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, Volume 1, Issue 2, pp 113–126 | Cite as

Balancing poset extensions

  • Jeff Kahn
  • Michael Saks
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 log2e (X) comparisons whereC is approximately 2.17.

AMS (MOS) subject classifications (1980)

06A10 68E05 

Key words

Sorting comparison information theoretic bound linear extension 

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References

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    H. Buseman (1958)Convex Surfaces, Interscience, New York.Google Scholar
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    T. Bonneson and W. Fenchel (1934)Theorie der konvexen Körper, Springer, Berlin, 1934; Chelsea, New York, 1948 and 1971.Google Scholar
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    M. Fredman (1976) How good is the information theory bound in sorting?,Theoretical Computer Science 1, 355–361.Google Scholar
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    N. Linial, The information theoretic bound is good for merging,SIAM J. Comp., to appear.Google Scholar
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    R. P. Stanley (1987) Two combinatorial applications of the Aleksandrov-Fenchel inequalities,J. Comb. Theory (A) 31, 56–65.Google Scholar

Copyright information

© D. Reidel Publishing Company 1984

Authors and Affiliations

  • Jeff Kahn
    • 1
    • 2
  • Michael Saks
    • 3
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.Rutgers UniversityNew BrunswickUSA
  3. 3.Rutgers UniversityNew BrunswickUSA

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