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Balancing poset extensions

Order volume 1, pages 113–126 (1984)Cite this article

Abstract

We show that any finite partially ordered setP (not a total order) contains a pair of elementsx andy such that the proportion of linear extensions ofP in whichx lies belowy is between 3/11 and 8/11. A consequence is that the information theoretic lower bound for sorting under partial information is tight up to a multiplicative constant. Precisely: ifX is a totally ordered set about which we are given some partial information, and ife(X) is the number of total orderings ofX compatible with this information, then it is possible to sortX using no more thanC log2 e (X) comparisons whereC is approximately 2.17.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Jeff Kahn

  2. Rutgers University, 08903, New Brunswick, NJ, USA

    Jeff Kahn

  3. Rutgers University, 08903, New Brunswick, NJ, USA

    Michael Saks

Authors
  1. Jeff Kahn
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  2. Michael Saks
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Additional information

Communicated by B. Sands

Supported in part by NSF Grant MCS83-01867.

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Kahn, J., Saks, M. Balancing poset extensions. Order 1, 113–126 (1984). https://doi.org/10.1007/BF00565647

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  • DOI: https://doi.org/10.1007/BF00565647

AMS (MOS) subject classifications (1980)

  • 06A10
  • 68E05

Key words

  • Sorting
  • comparison
  • information theoretic bound
  • linear extension