Searching Using Partial Ordering

Given a set U of n distinct numbers, we are interested in preprocessing them, so that subsequent rrembership queries of the form ‘Is y E U?’ can be answered quickly. For example, if one sorts the elements of U, then each query can be answerred in pg(n + I)1 comparisons by a binary search. This may be contrasted with the situation where no preprocessing is done, when n comparisons are needed to answer the query. In general, the esta.blishment of a partial order over U will facilitate answering the queries. In this paper we examine the trade-off between the preprocessing cost and the subsequent search cost for each query, in a model using pairwise comparisons among the numbers as the basic operations. Suppose we wish to be able to answer any membership query in at most S(n) comparisons. Can we put a lower bound on P(n), the worst-case cost of a preprocessing algorithm which builds some suitable partial orders on U? We shall show that P(n) + n lg S(n) 2 (1 + o( 1))n lg n for any comparison-based algorithm ’ This result can be extended in a straightforward manner to the case where some numbers in U may be identical. A simple constructive argument will also show that it is best possible. For some related work, see the papers of Detig et al. [l], and Munro and Suwanda [6].