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The Information Theoretic Bound for Problems on Ordered Sets and Graphs

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Graphs and Order

Part of the book series: NATO ASI Series ((ASIC,volume 147))

Abstract

One of the most basic problems arising in computer science is to sort a set X with unknown total order. The objective is to minimize the number of steps needed in worst case, where a step consists of a comparison of two elements x and y (this comparison is denoted x:y). The result of each comparison reduces the set of possible orderings of X to one of two sets: those in which x < y and those in which y < x. Since it is possible that the larger of these two sets remains, the number of possible orders on X is reduced by no more than half. In worst case we need at least log n! = n log n + 0(n) comparisons to sort the n-element set X. (We write log for log2 throughout this paper.)

This is an example of an information theoretic bound (ITB) for a combinatorial decision problem, one of the few general techniques known for obtaining lower bounds on the computational complexity of a problem. For those problems where this argument is applicable, a natural question to ask is: “can this bound be achieved?” or “how close can we come to achieving the bound?” For the sorting problem above there are several well-known algorithms that essentially attain the ITB (see [Kn]), but of course this is not the case for all such problems.

This paper is a survey of a number of computational problems involving ordered sets and graphs in which the ITB can be applied. As will be seen, the derivation of these bounds and the question of how good they are often lead to combinatorial questions that are interesting in their own right.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, NJ, USA, 08903

    Michael Saks

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  1. Michael Saks
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Editors and Affiliations

  1. Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada

    Ivan Rival

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© 1985 D. Reidel Publishing Company

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Saks, M. (1985). The Information Theoretic Bound for Problems on Ordered Sets and Graphs. In: Rival, I. (eds) Graphs and Order. NATO ASI Series, vol 147. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5315-4_4

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  • DOI: https://doi.org/10.1007/978-94-009-5315-4_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8848-0

  • Online ISBN: 978-94-009-5315-4

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