Tradeoffs in Depth-Two Superconcentrators

Abstract

An N-superconcentrator is a directed graph with N input vertices and N output vertices and some intermediate vertices, such that for k=1, 2, ..., N, between any set of k input vertices and any set of k output vertices, there are k vertex disjoint paths. In a depth-twoN-superconcentrator each edge either connects an input vertex to an intermediate vertex or an intermediate vertex to an output vertex. We consider tradeoffs between the number of edges incident on the input vertices and the number of edges incident on the output vertices in a depth-two N-superconcentrator. For an N-superconcentrator G, let a(G) be the average degree of the input vertices and b(G) be the average degree of the output vertices. Assume that b(G) ≥ a(G). We show that there is a constant k ₁ > 0 such that

$a(G)log (\frac{2b(G)}{a(G)}) log b(G) \geq k_1 \cdot log^2 N$.