Superconcentrators of depths 2 and 3; odd levels help (rarely)
Journal of Computer and System Sciences
On dynamic algorithms for algebraic problems
Journal of Algorithms
Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators
SIAM Journal on Discrete Mathematics
Lower bounds for dynamic algebraic problems
Information and Computation
Space bounds for a game on graphs
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
On non-linear lower bounds in computational complexity
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
Superconcentrators, generalizers and generalized connectors with limited depth
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Graph-theoretic properties in computational complexity
Journal of Computer and System Sciences
Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gates
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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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 k1 0 such that $a(G)log (\frac{2b(G)}{a(G)}) log b(G) \geq k_1 \cdot log^2 N$.