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

Abstract

Publisher's copyright statement: c 2013 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract-We study an interconnection network that we call 3T orus(m, n) obtained by pruning the 4m × 4n torus (of links) so that the resulting network is regular of degree 3. We show that 3T orus(m, n) retains many of the useful properties of tori (although, of course, there is a price to be paid due to the reduction in links). In particular: we show that 3T orus(m, n) is node-symmetric; we establish closedform expressions on the the length of a shortest path joining any two nodes of the network; we calculate the diameter precisely; we obtain an upper bound on the average internode distance; we develop an optimal distributed routing algorithm; we prove that 3T orus(m, n) has connectivity 3 and is Hamiltonian; we obtain a precise expression for (an upper bound on) the wide-diameter; and we derive optimal one-toall broadcast and personalized one-to-all broadcast algorithms under both a one-port and all-port communication model. We also undertake a preliminary performance evaluation of our routing algorithm. In summary, we find that 3T orus(m, n) compares very favourably with tori.

Variational networks of cube-connected cycles are recursive cubes of rings

“…Numerous interconnection networks have been proposed over the last fifty years or so and new designs continue to emerge. There are various reasons for this ongoing investigation and these include the following: the changing face of parallel and distributed systems, which encompass networks-on-chips, supercomputers, clusters and data centre networks, along with new and unforeseen applications, imposes new demands on the underlying interconnection networks (see, e.g., [14]); interconnection networks also feature in peer-to-peer and overlay networks (see, e.g., [15]), social networks (see, e.g., [16]) and wireless sensor networks (see, e.g., [2]); and the 'structured' graphs into which interconnection networks sit feature in combinatorial chemistry (see, e.g., [1]), coding theory (see, e.g., [3]), mathematical physics (see, e.g., [12]) and discrete mathematics in general (often purely as interesting combinatorial objects as regards the latter instantiation; see, e.g., [6,8,18]).…”