Abstract
We present an O(n log n)-time algorithm to solve the threedimensional layers-of-maxima problem, an improvement over the prior O(n log n log log n)-time solution. A previous claimed O(n log n)-time solution due to Atallah, Goodrich, and Ramaiyer [SCG’94] has technical flaws. Our algorithm is based on a common framework underlying previous work, but to implement it we devise a new data structure to solve a special case of dynamic planar point location in a staircase subdivision. Our data structure itself relies on a new extension to dynamic fractional cascading that allows vertices of high degree in the control graph.
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Supported by DARPA Grant F30602-00-2-0509 and NSF Grant CCR-0098068.
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References
P.K. Agarwal. Personal communication, 1992.
M. J. Atallah, M. T. Goodrich, and K. Ramaiyer. Biased finger trees and threedimensional layers of maxima. In Proc. 10th ACM SCG, pages 150–9, 1994.
S.W. Bent, D.D. Sleator, and R.E. Tarjan. Biased search trees. SIAM J. Comp., 14(3):545–68, 1985.
B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inf. Thy., IT-31:509-17, 1985.
B. Chazelle and L. J. Guibas. Fractional cascading: I. A data structure technique. Algorithmica, 1(2):133–62, 1986.
P. F. Dietz and R. Raman. Persistence, amortization and randomization. In Proc. 2nd ACM-SIAM SODA, pages 78–88, 1991.
P.G. Franciosa, C. Gaibisso, and M. Talamo. An optimal algorithm for the maxima set problem for data in motion. In Abs. CG’92, pages 17–21, 1992.
L. J. Guibas and R. Sedgewick. A dichromatic framework for balanced trees. In Proc. 19th IEEE FOCS, pages 8–21, 1978.
J.-W. Jang, M. Nigam, V.K. Prasanna, and S. Sahni. Constant time algorithms for computational geometry on the reconfigurable mesh. IEEE Trans. Par. and Dist. Syst., 8(1):1–12, 1997.
S. Kapoor. Dynamic maintenance of maximas of 2-d point sets. In Proc. 10th ACM SCG, pages 140–149, 1994.
H.T. Kung, F. Luccio, and P. Preparata. On finding the maxima of a set of vector. J. ACM, 22(4):469–76, 1975.
K. Mehlhorn and S. Näher. Bounded ordered dictionaries in O(log logN) time and O(n) space. IPL, 35(4):183–9, 1990.
K. Mehlhorn and S. Näher. Dynamic fractional cascading. Algorithmica, 5(2):215–41, 1990.
F.P. Preparata. A new approach to planar point location. SIAM J. Comp., 10(3):473–482, 1981.
R. Raman. Eliminating Amortization: On Data Structures with Guaranteed Fast Response Time. PhD thesis, U. Rochester, 1992.
P. van Emde Boas. Preserving order in a forest in less than logarithmic time and linear space. IPL, 6(3):80–2, June 1977.
D. E. Willard. New trie data structures which support very fast search operations. JCSS, 28(3):379–94, 1984.
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© 2002 Springer-Verlag Berlin Heidelberg
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Buchsbaum, A.L., Goodrich, M.T. (2002). Three-Dimensional Layers of Maxima. In: Möhring, R., Raman, R. (eds) Algorithms — ESA 2002. ESA 2002. Lecture Notes in Computer Science, vol 2461. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45749-6_26
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DOI: https://doi.org/10.1007/3-540-45749-6_26
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