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A Multipath Analysis of Biswapped Networks

Biswapped networks of the form Bsw(G) have recently been proposed as interconnection networks to be implemented as optical transpose interconnection systems. We provide a systematic construction of κ+1 vertex-disjoint paths joining any two distinct vertices in Bsw(G), where κ≥1 is the connectivity of G. In doing so, we obtain an upper bound of max{2Δ(G)+5, Δκ(G)+Δ(G)+2} on the (κ+1)-diameter of Bsw(G), where Δ(G) is the diameter of G and Δκ(G) the κ-diameter. Suppose that we have a deterministic multipath source routing algorithm in an interconnection network G that finds κ mutually vertex-disjoint paths in G joining any two distinct vertices and does this in time polynomial in Δκ(G), Δ(G) and κ (and independently of the number of vertices of G). Our constructions yield an analogous deterministic multipath source routing algorithm in the interconnection network Bsw(G) that finds κ+1 mutually vertex-disjoint paths joining any two distinct vertices in Bsw(G) so that these paths, all have length bounded as above. Moreover, our algorithm has time complexity polynomial in Δκ(G), Δ(G) and κ. We also show that if G is Hamiltonian, then Bsw (G) is Hamiltonian, and that if G is a Cayley graph, then Bsw(G) is a Cayley graph.