cyclic-cubes and wrap-around butterflies
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Information Processing Letters 75 (2000) 2527
Cyclic-cubes and wrap-around butterfliesMeghanad D. Wagh , Prakash Math, Osman Guzide
Department of Electrical Engineering and Computer Science Lehigh University, Bethlehem, PA 18015, USAReceived 4 January 2000; received in revised form 7 April 2000
Communicated by F. Dehne
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-around k-ary butterflies. 2000 ElsevierScience B.V. All rights reserved.Keywords: Cayley graph; Interconnection networks; Butterfly networks
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 , 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 .They call these graphs the Cyclic-cubes. We show inthis paper that this class of graphs is the same as thewrap-around k-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 , searchalgorithms , data selection and broadcast , etc. Corresponding author. Email: mdw0@Lehigh.Edu.
There are nkn nodes in Gkn, each labeled with acyclic permutation of n symbols t0, t1, . . . , tn1 in thatorder. In addition, each symbol is attached with a rankbetween 0 and k 1 written as the superscript of thesymbol. Examples of node labels ofG34 are (t
(t12 t13 t
21 ) and (t
23 ). To specify the edges of the
graph, define functions f (i), f (i) :Gkn Gkn, 0 6i < n as follows: 1
f (i)(tjee . . . t
j11 . . . t
)= (tje+1e+1 . . . tjn1n1 tj00 tj11 . . . tje+ie ),
f (i)(tjee . . . t
j11 . . . t
)= (tje1ie1 . . . tjn1n1 tj00 tj11 . . . tje2e2 ).
(Note that the operations on the rank of a symbol areperformed modulo k.) Then u Gkn is connected tof (0)(u), f (1)(u), . . . , f (k1)(u),f (0)(u), f (1)(u), . . . , f ((k1)))(u).
1 In  authors use ranks 1 through k and call the functions f (0)and f (0) as g and g1, respectively. Our notation simplifies therepresentation.
0020-0190/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0020-0190(00) 00 07 4- 0
26 M.D. Wagh et al. / Information Processing Letters 75 (2000) 2527
Clearly each of these edges are bidirectional. Inparticular, if f (i)(u) = v then f (i)(v) = u. Further,for n > 3, all the destinations of u are distinct.Therefore the node degree of Gkn is 2k.G2n is identicalto the graph defined in . It has been shown in  thatGkn is a Cayley graph and supports Hamilton cycle, andembedding of hypercube and mesh.
2. Isomorphism to the wrap-around k-arybutterfly network
Previously Chen and Lau  have shown that G2ndefined in  is identical to a binary butterfly. We willnow prove that the graph Gkn is the same as the wrap-around k-ary butterfly Bkn . The definition of the k-arybutterfly network can be found in .
Definition 1. The n-dimensional wrap-around k-arybutterfly, n > 3, k > 2, is defined to be a graph withnkn nodes labeled l,w where l Zn and w Znk ,and having edges 2
l,w0w1 . . .wn1l 1,w0w1 . . .wn1
where wi =wi for i 6= (l 1), and wl1 Zk ,l,w0w1 . . .wn1
l + 1,w0w1 . . .wn1
where wi =wi for i 6= l, and wl Zk .
Thus the 2k nodes connected to any node l,ware obtained by either decreasing its first index l by1 (modulo n) and replacing the (l 1)th component,wl1, of the k-ary representation of w, by an integerbetween 0 and k 1 or by increasing l by 1 (modulon) and replacingwl by an integer between 0 and k1.Bkn can be visualized as a knn array of nodes with
node l,w placed in the lth column and the wth 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. B33 is shown in Fig. 1.
2 Here, Zn denotes integers 0 through n 1 and Znk denotes ncopies of Zk . Thus w Znk may be expressed as w0w1 . . .wn1where wi Zk . The string w0w1 . . . wn1 may be thought of as therepresentation of integer w in radix k system.
Fig. 1. Wrap-around k-ary butterfly B33 .
To demonstrate the isomorphism between Gkn andBkn , define a function :Gkn Bkn as(tjee . . . t
j11 . . . t
)= e, j0j1 . . . jn1. (1)Clearly is one-one and onto. We now show that it
also preserves the graph connectivity. Let
u= (tjee . . . tjn1n1 tj00 tj11 . . . tje1e1 ).Its image (u) is given in (1). The connectivity in Gknshows that u is connected to f (i)(u), 06 i 6 (n 1).Now,
M.D. Wagh et al. / Information Processing Letters 75 (2000) 2527 27
)=(f (i)(tjee . . . tjn1n1 tj00 tj11 . . . tje1e1 ))=(tje+1e+1 . . . tjn1n1 tj00 tj11 . . . tje+ie )= e+ 1, j0j1 . . . je1 je + i je+1 . . . jn1. (2)By comparing the second indices of (u) in (1)
and (f (i)(u)) in (2), one can see that all theircomponents match, except the eth component, je + i ,which takes a value between 0 and (n 1) since therank operations are always modulo n. Therefore fromthe definition of the connectivity of Bkn , (u) and(f (i)(u)), 0 6 i 6 (n 1) are connected. Further,since f (i) is simply the inverse of f (i), (f (i)(u)),and (u), 06 i 6 (n 1) are also connected.
This proves that the even degree Cayley graph Gknis isomorphic to the wrap-around k-ary butterfly Bkn .
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