Hilbert graph: An expandable interconnection for clusters

This paper presents a new interconnection network inspired by chordal graphs and the small-world phenomenon, with an emphasis in providing an incrementally expandable network with a fixed degree and a small diameter. Whereas most small-world graphs are constructed by a random process, the proposed graphs are completely deterministic and regular. The Hilbert graph is constructed by a number of nodes laid down in a Hilbert curve to which a number of additional edges forming an extended mesh are incorporated. It is shown that the resulting structures are quasi planar, and posses the same cutwidth complexity of a two dimensional torus. Furthermore the Hilbert graph posses the same fixed degree and incrementally expandable properties of the torus and mesh; however it is shown that the Hilbert graph has a much larger tolerance to an increase in traffic, and can therefore be expanded to much larger systems than the mesh or torus. In particular it is shown that with an equal congestion, the number of nodes on a Hilbert network is at least O(N2.477) larger than that of a torus. Furthermore, while broadcast in the two dimensional torus takes at least N1/2 steps, it is shown that the same operation can be accommodated in O(N1/4) steps in a Hilbert graph. © 2005 Springer Science + Business Media, Inc.