A Group Construction Method with Applications to Deriving Pruned Interconnection Networks

Abstract

Abstract-A number of low degree and, thus, low complexity, Cayley-graph interconnection structures, such as honeycomb and diamond networks, are known to be derivable by systematic pruning of 2D or 3D tori. In this paper, we extend these known pruning schemes via a general algebraic construction based on commutative groups. We show that, under certain conditions, Cayley graphs based on the constructed groups are pruned networks when Cayley graphs of the original commutative groups are kD tori. Thus, our results offer a general mathematical framework for synthesizing and exploring pruned interconnection networks that offer lower node degrees and, thus, smaller VLSI layout and simpler physical packaging. Our constructions also lead to new insights, as well as new concrete results, for previously known interconnection schemes such as honeycomb and diamond networks.

Using semidirect products of groups to build classes of interconnection networks

“…We call our general class of graphs toroidal semidirect product graphs. We explain how our framework not only encompasses cubeconnected cycles, recursive cubes of rings and cube-connected circulants but also the dual-cubes from [23], (some of the) multiswapped networks from [32], pruned tori similar to those in [38], and (some of the) biswapped networks from [39]. Our framework is such that there is a range of parameters at our disposal so that we might vary the graphs obtained; indeed, there is also considerable scope for defining brand new classes of graphs.…”

“…In so far as we are aware, such a graph S has not featured before in the literature but it demonstrates the variety of graphs that can be defined within our framework. More on the pruning of interconnection networks using the semidirect product can be found in [38]. As our final illustration of the graphs that can be defined within our framework, let us involve the group H which has hitherto not featured.…”

An Optimal Single-Path Routing Algorithm in the Datacenter Network DPillar

“…The first paper to establish that being a Cayley graph is a useful property for an interconnection network is [4] and since then, there has been much research into representing interconnection networks using finite groups. Not only do we immediately obtain that any Cayley graph is node-symmetric (which is a fundamental property of interconnection networks [7]) but Cayley graphs have been shown to be relevant to various networks in a variety of ways; for example, with regard to the design of interconnection networks by pruning nodes and edges from tori [24], the design of wireless DCNs [22], and the design of high-dimensional mesh-based interconnection networks [6].…”

Interconnection Networks of Degree Three Obtained by Pruning Two-Dimensional Tori

“…Incomplete (or pruned) k-ary n-cubes were derived by Parhami and Kwai in [34] and properties relating to symmetry, shortest paths, connectivity and Hamiltonicity were established. Certain pruned 3-dimensional tori were also studied by Xiao and Parhami in [40], and in [41] Xiao and Parhami established general algebraic constructions (based on commutative groups) to develop pruning techniques, which were used to improve known results relating to honeycomb networks and to diamond networks. The algebraic approach of Parhami, Kwai and Xiao was subsequently continued: in [35] where Rahman, Jiang, Masud and Horiguchi applied pruning techniques to the hierarchical torus network and studied properties relating to shortest paths, average inter-node distance, bisection width and VLSI layout area; and in [9] where an algebraic construction related to group semidirect products was developed and used to provide a generalization of earlier pruning schemes.…”