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On the average number of maximal in a set of vectors. (English) Zbl 0682.68041
Summary: Any of n vectors in d-space is called maximal if none of the remaining vectors dominates it in every component. Assuming that n vectors are distributed identically and that the d components of each vector are distributed independently and continuously, we determine the expected number of maximal vectors explicitly for any n and d. The asymptotic behaviour of this quantity as n tends to infinity, which was investigated by Bentley, Kung, Schkolnick, Thompson and Devroye, follows immediately from our result.

MSC:
68Q25 Analysis of algorithms and problem complexity
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[1] Bentley, J.L.; Kung, H.T.; Schkolnick, M.; Thompson, C.D., On the average number of maxima in a set of vectors and applications, J. ACM, 25, 536-543, (1978) · Zbl 0388.68056
[2] Bentley, J.L.; Shamos, M.I., Divide and conquer for linear expected time, Inform. process. lett., 7, 87-91, (1978) · Zbl 0404.68046
[3] Devroye, L., A note on finding convex hulls via maximal vectors, Inform. process. lett., 11, 53-56, (1980) · Zbl 0444.68063
[4] Preparata, F.P.; Shamos, M.I., Computational geometry, (1985), Springer New York · Zbl 0759.68037
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