Searching in random partially ordered sets

Abstract

We consider the problem of searching for a given element in a partially ordered set. More precisely, we address the problem of computing efficiently near-optimal search strategies for typical partial orders. We consider two classical models for random partial orders, the random graph model and the uniform model. We shall show that certain simple, fast algorithms are able to produce nearly-optimal search strategies for typical partial orders under the two models of random partial orders that we consider. For instance, our algorithm for the random graph model produces, in linear time, a search strategy that makes O((log n)1/2 log log n) more queries than the optimal strategy, for almost all partial orders on n elements. Since we need to make at least lg n = log2 n queries for any n-element partial order, our result tells us that one may efficiently devise near-optimal search strategies for almost all partial orders in this model (the problem of determining an optimal strategy is NP-hard, as proved recently in [1]).