Sorting with networks of data structures

We consider the problem of sorting a permutation using a network of data structures as introduced by Knuth and Tarjan. In general the model as considered previously was restricted to networks that are directed acyclic graphs (DAGs) of stacks and/or queues. In this paper we study the question of which are the smallest general graphs that can sort an arbitrary permutation and what is their efficiency. We show that certain two-node graphs can sort in time Θ(n2)Θ(n2) and no simpler graph can sort all permutations. We then show that certain three-node graphs sort in time Ω(n3/2)Ω(n3/2), and that there exist graphs of kk nodes which can sort in time Θ(nlogkn)Θ(nlogkn), which is optimal.