On the isomorphism between cyclic-cubes and wrapped butterfly networks

Correspondence Papers

On the Isomorphism between Cyclic-Cubes

and Wrapped Butterfly Networks

Chun-Nan Hung, Jeng-Jung Wang, Ting-Yi Sung,

Member, IEEE Computer Society, and

Lih-Hsing Hsu

AbstractÐWe show that the cyclic-cubes defined by Ada W.C. Fu and S.C. Chau [1] are isomorphic to k-ary wrapped butterfly networks.

Index TermsÐCyclic-cubes, wrapped butterfly networks.

æ

FUand Chau [1] proposed a new family of Cayley graphs, called cyclic-cubes, which have even fixed degrees. Let Gk

n denote k-ary

n-dimensional cyclic-cubes. In [1], Fu and Chou also proposed optimal routing algorithms for Gk

n. Moreover, they showed that Gkn

has a Hamiltonian cycle, a diameter of b3n

2c, and connectivity of 2k

if n  3. In this short comment, we show that this family of graphs are indeed isomorphic to k-ary wrapped butterfly networks WB…n; k† which are defined in [2, pp. 442-446].

For a graph G, we use V …G† and E…G† to denote the vertex set and the edge set of G, respectively. To define Gk

n, let t1; t2;


; tnbe

n distinct symbols with ordering t1> t2


> tn. Each symbol tjis

assigned a rank i for 1  i  k, and this ranked symbol is denoted by ti

j. The graph Gkn has n
kn vertices, and each vertex of Gkn is

represented by an n-bit vector which is a circular permutation of ti1

1ti22


tinn for 1  i1; i2;


; in k. For example, in G24t23t14t11t22 is a

vertex and t2

3t14t22t11is not. In other words,

V …Gk

n† ˆ ftijjtij‡1j‡1


tninti11


tijÿ1jÿ1j for 1  j  n

and 1  i1; i2;


; in kg:

To define edges in Gk

n, we first define function fs, for every

1  s  k, mapping V …Gk

n† onto itself as follows:

fs…tijjtj‡1ij‡1


tinnti11


tjÿ1ijÿ1† ˆ tij‡1j‡1


tinnt1i1


tijÿ1jÿ1tsj for any 1  s  k:

Note that all fs are bijective functions. Each vertex x 2 V …Gkn† is

linked to exactly 2k vertices fs…x† and fsÿ1…x† for all 1  s  k. For

example, in G2

4 the vertex t23t14t11t22 is linked to t14t11t22t13, t14t11t22t23,

t1

2t23t14t11, and t22t23t14t11.

Now we introduce the definition of wrapped butterfly net-works WB…n; k†. The network WB…n; k† has n
kn_{vertices and each}

vertex is represented by an …n ‡ 1†-bit vector a0a1


anÿ1i, where

0  i  n ÿ 1 and 1  aj k for all 0  j  n ÿ 1. Two vertices

a0a1


anÿ1i and b0b1


bnÿ1j are adjacent in WB…n; k† if and only

if j ÿ i ˆ 1…mod n† and atˆ btfor all 0  t 6ˆ j  n ÿ 1.

In fact, Gk

n is isomorphic to WB…n; k†, as stated in the following

theorem. Theorem 1. Gk

nis isomorphic to WB…n; k†.

Proof. For each vertex a0a1


anÿ1i in WB…n; k†, we define a

function  mapping V …WB…n; k†† to V …Gk

n† as follows:

…a0a1


anÿ1i† ˆ tai‡2i‡1tai‡3i‡2


tnanÿ1ta10ta21


tai‡1i :

The function  is obviously bijective.

Let u ˆ a0a1


anÿ1i and v ˆ b0b1


bnÿ1j be two distinct

vertices in WB…n; k†. It follows that …u† and …v† are two distinct vertices in Gk

ngiven as follows:

…u† ˆ tai‡1

i‡2tai‡3i‡2


tannÿ1t1a0ta21


tai‡1i ;

…v† ˆ tbj‡1

j‡2tbj‡3j‡2


tbnnÿ1t1b0tb21


tbj‡1j :

Suppose that u and v are adjacent in WB…n; k†. Without loss of generality, we may assume that j ˆ i ‡ 1…mod n†. It follows that atˆ btfor all 0  t 6ˆ j  n ÿ 1, i.e.,

v ˆ a0a1


aibi‡1ai‡2


anÿ1…i ‡ 1†:

Therefore, …v† ˆ tai‡2

i‡3tai‡4i‡3


tannÿ1ta10t2a1


tai‡1i tbi‡2i‡1ˆ fbi‡1……u††:

Thus, …u† and …v† are adjacent in Gk

n. Hence, …u; v† 2

E…WB…n; k†† implies ……u†; …v†† 2 E…Gk n†.

On the other hand, let …u† and …v† be adjacent in Gk n.

It follows that …v† can be fs……u†† or fsÿ1……u†† for

some 1  s  k. Consider …v† ˆ fs……u†† for some 1  s  k.

It follows that …v† ˆ tai‡2

i‡3tai‡4i‡3


tannÿ1t1a0ta21


tai‡1i tsi‡2;

and v ˆ a0a1


aisai‡2


anÿ1…i ‡ 1†. Therefore, u; v are adja±

cent in WB…n; k† and furthermore, …v† ˆ fs……u†† implies

…u; v† 2 E…WB…n; k††. Since every fs is a bijective function, it

follows that …v† ˆ fÿ1

s ……u†† also implies …u; v† 2 E…WB…n; k††.

Hence, ……u†; …v†† 2 E…Gk

n† implies …u; v† 2 E…WB…n; k††.

Since …u; v† 2 E…WB…n; k†† if and only if ……u†; …v†† 2 E…Gk

n†;

the two graphs WB…n; k† and Gk

nare isomorphic. This theorem

is proven. tu

R

[1] A.W. Fu and S.-C. Chau, ªCyclic-Cubes: A New Family of Interconnection Networks of Even Fixed-Degrees,º IEEE Trans. Parallel and Distributed System, vol. 9, no. 12, pp. 1,253±1,268, Dec. 1998.

[2] F.T. Leighton, Introduction to Parallel Algorithms and Architecture: Arrays, Trees, Hypercubes. San Mateo: Morgan Kaufmann, 1992.

864 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 11, NO. 8, AUGUST 2000

. C.-N. Hung and L.-H. Hsu are with the Department of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan 30050, Republic of China.

E-mail: [email protected], [email protected].

. J.-J. Wang and T.-Y Sung are with the Institute of Information Science, Academia Sinica, Taipei, Taiwan 115, Republic of China.

E-mail: {jjwang, tsung}@iis.sinica.edu.tw.

Manuscript received 19 Apr. 1999; accepted 3 Feb. 2000.

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