Abstract
The problem of finding the maxima of a point set plays a fundamental role in computational geometry. Based on the idea of the certificates of exclusion, two algorithms are presented to solve the maxima problem under the assumption that N points are chosen from a d-dimensional hypercube uniformly and each component of a point is independent of all other components. The first algorithm runs in O(N) expected time and finds the maxima using dN + dln N + d2N 1 − 1/d(lnN)1/d + O(dN1 − 1/d) expected scalar comparisons. The experiments show the second algorithm has a better expected running time than the first algorithm while a tight upper bound of the expected running time is not obtained. A third maxima-finding algorithm is presented for N points with a d-dimensional component independence distribution, which runs in O(N) expected time and uses 2dN + O(ln N(ln(ln N))) + d 2 N 1 − 1/d(lnN)1/d + O(dN1 − 1/d) expected scalar comparisons. The substantial reduction of the expected running time of all three algorithms, compared with some known linear expected-time algorithms, has been attributed to the fact that a better certificate of exclusion has been chosen and more non-maximal points have been identified and discarded.
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References
Bentley, J.L., Clarkson, K.L., Levine, D.B.: Fast linear expectedtime algorithms for computing maxima and convex hulls. In: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 179–187 (1990)
Bentley, J.L.: Multidimensional divide-and-conquer. Communications of the Association for Computing Machinery 23(4), 214–229 (1980)
Bentley, J.L., Kung, H.T., Schkolnick, M., Thompson, C.D.: On the average number of maxima in a set of vectors and applications. Journal of the Association for Computing Machinery 25, 536–543 (1978)
Floyd, R.W., Rivest, R.L.: Expected time bounds for selection. Communications of the ACM 18, 165–172 (1975)
Fredrickson, G.N., Rodger, S.: A new approach to the dynamic maintenance of maximal points in the plane. Discrete and Computational Geometry 5, 365–374 (1990)
Janardan, R.: On the dynamic maintenance of maximal points in the plane. Information Processing Letters 40(2), 59–64 (1991)
Kapoor, S.: Dynamic maintenance of maxima of 2 − d point sets. In: Proceedings of the Tenth Computational Geometry, pp. 140–149 (1994)
Kung, H.T., Luccio, F., Preparata, F.P.: On finding the maxima of a set of vectors. Journal of the ACM 22(4), 469–476 (1975)
L’Ecuyer, P.: Efficient and portable combined random number generators. Communications of the ACM 31(6), 742–751 (1988)
Overmars, M.H., Van Leeuwen, J.L.: Maintenance of configuration in the plane. Journal of Computer and System Sciences 23, 253–257 (1981)
Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, New York (1985)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge (1993)
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Dai, H.K., Zhang, X.W. (2004). Improved Linear Expected-Time Algorithms for Computing Maxima. In: Farach-Colton, M. (eds) LATIN 2004: Theoretical Informatics. LATIN 2004. Lecture Notes in Computer Science, vol 2976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24698-5_22
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DOI: https://doi.org/10.1007/978-3-540-24698-5_22
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