Incomplete k-ary n-cube and its derivatives

Abstract

Incomplete or pruned k-ary n-cube, nX3; is derived as follows. All links of dimension n À 1 are left in place and links of the remaining n À 1 dimensions are removed, except for one, which is chosen periodically from the remaining dimensions along the intact dimension n À 1: This leads to a node degree of 4 instead of the original 2n and results in regular networks that are Cayley graphs, provided that n À 1 divides k: For n ¼ 3 ðn ¼ 5Þ; the preceding restriction is not problematic, as it only requires that k be even (a multiple of 4). In other cases, changes to the basis network to be pruned, or to the pruning algorithm, can mitigate the problem. Incomplete k-ary n-cube maintains a number of desirable topological properties of its unpruned counterpart despite having fewer links. It is maximally connected, has diameter and fault diameter very close to those of k-ary n-cube, and an average internode distance that is only slightly greater. Hence, the cost/performance tradeoffs offered by our pruning scheme can in fact lead to useful, and practically realizable, parallel architectures. We study pruned k-ary n-cubes in general and offer some additional results for the special case n ¼ 3: r

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

“…They also derived results relating to the diameter, the average inter-node distance and the layout of these networks. 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.…”

A Group Construction Method with Applications to Deriving Pruned Interconnection Networks

“…Let S be as above and AE ¼ CayðH; T Þ, where T ¼ fs 1 ; s qþ1 ; s À1 qþ1 g. Then, AE is known as cube-connected cycles CCC q . It is the pruned network obtained from the graph À ¼ CayðG; SÞ, which is a ðq þ 1ÞD torus [10]. Fig.…”

“…Under certain conditions, when the original networks are Cayley graphs, so are the pruned versions [9], [10]. Thus, the pruned networks maintain node symmetry while benefiting from lower node degree, sparser wiring, and simpler layout.…”

“…The two cases together establish the result that DiamðAEÞ ¼ 3k=2 for any even k. t u Remark 3. In [9], the authors showed that the diameter of the pruned 3D torus [10] is 3k=2. Theorem 2 establishes that the pruned 3D torus, which has the same node and wiring complexity as the 3D diamond network, is also identical to the latter in terms of diameter.…”

Further Properties of Cayley Digraphs and Their Applications to Interconnection Networks

“…It is easily shown that a number of pruning schemes, in-cluding the one studied in [8], are equivalent to the construction above. Pruning of interconnection networks constitutes a way of obtaining variants with lower implementation cost, and greater scalability and packageability [9]. If pruning is done with care, and in a systematic fashion, many of the desirable properties of the original (unpruned) network, including (node, edge) symmetry and regularity, can be maintained while reducing both the node degree and wiring density which influence the network cost.…”