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Packing random intervals
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  • Published: 01 March 1995

Packing random intervals

  • E. G. Coffman Jr.1,
  • Bjorn Poonen2 &
  • Peter Winkler1 

Probability Theory and Related Fields volume 102, pages 105–121 (1995)Cite this article

  • 112 Accesses

  • 8 Citations

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Summary

Letn random intervalsI 1, ...,I n be chosen by selecting endpoints independently from the uniform distribution on [0.1]. Apacking is a pairwise disjoint subset of the intervals; itswasted space is the Lebesgue measure of the points of [0,1] not covered by the packing. In any set of intervals the packing with least wasted space is computationally easy to find; but its expected wasted space in the random case is not obvious. We show that with high probability for largen, this “best” packing has wasted space\(O(\frac{{\log ^2 n}}{n})\). It turns out that if the endpoints 0 and 1 are identified, so that the problem is now one of packing random arcs in a unit-circumference circle, then optimal wasted space is reduced toO(1/n). Interestingly, there is a striking difference between thesizes of the best packings: about logn intervals in the unit interval case, but usually only one or two arcs in the circle case.

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References

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Author information

Authors and Affiliations

  1. AT&T Bell Laboratories, 07974, Murray Hill, NJ, USA

    E. G. Coffman Jr. & Peter Winkler

  2. Mathematical Sciences Research Institute, 94720, Berkeley, CA, USA

    Bjorn Poonen

Authors
  1. E. G. Coffman Jr.
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  2. Bjorn Poonen
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  3. Peter Winkler
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Cite this article

Coffman, E.G., Poonen, B. & Winkler, P. Packing random intervals. Probab. Theory Relat. Fields 102, 105–121 (1995). https://doi.org/10.1007/BF01295224

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  • Received: 03 October 1994

  • Revised: 29 December 1994

  • Published: 01 March 1995

  • Issue Date: March 1995

  • DOI: https://doi.org/10.1007/BF01295224

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Mathematics Subject Classification

  • 60D05
  • 05B40
  • 52A22
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