Constructions of Sparse Asymmetric Connectors

Abstract

We 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 each 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 + log ² n) Oruç[5] gave the presently best (n,N) -connector with O(N + n ·log n) edges. For k=2 we show by a probabilistic argument that an optimal (n,N) -connector has Θ(N) edges, if $n \leq N^{\frac{1}{2}-\epsilon}$ for some ε > 0 . Moreover, we give explicit constructions based on a new number theoretic approach that need $O(N n^{\frac{1}{2}} + N^{\frac{1}{2}}n^{\frac{3}{2}})$ edges.