 
Summary: Superconcentrators of depth 2 and 3; odd levels help (rarely)
Noga Alon
Bellcore, Morristown, NJ, 07960, USA
and Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
and
Pavel Pudlak
Mathematical Institute
CSAV, Prague
Abstract
It is shown that the minimum possible number of edges in an n superconcentrator of depth
3 is (n log log n), whereas the minimum possible number of edges in an nsuperconcentrator of
depth 2 is (n(log n)3/2
) (and is O(n(log n)2
)).
1 Introduction
An nsuperconcentrator is a directed acyclic graph S with the following properties.
(i) There are two disjoint subsets of vertices of S, U (called the set of inputs) and V (called the set
of outputs), each of cardinality n, where the indegree of each vertex in U is 0 and the outdegree of
