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 vertexdisjoint 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{2∆(G)+5,∆κ(G)+∆(G)+2} on the (κ+1)-diameter of Bsw(G), where ∆(G) is the diameter of G and ∆κ(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 2 distinct vertices and does this in time polynomial in ∆κ(G), ∆(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 2 distinct vertices in Bsw(G) so that these paths all have length bounded as above. Moreover, our algorithm has time complexity polynomial in ∆κ(G), ∆(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.