04-12 | Berichtsreihe des Mathematischen Seminars der Universität Kiel | |
Andreas Baltz, Gerold Jäger, Anand Srivastav:Constructions of Sparse Asymmetric Connectors with Number Theoretic MethodsWe consider the problem of connecting a set I of n inputs to a set O of N outputs (n ≤ N) by as few edges as possible such that for every injective mapping f:I → O there are n vertex disjoint paths from i to f(i) of length k for a given k ∈ N. For k = Ω(log N + log2 n) Oruç (1994) gave the presently best (n,N)-connector with O(N + n log n) edges. For k=2 and N the square of a prime, Hwang and Richards (1985) described a construction using We show by a probabilistic argument that an optimal (n,N)-connector has Θ(N) edges, if n ≤ N1/2 - ε for some ε > 0. Moreover, we give explicit constructions based on a new number theoretic approach that need at most edges for arbitrary choices of n and N. The improvement we achieve is based on applying a generalization of the Erdös-Heilbronn conjecture on the size of restricted sums. Mathematics Subject Classification (1991): 05C35, 11P81 Keywords: connector, rearrangeable network, restricted sums
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