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Tight bound of random interval packing

Part of: Operations research and management science Limit theorems

Published online by Cambridge University Press:  14 July 2016

Wansoo T. Rhee
Wansoo T. Rhee*
Affiliation:
Ohio State University
*
∗Postal address: Department of Management Science, The Ohio State University, Columbus, OH 43210, USA.
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Abstract

Consider n random intervals I1, …, IN chosen by selecting endpoints independently from the uniform distribution. A packing of I1, …, IN is a disjoint sub-collection of these intervals: its wasted space is the measure of the set of points not covered by the packing. We investigate the random variable WN equal to the smallest wasted space among all packings. Coffman, Poonen and Winkler proved that EWN is of order (log N)2/N. We provide a sharp estimate of log P(WN ≥ t (log N)2 / N) and log P(WN ≤ t (log N)2 / N) for all values of t.

Keywords

Interval packingwasted spaceuniform distribution

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 90B35: Scheduling theory, deterministic
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 990 - 997
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Research supported in part by NSF contract DMS-9303188.

References

Coffman, E. G. Jr, and Lueker, G. S. (1991). Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley, New York.Google Scholar
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Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13–30.CrossRefGoogle Scholar
Rhee, W. T., and Talagrand, M. (1996). Packing random intervals. Ann. Appl. Prob. 6, 572–576.CrossRefGoogle Scholar