Sparse asymmetric connectors in communication networks

Authors:
R. Ahlswede;H. Aydinian
Affiliations:
Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany;Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany
Venue:
General Theory of Information Transfer and Combinatorics
Year:
2006

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Abstract

An (n,N,d)–connector is an acyclic digraph with n inputs and N outputs in which for any injective mapping of input vertices into output vertices there exist n vertex disjoint paths of length d joining each input to its corresponding output. We consider the problem of construction of sparse (n,N,2)–connectors (depth 2 connectors) when n≪N. The probabilistic argument in [1] shows the existence of (n,N,2)–connectors of size (number of edges) O(N) if $n\leq N^{1/2-\varepsilon},\ \varepsilon 0$. However, the known explicit constructions with $n\leq\sqrt{N}$ in [6],[1],[2] are of size O$(N\sqrt{n})$. Here we present a simple combinatorial construction for (n,N,2)–connectors of size O(N log2n). We also consider depth 2 fault–tolerant connectors under arc or node failures.