Information Processing Letters 75 (2000) 25–27

Cyclic-cubes and wrap-around butterflies

Meghanad D. Wagh∗, Prakash Math, Osman GuzideDepartment of Electrical Engineering and Computer Science Lehigh University, Bethlehem, PA 18015, USA

Received 4 January 2000; received in revised form 7 April 2000Communicated by F. Dehne

Abstract

Recently a new family of Cayley graphs of fixed even node degree called the Cyclic-cubes were proposed as interconnectionnetworks by Fu and Chau (1998). We show that these networks are simply wrap-aroundk-ary butterflies. 2000 ElsevierScience B.V. All rights reserved.

Keywords:Cayley graph; Interconnection networks; Butterfly networks

1. Introduction

Interconnection networks form the backbone of dis-tributed memory parallel architecture. One of the de-sirable properties of these networks is their symmetry.Symmetry often implies simpler mapping algorithmsas well as better fault tolerance. Since symmetry isan inherent property of Cayley graphs [2], there havebeen many attempts to base new interconnection net-works on Cayley graphs. Fixed node degree is also adesirable property of interconnection networks. It low-ers hardware costs as well as allows unbounded sizearchitectures using the same processors.

Fu and Chau have recently proposed a class,Gkn,of Cayley graphs with fixed even node degrees [1].They call these graphs the Cyclic-cubes. We show inthis paper that this class of graphs is the same as thewrap-aroundk-ary butterflies. This identification willhelp visualization of these graphs and their mappings.It will also enable porting of results developed forbinary butterflies regarding tree mappings [3], searchalgorithms [4], data selection and broadcast [5], etc.

∗ Corresponding author. Email: mdw0@Lehigh.Edu.

There arenkn nodes inGkn, each labeled with acyclic permutation ofn symbolst0, t1, . . . , tn−1 in thatorder. In addition, each symbol is attached with arankbetween 0 andk − 1 written as the superscript of thesymbol. Examples of node labels ofG3

4 are(t01 t22 t

03 t

10),

(t12 t13 t

20 t

21) and(t10 t

21 t

02 t

23). To specify the edges of the

graph, define functionsf (i), f (−i) :Gkn → Gkn, 0 6i < n as follows:1

f (i)(tjee . . . t

jn−1n−1 t

j00 t

j11 . . . t

je−1e−1

)= (tje+1

e+1 . . . tjn−1n−1 t

j00 t

j11 . . . t

je+ie

),

f (−i)(tjee . . . t

jn−1n−1 t

j00 t

j11 . . . t

je−1e−1

)= (tje−1−i

e−1 . . . tjn−1n−1 t

j00 t

j11 . . . t

je−2e−2

).

(Note that the operations on the rank of a symbol areperformed modulok.) Thenu ∈Gkn is connected to

f (0)(u), f (1)(u), . . . , f (k−1)(u),

f (0)(u), f (−1)(u), . . . , f (−(k−1)))(u).

1 In [1] authors use ranks 1 throughk and call the functionsf (0)

andf (−0) asg andg−1, respectively. Our notation simplifies therepresentation.

0020-0190/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0020-0190(00)00074-0

26 M.D. Wagh et al. / Information Processing Letters 75 (2000) 25–27

Clearly each of these edges are bidirectional. Inparticular, iff (i)(u) = v thenf (−i)(v) = u. Further,for n > 3, all the destinations ofu are distinct.Therefore the node degree ofGkn is 2k.G2

n is identicalto the graph defined in [6]. It has been shown in [1] thatGkn is a Cayley graph and supports Hamilton cycle, andembedding of hypercube and mesh.

2. Isomorphism to the wrap-aroundk-arybutterfly network

Previously Chen and Lau [7] have shown thatG2n

defined in [6] is identical to a binary butterfly. We willnow prove that the graphGkn is the same as the wrap-aroundk-ary butterflyBkn . The definition of thek-arybutterfly network can be found in [8].

Definition 1. The n-dimensional wrap-aroundk-arybutterfly,n > 3, k > 2, is defined to be a graph withnkn nodes labeled〈l,w〉 where l ∈ Zn andw ∈ Znk ,and having edges2

〈l,w0w1 . . .wn−1〉→⟨l − 1,w′0w′1 . . .w′n−1

⟩,

wherew′i =wi for i 6= (l − 1), andw′l−1 ∈ Zk ,〈l,w0w1 . . .wn−1〉→

⟨l + 1,w′0w′1 . . .w′n−1

⟩,

wherew′i =wi for i 6= l, andw′l ∈ Zk .

Thus the 2k nodes connected to any node〈l,w〉are obtained by either decreasing its first indexl by1 (modulon) and replacing the(l − 1)th component,wl−1, of thek-ary representation ofw, by an integerbetween 0 andk − 1 or by increasingl by 1 (modulon) and replacingwl by an integer between 0 andk−1.Bkn can be visualized as akn×n array of nodes with

node〈l,w〉 placed in thelth column and thewth row.Clearly, nodes in any column have connections only tothe nodes in the neighboring columns (except for thewrap-around links between the(n− 1)th column andthe 0th column). To simplify the drawing the wrap-around links, 0th column is redrawn following the(n− 1)th column.B3

3 is shown in Fig. 1.

2 Here,Zn denotes integers 0 throughn − 1 andZnk denotesncopies ofZk . Thusw ∈ Znk may be expressed asw0w1 . . .wn−1wherewi ∈ Zk . The stringw0w1 . . . wn−1 may be thought of as therepresentation of integerw in radix k system.

Fig. 1. Wrap-aroundk-ary butterflyB33.

To demonstrate the isomorphism betweenGkn andBkn , define a functionψ :Gkn→ Bkn as

ψ(tjee . . . t

jn−1n−1 t

j00 t

j11 . . . t

je−1e−1

)= 〈e, j0j1 . . . jn−1〉. (1)

Clearlyψ is one-one and onto. We now show that italso preserves the graph connectivity. Let

u= (tjee . . . tjn−1n−1 t

j00 t

j11 . . . t

je−1e−1

).

Its imageψ(u) is given in (1). The connectivity inGknshows thatu is connected tof (i)(u), 06 i 6 (n− 1).Now,

M.D. Wagh et al. / Information Processing Letters 75 (2000) 25–27 27

ψ(f (i)(u)

)=ψ(f (i)(tjee . . . tjn−1

n−1 tj00 t

j11 . . . t

je−1e−1

))=ψ(tje+1

e+1 . . . tjn−1n−1 t

j00 t

j11 . . . t

je+ie

)= 〈e+ 1, j0j1 . . . je−1 je + i je+1 . . . jn−1〉. (2)

By comparing the second indices ofψ(u) in (1)and ψ(f (i)(u)) in (2), one can see that all theircomponents match, except theeth component,je + i,which takes a value between 0 and(n− 1) since therank operations are always modulon. Therefore fromthe definition of the connectivity ofBkn , ψ(u) andψ(f (i)(u)), 06 i 6 (n − 1) are connected. Further,sincef (−i) is simply the inverse off (i), ψ(f (−i)(u)),andψ(u), 06 i 6 (n− 1) are also connected.

This proves that the even degree Cayley graphGknis isomorphic to the wrap-aroundk-ary butterflyBkn .

References

[1] A.W.C. Fu, S.C. Chau, Cyclic-cubes: A new family of inter-connection networks of even fixed-degrees, IEEE Trans. ParallelDistrib. Systems 9 (1998) 1254–1268.

[2] S. Akers, B. Krishnamurthy, A group-theoretic model for sym-metric interconnection networks, IEEE Trans. Comput. 38 (4)(1989) 555–566.

[3] A. Gupta, S.E. Hambrusch, Embedding complete binary treesinto butterfly networks, IEEE Trans. Comput. 40 (1991) 853–863.

[4] A. Ranade, Optimal speedup for backtrack search on a butterflynetwork, Math. Systems Theory 27 (1994) 85–102.

[5] W. Lin, T. Sheu, C.R. Das, T. Feng, C. Wu, Fast data selectionand broadcast on the butterfly network, in: Proc. WorkshopFuture Trends Distrib. Comput. Syst. 1990s, 1990, pp. 65–72.

[6] P. Vadapalli, K. Srimani, A new family of cayley graph intercon-nection networks of constant degree four, IEEE Trans. ParallelDistrib. Systems 17 (1) (1996) 26–32.

[7] G. Chen, C.M. Lau, Comments on a new family of Cayley graphinterconnection networks of constant degree four, IEEE Trans.Parallel Distrib. Systems 17 (1) (1997) 1299–1300.

[8] F. Leighton, Introduction to Parallel Algorithms and Architec-tures: Arrays, Trees, Hypercubes, Morgan Kaufmann, Los Al-tos, CA, 1992.