Abstract
We present efficient parallel algorithms for two geometric k-clustering problems in the CREW PRAM model of parallel computation. Given a point set P of n points in two dimensions, these problems are to find a k-point subset such that some measure for this subset is minimized. We consider the problems of finding a k-point subset with minimum L ∞ perimeter and minimum L ∞ diameter. For the L ∞ perimeter case, our algorithm runs in O(log2 n) time and O(n log2 n + nk 2 log2 k) work. For the L ∞ diameter case, our algorithm runs in O(log2 n + log2 k loglog k log* k) time and O(n log2 n) work. The work done (processor-time product) by our algorithms is close to the time complexity of best known sequential algorithms. Previously, no parallel algorithm was known for either of these problems.
This work was done when the author was working as a Post Doctoral Fellow in Max Planck Institut für Informatik, Saarbrücken,Germany. This work was supported by the ESPRIT Basic Research Actions Program, under contract No. 7141 (project ALCOM II).
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References
H. C. Andrews. Introduction to Mathematical Techniques in Pattern Recognition. Wiley-Interscience, New York, 1972.
A. Aggarwal, H. Imai, N. Katoh and S. Suri. Finding k points with minimum diameter and related problems. J. Algorithms 12 (1991), pp. 38–56.
O. Berkman, D. Breslauer, Z. Galil, B. Scheiber and U. Vishkin. Highly parallelizable problems, Proc. 21st Annual ACM Symp. on Theory of Computing, 1989, pp. 309–319.
S. Chandran, S. K. Kim and D. M. Mount. Parallel computational geometry of rectangles. Algorithmica, 7, (1992), pp. 25–49.
R. Cole. Parallel merge sort. SIAM J. Comput. 17, (1988), pp. 770–785.
A. Datta, H. P. Lenhof, C. Schwarz and M. Smid. Static and dynamic algorithms for k-point clustering problems. Proceedings of WADS '93, Lecture Notes in Computer Science, Springer-Verlag, Vol. 709, pp. 265–276.
D. Eppstein and J. Erickson. Iterated nearest neighbours and finding minimal polytopes. Proc. 4th ACM-SIAM Symp. on Discrete Algorithms, (1993), pp. 64–73.
G. N. Prederickson and D. B. Johnson. The complexity of selection and ranking in X + Y and matrices with sorted columns. Journal of Computer and System Sciences, 24, (1982), pp. 197–208.
J. A. Hartigan. Clustering Algorithms, John-Wiley, New York, 1975.
J. JáJá. An introduction to Parallel Algorithms. Addison-Wesley, 1992.
D. B. Johnson and T. Mizoguchi. Selecting the Kth element in X + Y and X1 + X2 + + X m . SIAM J. Comput. 7, (1978), pp. 147–157.
H. P. Lenhof and M. Smid. Sequential and parallel algorithms for the k-closest pairs problem. Max Planck Institut für Informatik, Technical Report, MPI-I-92-134, August 1992. A preliminary version appears in Proc. 33rd Annual IEEE Symp. on Foundations of Computer Science, pp. 380–386.
F. P. Preparata and M. I. Shamos. Computational Geometry: an Introduction. Springer-Verlag, New York, 1985.
R. Sarnath and X. He. Efficient parallel algorithms for selection and searching on sorted matrices. Proc. 6th International Parallel Processing Symposium, (1992), pp. 108–111.
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© 1994 Springer-Verlag Berlin Heidelberg
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Datta, A. (1994). Efficient parallel algorithms for geometric k-clustering problems. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds) STACS 94. STACS 1994. Lecture Notes in Computer Science, vol 775. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57785-8_164
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DOI: https://doi.org/10.1007/3-540-57785-8_164
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