997.MATHMathSciNetGoogle Scholar - 13.P.J. de Rezende and D.T. Lee. Point set pattern matching in d-dimensions.
*Algorithmica*, 13:387–404, 1995.MATHCrossRefMathSciNetGoogle Scholar - 14.A. Efrat, A. Itai, and M. Katz. Geometry helps in bottleneck matching and related problems. To appear in Algorithmica.Google Scholar
- 15.P.J. Heffernan. The translation square map and approximate congruence.
*Information Processing Letters*, 39:153–159, 1991.MATHCrossRefMathSciNetGoogle Scholar - 16.P.J. Heffernan. Generalized approximate algorithms for point set congruence. In
*Proc. 3rd. Workshop on Algorithms and Data Structures*, LNCS 709, pages 373–384, Montréal, Canada, 1993.Google Scholar - 17.P.J. Heffernan and S. Schirra. Approximate decision algorithms for point set congruence. In
*Proc. 8th. Annual ACM Symp. on Computational Geometry*, pages 93–101, 1992.Google Scholar - 18.D.P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications.
*Discrete and Computational Geometry*, 9:267–291, 1993.MATHCrossRefMathSciNetGoogle Scholar - 19.P. Indyk, R. Motwani, and S. Venkatasubramanian. Geometric matching under noise: Combinatorial bounds and algorithms. In
*Proc. 10th. Annual ACM-SIAM Symp. on Discrete Algorithms*, pages 457–465, 1999.Google Scholar - 20.P. Indyk and S. Venkatasubramanian. Approximate congruence in nearly linear time. In
*Proc. 11th. Annual ACM-SIAM Symp. on Discrete Algorithms*, 2000.Google Scholar - 21.S. Irani and P. Raghavan. Combinatorial and experimental results for randomized point matching algorithms.
*Computational Geometry: Theory and Applications*, 12:17–31, 1999.MATHMathSciNetGoogle Scholar - 22.S. Schirra. Approximate decision algorithms for approximate congruence.
*Information Processing Letters*, 43:29–34, 1992.MATHCrossRefMathSciNetGoogle Scholar - 23.N. Sleumer. Output-sensitive cell enumeration in hyperplane arrangements. In
*Proc. 6th. Scandinavian Workshop on Algorithm Theory*, LNCS 1432, pages 300–309, 1998.Google Scholar