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Bipartite Diameter and Other Measures Under Translation

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Acknowledgements

We would like to thank Pankaj K. Agarwal, Mark de Berg, and Timothy Chan for their assistance in the preparation of this manuscript.

Cite As

Boris Aronov, Omrit Filtser, Matthew J. Katz, and Khadijeh Sheikhan. Bipartite Diameter and Other Measures Under Translation. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.STACS.2019.8

Abstract

Let A and B be two sets of points in R^d, where |A|=|B|=n and the distance between them is defined by some bipartite measure dist(A, B). We study several problems in which the goal is to translate the set B, so that dist(A, B) is minimized. The main measures that we consider are (i) the diameter in two and three dimensions, that is diam(A,B) = max {d(a,b) | a in A, b in B}, where d(a,b) is the Euclidean distance between a and b, (ii) the uniformity in the plane, that is uni(A,B) = diam(A,B) - d(A,B), where d(A,B)=min{d(a,b) | a in A, b in B}, and (iii) the union width in two and three dimensions, that is union_width(A,B) = width(A cup B). For each of these measures we present efficient algorithms for finding a translation of B that minimizes the distance: For diameter we present near-linear-time algorithms in R^2 and R^3, for uniformity we describe a roughly O(n^{9/4})-time algorithm, and for union width we offer a near-linear-time algorithm in R^2 and a quadratic-time one in R^3.

Subject Classification

ACM Subject Classification
• Theory of computation → Computational geometry
Keywords
• Translation-invariant similarity measures
• Geometric optimization
• Minimum-width annulus

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References

1. Pankaj K. Agarwal, Boris Aronov, Marc van Kreveld, Maarten Löffler, and Rodrigo Silveira. Computing Correlation between Piecewise-Linear Functions. SIAM J. Comput., 42:1867-1887, 2013.
2. Pankaj K. Agarwal, Sariel Har-Peled, Micha Sharir, and Yusu Wang. Hausdorff distance under translation for points and balls. In Proc. 19th Ann. Symp. Comput. Geometry, pages 282-291. ACM, 2003.
3. Pankaj K. Agarwal and Micha Sharir. Efficient randomized algorithms for some geometric optimization problems. Discr. Comput. Geometry, 16(4):317-337, 1996.
4. Pankaj K. Agarwal, Micha Sharir, and Sivan Toledo. Applications of Parametric Searching in Geometric Optimization. J. Algorithms, 17(3):292-318, 1994. URL: http://dx.doi.org/10.1006/jagm.1994.1038.
5. Hee-Kap Ahn, Peter Brass, and Chan-Su Shin. Maximum overlap and minimum convex hull of two convex polyhedra under translations. Computational Geometry, 40(2):171-177, 2008.
6. Helmut Alt, Kurt Mehlhorn, Hubert Wagener, and Emo Welzl. Congruence, similarity, and symmetries of geometric objects. Discr. Comput. Geometry, 3(3):237-256, 1988.
7. Rinat Ben-Avraham, Matthias Henze, Rafel Jaume, Balázs Keszegh, Orit E. Raz, Micha Sharir, and Igor Tubis. Minimum partial-matching and Hausdorff RMS-distance under translation: Combinatorics and algorithms. In European Symp. Algorithms, pages 100-111. Springer, 2014.
8. Arijit Bishnu, Sandip Das, Subhas C. Nandy, and Bhargab B. Bhattacharya. Simple algorithms for partial point set pattern matching under rigid motion. Pattern Recognition, 39(9):1662-1671, 2006.
9. K. Q. Brown. Geometric Transforms for Fast Geometric Algorithms. Ph.D. thesis, Carnegie-Mellon University, Pittsburgh, PA, 1980.
10. Bernard Chazelle, Richard Cole, Franco P. Preparata, and Chee Yap. New upper bounds for neighbor searching. Information and Control, 68(1-3):105-124, 1986.
11. Bernard Chazelle, Herbert Edelsbrunner, Leonidas J. Guibas, and Micha Sharir. Algorithms for bichromatic line segment problems and polyhedral terrains. Algorithmica, 11:116-132, 1994.
12. Kenneth L. Clarkson. Las Vegas algorithms for linear and integer programming when the dimension is small. J. ACM, 42:488-499, 1995.
13. Kenneth L. Clarkson and Peter W. Shor. Applications of random sampling in computational geometry, II. Discr. Comput. Geometry, 4(1):387-421, 1989.
14. M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, third edition, 2008.
15. Alon Efrat, Alon Itai, and Matthew J. Katz. Geometry helps in bottleneck matching and related problems. Algorithmica, 31(1):1-28, 2001.
16. Leonidas J. Guibas and Raimund Seidel. Computing convolutions by reciprocal search. Discr. Comput. Geometry, 2(2):175-193, 1987.
17. Paul J. Heffernan and Stefan Schirra. Approximate decision algorithms for point set congruence. Computational Geometry, 4(3):137-156, 1994.
18. M. E. Houle and G. T. Toussaint. Computing the width of a set. IEEE Trans. Pattern Anal. Mach. Intell., PAMI-10(5):761-765, 1988.
19. Daniel P. Huttenlocher, Klara Kedem, and Micha Sharir. The Upper Envelope of Voronoi Surfaces and Its Applications. Discr. Comput. Geometry, 9:267-291, 1993. URL: http://dx.doi.org/10.1007/BF02189323.
20. Piotr Indyk and Suresh Venkatasubramanian. Approximate congruence in nearly linear time. Computational Geometry, 24(2):115-128, 2003.
21. Nimrod Megiddo. Linear-time algorithms for linear programming in R³ and related problems. SIAM J. Comput., 12(4):759-776, 1983.
22. Edgar A. Ramos. Deterministic algorithms for 3-D diameter and some 2-D lower envelopes. In Proc. 16th Ann. Symp. Comput. Geometry, pages 290-299. ACM, 2000.
23. M. I. Shamos. Computational Geometry. PhD thesis, Yale University, 1978.
24. Emo Welzl. Smallest Enclosing Disks (balls and Ellipsoids). In Results and New Trends in Computer Science, pages 359-370. Springer-Verlag, 1991.
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