Average Cost to Produce Partial Orders

We study the average-case complexity of partial order productions. Any comparison-based algorithm for solving problems in computer science can be viewed as the partial order production. The productions of some specific partial orders, such as sorting, searching, and selection, have received much attention in the past decades. As to arbitrary partial orders, very little is known about the inherent complexity of their productions. In particular, no non-trivial average-case lower bounds were known. By combining information-theoretic lower bounds with adversary-based arguments, we present some non-trivial average-case lower bounds for the productions of arbitrary partial orders. More precisely, we first demonstrate a counter-example to some intuition about lower bounds on partial order productions, and then present some simple lower bounds. By utilizing adversaries which were originally constructed for deriving worst-case lower bounds, we prove non-trivial average-case lower bounds for partial order productions. Our lower-bound techniques of mixing the information-theoretical and adversary-based approaches are interesting, as well as the lower-bound results obtained. Moreover, several conjectures concerning the production complexity of partial orders are answered. Motivating from the selection problem and from the design of efficient algorithms, we also investigate average-case cost for producing many isomorphic copies simultaneously of some partial order.