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

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.