Abstract
This paper considers the parameters height, width, number of comparabilities, and number of linear extensions of a finite partially ordered set viewing them as measures which capture the informal idea of ‘orderliness’. The first three of these measures are all easy to compute and a survey of methods for calculating the fourth measure is given. Then orientations of a tree are studied. Every orientation of a given tree can be regarded as the diagram of an ordered set; in this class of orders those for which height, width, number of comparabilities, and number of linear extensions are greatest and least are considered. Finally the problem of producing order information by comparison algorithms is discussed. Algorithms which use only n-1 comparisons are considered and the question of how much orderliness can be generated with respect to the four measures is studied. In particular, the class of perfect posets is introduced; these are posets with connected diagrams which can be generated by an algorithm which uses n-1 comparisons.
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© 1989 Kluwer Academic Publishers
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Atkinson, M.D. (1989). The Complexity of Orders. In: Rival, I. (eds) Algorithms and Order. NATO ASI Series, vol 255. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2639-4_5
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DOI: https://doi.org/10.1007/978-94-009-2639-4_5
Publisher Name: Springer, Dordrecht
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