NDLI: Average case lower bounds on the construction and searching of partial orders

Content Provider	IEEE Xplore Digital Library
Author	Mairson, Harry G.
Copyright Year	1985
Abstract	It is very well known in computer science that partially ordered files are easier to search. In the worst case, for example, a totally unordered file requires no preprocessing, but ¿(n) time to search, while a totally ordered file requires ¿(n log n) preprocessing time to sort, but can be searched in O(log n) time. Behind the casual observation, then, lurks the notion of a computational tradeoff between sorting and searching. We analyze this tradeoff in the average case, using the decision tree model. Let P be a preprocessing algorithm that produces partial orders given a set U of n elements, and let S be a searching algorithm for these partial orders. Assuming any of the n! permutations of the elements of U are equally likely, and that we search for any y isin; U with equal probability (in unsuccessful search, all "gaps" are considered equally likely), the average costs P(n) of preprocessing and S(n) of searching may be computed. We demonstrate a tradeoff of the form P(n) + n log S(n) = ¿(n log n), for both successful and unsuccessful search. The bound is tight up to a constant factor. In proving this tradeoff, we show a lower bound on the average case of searching a partial order. Let A be a partial order on n elements consistent with Π permutations. We show S(n) = ¿(Π3/n/n2) for successful search of A, and S(n) = ¿(Π2/n/n) for unsuccessful search. These lower bounds show, for example, that heaps require linear time to search on the average.
Starting Page	303
Ending Page	311
File Size	870469
Page Count	9
File Format	PDF
ISBN	0818606444
ISSN	02725428
DOI	10.1109/SFCS.1985.13
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	1985-10-21
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Sorting Decision trees Computer science Costs Application software Merging Data structures Particle measurements Information analysis Binary trees
Content Type	Text
Resource Type	Article

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