Abstract
This paper considers the following problem: given two point sets A and B (¦A¦= ¦B¦ = n) in d-dimensional Euclidean space, determine whether or not A is congruent to B. First, this paper presents a randomized algorithm which works in O(n (d−1)/2(logn)2) time. This improves the previous result (an O(n (d−2) log n) time deterministic algorithm). The birthday paradox, which is a well-known property in combinatorics, is used effectively in our algorithm. Next, this paper shows that if d is not bounded, the problem is at least as hard as the graph isomorphism problem in the sense of the polynomiality. Several related results are described too.
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References
T. Akutsu. Algorithms for determining geometrical congruity in two and three dimensions. Proc. 3rd International Symp. on Algorithms and Computation, pp. 279–288, 1992.
T. Akutsu. A parallel algorithm for determining the congruity of point sets in three dimensions. Technical Report AL31-3, Information Processing Society of Japan, 1993.
H. Alt and M. Godau. Measuring the resemblance of polygonal curves. Proc. ACM Symp. Computational Geometry, pp. 102–109, 1992.
H. Alt, K. Melhorn, H. Wagener and E. Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, Vol. 3, pp. 237–256, 1988.
M. J. Atallah. On symmetry detection. IEEE Trans. Computers, Vol. C-34, pp. 663–666, 1985.
M. D. Atkinson. An optimal algorithm for geometrical congruence. Journal of Algorithms, Vol. 8, pp. 159–172.
L. Babai and L. Kučera. Canonical labeling of graphs in linear average time. Proc. IEEE Symp. Foundations of Computer Science, pp. 39–46, 1979.
R. Cole. Parallel merge sort. Proc. IEEE Symp. Foundations of Computer Science, pp. 511–516, 1986.
P. Flajolet, D. Gardy and L. Thimonier. Birthday paradox, coupon collectors, cashing algorithms and self-organizing search. Discrete Applied Mathematics, Vol. 39, pp. 207–229, 1992.
H. Gazit and J. Reif. A randomized parallel algorithm for planar garph isomorphism. Proc. ACM Symp. Parallel Algorithms and Architectures, pp. 210–219, 1990.
P. J. Heffernan and S. Schirra. Approximate decision algorithm for point sets congruence. Proc. ACM Symp. Computational Geometry, pp. 93–101, 1992.
P. T. Highnam. Optimal algorithms for finding the symmetries of a planar point set. Information Processing Letters, Vol. 22, pp. 219–222, 1986.
G. Manacher. An application of pattern matching to a problem in geometrical complexity. Information Processing Letters, Vol. 5, pp. 6–7, 1976.
K. Sugihara. An n log n algorithm for determining the congruity of polyhedra. Journal of Computer and System Sciences, Vol. 29, pp. 36–4, 1984.
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© 1994 Springer-Verlag Berlin Heidelberg
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Akutsu, T. (1994). On determining the congruity of point sets in higher dimensions. In: Du, DZ., Zhang, XS. (eds) Algorithms and Computation. ISAAC 1994. Lecture Notes in Computer Science, vol 834. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58325-4_164
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DOI: https://doi.org/10.1007/3-540-58325-4_164
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Online ISBN: 978-3-540-48653-4
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