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Computer Science Technical Report Computer Science Technical Report A New Fixed Degree Regular Network for Parallel Processing Shahram Lati® Pradip K. Srimani Department of Electrical Engineering Department of Computer Science University of Nevada Colorado State University Las Vegas, NV 89154 Ft. Collins, CO 80523 lati®@ee.unlv.edu [email protected] Technical Report CS-96-126 Computer Science Department Colorado State University Fort Collins, CO 80523-1873 Phone: (970) 491-5792 Fax: (970) 491-2466 WWW: http://www.cs.colostate.edu oceedings of Eighth IEEE Symposium on Paral lel To appear in the Pr Distributed Processing and , New orleans, October 1996. A New Fixed Degree Regular Network for Parallel Processing Shahram Lati® Pradip K. Srimani Department of Electrical Engineering Department of Computer Science University of Nevada Colorado State University Las Vegas, NV 89154 Ft. Collins, CO 80523 lati®@ee.unlv.edu [email protected] Abstract ameter, low degree, high fault tolerance etc. with the well known hypercubes (which are also Cayley graphs). We propose a family of regular Cayley network graphs All of these Cayley graphs are regular, i.e., each vertex of degree three based on permutation groups for design of has the same degree, but the degree of the vertices increases massively parallel systems. These graphs are shown to be with the size of the graph (the number of vertices) either based on the shuf¯e exchange operations, to have logarith- logarithmically or sub logarithmically. This property can mic diameter in the number of vertices, and to be maximally make the use of these graphs prohibitive for networks with fault tolerant. We investigate different algebraic properties large number of vertices [6]. Fixed vertex-degree networks of these networks (including fault tolerance) and propose are also very important from VLSI implementation point of a simple routing algorithm. These graphs are shown to be view [7]; there are applications where the computing ver- able to ef®ciently simulate other permutation group based tices in the interconnection network can have only a ®xed graphs; thus they seem to be very attractive for VLSI imple- number of I/O ports [6]. As a matter of fact a large multipro- mentation and for applications requiring bounded number cessor (with 8096 vertices) is being built by JPL around the of I/O ports as well as to run existing applications for other binary De Bruijn graphs [8], which are almost regular net- permutation group based architectures. work graphs of ®xed vertex degree 4. There are also other graphs in the literature that have almost constant vertex de- grees like the Moebius graphs [9] of degree 3. But neither 1 Introduction the De Bruijn graphs [8] nor the Moebius graphs [9] are reg- ular (and none are Cayley graphs). Also, these graphs have a low vertex connectivity of only 2 although most of the ver- Performance of any distributed system is signi®cantly de- tices in those graphs have degree larger than 2. Some ®xed termined by the choice of the underlying network topology. degree regular networks have been studied in [10, 11, 12]. One of the most ef®cient interconnection networks has been In this paper, we introduce a Cayley graph for intercon- the well known binary n-cubes or hypercubes; they have necting a large number of processors but with a constant de- been used to design various commercial multiprocessor ma- gree of the vertex. The proposed family of graphs is reg- chines and they have been extensively studied. For the past ular of degree 3 independent of the size of the graph, has several years, there has been a spurt of research on a class a logarithmic diameter in the number of vertices and has a of graphs called Cayley graphs which are very suitable for vertex connectivity of 3, i.e., the graphs are maximally fault designing interconnection networks. These Cayley graphs tolerant (vertex connectivity cannot exceed the vertex de- are based on permutation groups and they include a large gree). The proposed family of graphs offers a better and number of families of graphs like star graphs [1, 2], hy- more attractive alternative to De Bruijn graphs or Moebius percubes [3], pancake graphs [1, 4] and others [5]. These graphs for VLSI implementation in terms of regularity and graphs are symmetric (edges are bidirectional), regular and greater fault tolerance. Another interesting feature of the seem to share many of the desirable properties like low di- proposed graphs is that it can easily simulate other permu- tation group based graphs and hence parallel algorithms de- oceedings of Eighth IEEE Sym- To appear in the Pr signed for those topologies can be ef®ciently run on the new osium on Paral lel and Distributed Processing p , New orleans, October 1996. architecture with very minimal change. 2 Shuf¯e-Exchange Permutation (SEP) Proof : The proof follows from the fact that any basic 2- Graph cycle of a permutation can always be written in terms of the 2 swap and the n-cycle; for details, see [13]. ! A Cayley graph is a vertex-symmetric graph often with n SEP n 2 vertices, each assigned a label which is a distinct permuta- Lemma 2 n,, is a Cayley graph. 1; 2;:::;ng tion of the set f . The adjacency among the ver- SEP n 2 G = tices is de®ned based on a set of permutations referred to as Proof : The set of generators for n,, fg ;g ;g g L R generators; the neighbors of a vertex are obtained by com- 12 is closed under the inverse operation and the posing the generators with the label of the vertex. The set generator set can generate all the elements in the vertex set SEP 2 of generators de®ned for a Cayley graph must be closed un- of n. der the inverse operation to guarantee that the graph will be g g L ;2 undirected. Furthermore, the set of generators together with Remark 3 Note that the two generators 1 and could S n! the Identity permutation must produce all permutations produce all the elements of the group n and so could the g g n S R ;2 two generators 1 and . The third generator is in- on integers (which collectively belong to set n ), or a sub- S cluded with two objectives: to make the generator set closed group of n when the generators are repeatedly applied to the already generated vertices. under inversion (a requirement for the graph to be a Cay- SEP An n-dimensional graph is an undirected regular ley graph) and to make the graph undirected (bidirectional) = n! graph with N vertices, each vertex corresponding to since the bandwidth of a link incident on a vertex can be g g f1; 2; ;ng R a distinct permutation of the set . Two vertices shared between the incoming traf®c activated by L (or ) g g L are directly connected if and only if the label (permutation) and the outgoing traf®c activated by R (or ). of one is obtained from the label of the other by one of the n n 2 SEP following operations: Theorem 1 For any , , the graph n:(1)isa symmetric (undirected) regular graph of degree 3; (2) has Swapping the ®rst two digits (the leftmost digit is ! 3n!=2 n vertices; and (3) has edges. ranked as ®rst). Shifting cyclically to left (or right) by one digit. Proof : (1) follows from Remark 1. (2) follows from 2 Formally, consider a set of n distinct symbols (we use En- Lemma 1. (3) follows from (1) and (2). = 4 glish numerals as symbols; for example, when n ,the g De®nition 1 The edges which correspond to 12 are called ; 2; 3; 4 n symbols are 1 ). Thus, for distinct symbols, there E g g R Exchange or -edges and those corresponding to L (or ! are exactly n different permutation of the symbols or the S ) are referred to as Shuf¯e or S -edges; an -edge may be SEP n! a a a 1 2 n number of vertices in n is ;using as the symbolic representation of an arbitrary vertex, the edges of referred to either as an L-edge (corresponding to left shift) or an R -edge (corresponding to right shift). SEP n are de®ned by the following three generators in the graph: Remark 4 This is similar to the familiar shuf¯e-exchange g a a a =a a a a 1 2 n 2 3 n n L network where adjacent labels are obtained by either shift- g a a a =a a a a 1 2 n n 1 2 n1 R ing the label cyclically to left by one digit, or by complement- g a a a =a a a a 1 2 n 2 1 3 n 12 ing the rightmost digit of the label (instead of swapping). Remark 1 (i) The SEP is an undirected graph as swapping E ; L; R De®nition 2 We use the regular expression to the ®rst two digits is a symmetric operation; and if a vertex denote any sequence of generators; generator sequences v u is obtained from another vertex by shifting to left, vertex will be used to specify vertex to vertex paths in the graph. u u v can be obtained from by shifting to right; thus, if is v v u g adjacent to ,sois to , (ii) The generator 12 is its own 1 1 1 g g = g g = g = g R L inverse, i.e., 12 and ( ), (ii) 3 Simple Routing Algorithm 12 L R SEP Figure 1 shows the proposed trivalent Cayley graphs 3 SEP 3 4 4 and of dimensions and respectively.
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