Abstract
The problem of finding the closest pair among a collection of points in \(\Re^d\) is a well-known problem. There are better-than-naive-solutions for constant d and approximate solutions in general. We propose the first better-than-naive-solutions for the problem for large d. In particular, we present algorithms for the metrics L 1 and L ∞ with running times of O(n (ω + 3)/2) and O(n (ω + 3)/2logD) respectively, where O(n ω) is the running time of matrix multiplication and D is the diameter of the points.
This work was supported by the German-Israel Foundation (G.I.F.) young scientists program research grant agreement no. 2055-1168.6/2002.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abrahamson, K.: Generalized string matching. SIAM J. Computing 16(6), 1039–1051 (1987)
Amir, A., Apostolico, A., Lewenstein, M.: Inverse Pattern Matching. J. of Algorithms 24(2), 325–339 (1997)
Amir, A., Farach, M.: Efficient 2-dimensional approximate matching of half-rectangular figures. Information and Computation 118(1), 1–11 (1995)
Amir, A., Porat, E., Lewenstein, M.: Faster algorithms for string matching with k mismatches. J. of Algorithms, special SODA 2000 issue (2000) (to appear)
Borodin, A., Ostrovsky, R., Rabani, Y.: Subquadratic approximation algorithms for clustering problems in high dimensional spaces. In: Proceedings of the Symposium on Theory of Computing (1999)
Datar, M., Immorlica, N., Indyk, P., Mirrokni, V.: Locality-sensitive hashing scheme based on p-stable distributions. In: Proc. of Symposium of Computational Geometry, SOCG (2004) (to appear)
Dubiner, M., Galil, Z., Magen, E.: Faster tree pattern matching. J. of the ACM 41(2), 205–213 (1994)
Indyk, P., Motwani, R.: Approximate nearest neighbor: towards removing the curse of dimensionality. In: Proceedings of the Symposium on Theory of Computing (1998)
Indyk, P.: High-dimensional computational geometry. Ph.D. Thesis, Department of Computer Science, Stanford University (2001)
Kosaraju, R.: Efficient tree pattern matching. In: Proc. of Symosium on Foundation of Computer Science (1990)
Kleinberg, J.: Two algorithms for nearest-neighbor search in high dimensions. In: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing (1997)
Lipsky, O., Porat, E.: Approximated Pattern Matching with the L1 L2 and L ∞ Metrics. Submitted to SODA 2004 (2004) (manuscript)
Muthukrishnan, S.: New results and open problems related to non-standard stringology. In: Galil, Z., Ukkonen, E. (eds.) CPM 1995. LNCS, vol. 937, Springer, Heidelberg (1995)
Muthukrishnan, S., Palem, K.: Non-standard stringology: algorithms and complexity. In: Proc. of the ACM Symposium on Theory of Computing, pp. 770–779 (1994)
Rabin, M.O.: Probabilistic algorithms. In: Traub, J.F. (ed.) Algorithms and Complexity, pp. 21–39. Academic Press, London (1976)
Shamos, I., Bentley, J.: Divide-and-conquer in multidimensional space. In: Proceedings of the Symposium on Theory of Computing, pp. 220–230 (1976)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Indyk, P., Lewenstein, M., Lipsky, O., Porat, E. (2004). Closest Pair Problems in Very High Dimensions. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_66
Download citation
DOI: https://doi.org/10.1007/978-3-540-27836-8_66
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22849-3
Online ISBN: 978-3-540-27836-8
eBook Packages: Springer Book Archive