Construction of asymmetric connectors of depth two
Journal of Combinatorial Theory Series A - Special issue in honor of Jacobus H. van Lint
Interactive Communication, Diagnosis and Error Control in Networks
Algorithmics of Large and Complex Networks
Multicast Routing and Design of Sparse Connectors
Algorithmics of Large and Complex Networks
Sparse asymmetric connectors in communication networks
General Theory of Information Transfer and Combinatorics
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We consider the problem of connecting a set 1 of n inputs toa set O of N outputs (n ≤ N) by asfew edges as possible such that for every injective mappingf : I → O there are n vertexdisjoint paths from i to f(i) of lengthk for a given k ∈ IN. For k =Ω(logN + log2n) Oruç (JParallet Distributed Comput 1994, 359366(10) gave the presentlybest (n,N)-connector with O(N +n · logn) edges. For k = 2 and Nthe square of a prime, Richards and Hwang (1985) described aconstruction using $N\lceil\sqrt{n + 5/4} - 1/2\rceil +n\lceil\sqrt{n + 5/4} - 1/2 \rceil \sqrt{N}$ edges. We show by aprobabilistic argument that an optimal(n,N)-connector has Θ(N) edges, ifn ≤ N½-ε for some ∈≥ 0. Moreover, we give explicit constructions based on a newnumber theoretic approach that need at most $N\lceil\sqrt{3n/4}\rceil + 2n\lceil \sqrt{3n/4}\rceil\lceil\sqrt{N}\rceil$ edges for arbitrary choices ofn and N. The improvement we achieve is based onapplying a generalization of the Erdös-Heilbronn conjecture onthe size of restricted sums. © 2005 Wiley Periodicals, Inc.NETWORKS, Vol. 45(3), 119124 2005A preliminary version of thispaper appeared in the proceedings of the 23rd conference onfoundations of software technology and theoretical computer science(FSTTCS), Mumbai, India, December 2003, LNCS 2914, 1322.