- Email: [email protected]
A note on cyclic-cubes
Information Processing Letters 72 (1999) 131–135
A note on cyclic-cubes Shien-Ching Hwang, Gen-Huey Chen ∗ Department of Computer Science and Information Engineering, National Taiwan University, Taipei 10764, Taiwan Received 20 May 1999 Communicated by S.G. Akl and F. Dehne
Abstract Fu and Chau have proposed a new family of Cayley graphs, named cyclic-cubes. In this note we show that cyclic-cubes are isomorphic to the well-known wrap-around butterfly graphs. 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Cayley graph; Cyclic-cube; Graph isomorphism; Wrap-around butterfly graph
1. Introduction Recently, Fu and Chau [3] have proposed a new family of Cayley graphs with even fixed-degrees, named cyclic-cubes, which is a generalization of Vadapalli and Srimani’s work [5]. In [2], the latter was shown isomorphic to the binary wrap-around butterfly graphs. In this note we show that the cyclic-cubes are also isomorphic to the wrap-around butterfly graphs. In the following, we review the structures of the cycliccubes and the wrap-around butterfly graphs. Let t1 , t2 , . . . , tn be n symbols with an ordering of t1 t2 · · · tn , where represents a total ordering. Each symbol is assigned a rank from {1, 2, . . . , k}. We j use tl to denote a ranked symbol, indicating that tl is assigned with rank j . The vertex set of a cyclic-cube contains all circular permutations of ranked symbols ordered by . We use Gkn to represent a cycliccube whose vertex set contains nk n distinct circular j j j permutations of t11 t22 . . . tnn , where jl ∈ {1, 2, . . . , k} for all 1 6 l 6 n.
The edges of Gkn are generated by 2k generator functions, denoted by g, g −1 , f 1 , f −1 , f 2 , f −2 , . . . , f k−1 , f −(k−1) , which are defined as follows. jl+1 jl+2 j j j tl+2 . . . tnn t11 . . . tl l g tl+1 j
j
j j
j
l+2 l+1 . . . tnn t11 . . . tl l tl+1 ; = tl+2 jl+1 jl−1 jl j j . . . tnn t11 . . . tl−1 tl g −1 tl+1
j j
j
j
j
l+1 l−1 . . . tnn t11 . . . tl−1 ; = tl l tl+1 jl+1 jl+2 j j j tl+2 . . . tnn t11 . . . tl l f i tl+1
j
j
j j
j
l+2 l+1 . . . tnn t11 . . . tl l tl+1 , = tl+2 i = 1, 2, . . . , k − 1; jl−1 jl j j −i jl+1 tl f tl+1 . . . tnn t11 . . . tl−1
j −i j
j
j
j
l+1 l−1 . . . tnn t11 . . . tl−1 , = tl l tl+1 i = 1, 2, . . . , k − 1,
where 1 6 l 6 n, j
l+1 tl+1
+i
(j +i)−k
l = tl+1
if jl + i > k,
and ∗ Corresponding author. Email: [email protected].
j −i
tl l
(jl −i)+k
= tl
if jl − i 6 0.
0020-0190/99/$ – see front matter 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 2 0 - 0 1 9 0 ( 9 9 ) 0 0 1 3 8 - 6
132
S.-C. Hwang, G.-H. Chen / Information Processing Letters 72 (1999) 131–135
Fig. 1. A drawing of G33 with the outer two columns identical.
if and only if δ(x) = y, where δ ∈ {g, g −1 , f i , f −i | i = 1, 2, . . . , k − 1}.
Formally, Gkn can be defined as follows. j j
j
Definition 1. The vertex set of Gkn is {t11 t22 . . . tnn , jn−1 j j j j j | jl ∈ {1, 2, . . . , k} for t22 t33 . . . t11 , . . . , tnn t11 . . . tn−1 all 1 6 l 6 n}. Two vertices x and y of Gkn are adjacent
Fig. 1 shows a drawing of G33 , where a b c is assumed. The vertices of G33 are grouped into different columns according to their first symbol. For
S.-C. Hwang, G.-H. Chen / Information Processing Letters 72 (1999) 131–135
133
Fig. 2. A drawing of BF(3, 3) with level 0 replicated.
the convenience of drawing, the column with the first symbol a is duplicated. We use BF(k, n) to denote an n-dimensional kary wrap-around butterfly graph. BF(k, n) contains
n levels, numbered 0, 1, . . . , n − 1, each of k n vertices. Each vertex v is represented by a two-tuple hl, β0 β1 . . . βn−1 i, where 0 6 l 6 n − 1 is the level of v and β0 β1 . . . βn−1 is a k-ary sequence that is used to
134
S.-C. Hwang, G.-H. Chen / Information Processing Letters 72 (1999) 131–135
Fig. 3. Redrawing of G33 .
distinguish v from the others with the same level as v. The edges exist between adjacent levels. Formally, BF(k, n) can be defined as follows.
1} for all 0 6 i 6 n−1}. Two vertices hl, β0 β1 . . . βn−1 i 0 and hl 0 , β00 β10 . . . βn−1 i are adjacent if and only if l 0 ≡ l + 1 mod n and βi = βi0 for all 0 6 i 6 n − 1 and i 6= l.
Definition 2. The vertex set of BF(k, n) is {hl, β0 β1 . . . βn−1 i | l ∈ {0, 1, . . . , n − 1} and βi ∈ {0, 1, . . ., k −
Fig. 2 shows a drawing of BF(3, 3). It was shown in [1] that BF(k, n) is an instance of the Cayley graphs.
S.-C. Hwang, G.-H. Chen / Information Processing Letters 72 (1999) 131–135
2. The isomorphism Two graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ) are isomorphic if there exists a one-to-one correspondence φ from V1 to V2 so that for all a, b ∈ V1 , (a, b) ∈ E1 if and only if (φ(a), φ(b)) ∈ E2 . For example, it is easy to see that G33 is isomorphic to BF(3, 3) if we redraw G33 as shown in Fig. 3. The following theorem shows that Gkn is isomorphic to BF(k, n). Theorem 1. Gkn is isomorphic to BF(k, n). j
j
j
j
. . . tl l of Gkn , define y = φ(x) = hl, (j1 − 1)(j2 − 1) . . . (jn − 1)i. Clearly, φ is a one-to-one correspondence. Since Gkn and BF(k, n) have the same number of edges, they are isomorphic if and only if for each edge (x1 , x2 ) of Gkn , (φ(x1 ), φ(x2 )) is an edge of BF(k, n). Since the Cayley graphs are vertex symmetric, we jl+1 jl+2 j j j tl+2 . . . tnn t11 . . . tl l . If (x1 , x2 ) is inassume x1 = tl+1 j
j
j j
l+2 l+1 . . . tnn t11 . . . tl l tl+1 , and duced by g, then x2 = tl+2 (φ(x1 ), φ(x2 )) = (hl, (j1 − 1)(j2 − 1) . . . (jn − 1)i, hl + 1, (j1 − 1)(j2 − 1) . . . (jn − 1)i), which is an edge of BF(k, n). Similarly, (φ(x1 ), φ(x2 )) is an edge of BF(k, n) if (x1 , x2 ) is induced by g −1 . jl+2 jl+3 tl+3 . . . If (x1 , x2 ) is induced by f i , then x2 = tl+2
j
j
j j
+i
1)(j2 − 1) . . . (jn − 1)i, hl + 1, (j1 − 1)(j2 − 1) . . . (jl − 1)(jl+1 − 1 + i)(jl+2 − 1) . . . (jn − 1)i), which is an edge of BF(k, n). Similarly, (φ(x1 ), φ(x2 )) is an edge of BF(k, n) if (x1 , x2 ) is induced by f −i . 2 In [3], a shortest-path routing algorithm for Gkn was proposed. According to Theorem 1 the algorithm can be adapted to BF(k, n). A shortest-path routing algorithm for BF(k, n) was also proposed in [4], independently.
j
l+1 l+2 tl+2 . . . tnn t11 Proof. For an arbitrary vertex x = tl+1
j
135
l+1 , and (φ(x1 ), φ(x2 )) = (hl, (j1 − tnn t11 . . . tl l tl+1
References [1] F. Annexstein, M. Baumslag, A.L. Rosenberg, Group action graphs and parallel architectures, SIAM J. Comput. 19 (3) (1990) 544–569. [2] G. Chen, F.C.M. Lau, Comments on “A new family of Cayley graph interconnection networks of constant degree four”, IEEE Trans. Parallel Distributed Systems 8 (12) (1997) 1299–1300. [3] A.W. Fu, S.C. Chau, Cyclic-cubes: A new family of interconnection networks of even fixed-degrees, IEEE Trans. Parallel Distributed Systems 9 (12) (1998) 1253–1268. [4] S.C. Hwang, G.H. Chen, Geodesics of butterfly graphs, Submitted. [5] P. Vadapalli, P.K. Srimani, A new family of Cayley graph interconnection networks of constant degree four, IEEE Trans. Parallel Distributed Systems 7 (1) (1996) 26–32.
A note on cyclic-cubes Shien-Ching Hwang, Gen-Huey Chen ∗ Department of Computer Science and Information Engineering, National Taiwan University, Taipei 10764, Taiwan Received 20 May 1999 Communicated by S.G. Akl and F. Dehne
Abstract Fu and Chau have proposed a new family of Cayley graphs, named cyclic-cubes. In this note we show that cyclic-cubes are isomorphic to the well-known wrap-around butterfly graphs. 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Cayley graph; Cyclic-cube; Graph isomorphism; Wrap-around butterfly graph
1. Introduction Recently, Fu and Chau [3] have proposed a new family of Cayley graphs with even fixed-degrees, named cyclic-cubes, which is a generalization of Vadapalli and Srimani’s work [5]. In [2], the latter was shown isomorphic to the binary wrap-around butterfly graphs. In this note we show that the cyclic-cubes are also isomorphic to the wrap-around butterfly graphs. In the following, we review the structures of the cycliccubes and the wrap-around butterfly graphs. Let t1 , t2 , . . . , tn be n symbols with an ordering of t1 t2 · · · tn , where represents a total ordering. Each symbol is assigned a rank from {1, 2, . . . , k}. We j use tl to denote a ranked symbol, indicating that tl is assigned with rank j . The vertex set of a cyclic-cube contains all circular permutations of ranked symbols ordered by . We use Gkn to represent a cycliccube whose vertex set contains nk n distinct circular j j j permutations of t11 t22 . . . tnn , where jl ∈ {1, 2, . . . , k} for all 1 6 l 6 n.
The edges of Gkn are generated by 2k generator functions, denoted by g, g −1 , f 1 , f −1 , f 2 , f −2 , . . . , f k−1 , f −(k−1) , which are defined as follows. jl+1 jl+2 j j j tl+2 . . . tnn t11 . . . tl l g tl+1 j
j
j j
j
l+2 l+1 . . . tnn t11 . . . tl l tl+1 ; = tl+2 jl+1 jl−1 jl j j . . . tnn t11 . . . tl−1 tl g −1 tl+1
j j
j
j
j
l+1 l−1 . . . tnn t11 . . . tl−1 ; = tl l tl+1 jl+1 jl+2 j j j tl+2 . . . tnn t11 . . . tl l f i tl+1
j
j
j j
j
l+2 l+1 . . . tnn t11 . . . tl l tl+1 , = tl+2 i = 1, 2, . . . , k − 1; jl−1 jl j j −i jl+1 tl f tl+1 . . . tnn t11 . . . tl−1
j −i j
j
j
j
l+1 l−1 . . . tnn t11 . . . tl−1 , = tl l tl+1 i = 1, 2, . . . , k − 1,
where 1 6 l 6 n, j
l+1 tl+1
+i
(j +i)−k
l = tl+1
if jl + i > k,
and ∗ Corresponding author. Email: [email protected].
j −i
tl l
(jl −i)+k
= tl
if jl − i 6 0.
0020-0190/99/$ – see front matter 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 2 0 - 0 1 9 0 ( 9 9 ) 0 0 1 3 8 - 6
132
S.-C. Hwang, G.-H. Chen / Information Processing Letters 72 (1999) 131–135
Fig. 1. A drawing of G33 with the outer two columns identical.
if and only if δ(x) = y, where δ ∈ {g, g −1 , f i , f −i | i = 1, 2, . . . , k − 1}.
Formally, Gkn can be defined as follows. j j
j
Definition 1. The vertex set of Gkn is {t11 t22 . . . tnn , jn−1 j j j j j | jl ∈ {1, 2, . . . , k} for t22 t33 . . . t11 , . . . , tnn t11 . . . tn−1 all 1 6 l 6 n}. Two vertices x and y of Gkn are adjacent
Fig. 1 shows a drawing of G33 , where a b c is assumed. The vertices of G33 are grouped into different columns according to their first symbol. For
S.-C. Hwang, G.-H. Chen / Information Processing Letters 72 (1999) 131–135
133
Fig. 2. A drawing of BF(3, 3) with level 0 replicated.
the convenience of drawing, the column with the first symbol a is duplicated. We use BF(k, n) to denote an n-dimensional kary wrap-around butterfly graph. BF(k, n) contains
n levels, numbered 0, 1, . . . , n − 1, each of k n vertices. Each vertex v is represented by a two-tuple hl, β0 β1 . . . βn−1 i, where 0 6 l 6 n − 1 is the level of v and β0 β1 . . . βn−1 is a k-ary sequence that is used to
134
S.-C. Hwang, G.-H. Chen / Information Processing Letters 72 (1999) 131–135
Fig. 3. Redrawing of G33 .
distinguish v from the others with the same level as v. The edges exist between adjacent levels. Formally, BF(k, n) can be defined as follows.
1} for all 0 6 i 6 n−1}. Two vertices hl, β0 β1 . . . βn−1 i 0 and hl 0 , β00 β10 . . . βn−1 i are adjacent if and only if l 0 ≡ l + 1 mod n and βi = βi0 for all 0 6 i 6 n − 1 and i 6= l.
Definition 2. The vertex set of BF(k, n) is {hl, β0 β1 . . . βn−1 i | l ∈ {0, 1, . . . , n − 1} and βi ∈ {0, 1, . . ., k −
Fig. 2 shows a drawing of BF(3, 3). It was shown in [1] that BF(k, n) is an instance of the Cayley graphs.
S.-C. Hwang, G.-H. Chen / Information Processing Letters 72 (1999) 131–135
2. The isomorphism Two graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ) are isomorphic if there exists a one-to-one correspondence φ from V1 to V2 so that for all a, b ∈ V1 , (a, b) ∈ E1 if and only if (φ(a), φ(b)) ∈ E2 . For example, it is easy to see that G33 is isomorphic to BF(3, 3) if we redraw G33 as shown in Fig. 3. The following theorem shows that Gkn is isomorphic to BF(k, n). Theorem 1. Gkn is isomorphic to BF(k, n). j
j
j
j
. . . tl l of Gkn , define y = φ(x) = hl, (j1 − 1)(j2 − 1) . . . (jn − 1)i. Clearly, φ is a one-to-one correspondence. Since Gkn and BF(k, n) have the same number of edges, they are isomorphic if and only if for each edge (x1 , x2 ) of Gkn , (φ(x1 ), φ(x2 )) is an edge of BF(k, n). Since the Cayley graphs are vertex symmetric, we jl+1 jl+2 j j j tl+2 . . . tnn t11 . . . tl l . If (x1 , x2 ) is inassume x1 = tl+1 j
j
j j
l+2 l+1 . . . tnn t11 . . . tl l tl+1 , and duced by g, then x2 = tl+2 (φ(x1 ), φ(x2 )) = (hl, (j1 − 1)(j2 − 1) . . . (jn − 1)i, hl + 1, (j1 − 1)(j2 − 1) . . . (jn − 1)i), which is an edge of BF(k, n). Similarly, (φ(x1 ), φ(x2 )) is an edge of BF(k, n) if (x1 , x2 ) is induced by g −1 . jl+2 jl+3 tl+3 . . . If (x1 , x2 ) is induced by f i , then x2 = tl+2
j
j
j j
+i
1)(j2 − 1) . . . (jn − 1)i, hl + 1, (j1 − 1)(j2 − 1) . . . (jl − 1)(jl+1 − 1 + i)(jl+2 − 1) . . . (jn − 1)i), which is an edge of BF(k, n). Similarly, (φ(x1 ), φ(x2 )) is an edge of BF(k, n) if (x1 , x2 ) is induced by f −i . 2 In [3], a shortest-path routing algorithm for Gkn was proposed. According to Theorem 1 the algorithm can be adapted to BF(k, n). A shortest-path routing algorithm for BF(k, n) was also proposed in [4], independently.
j
l+1 l+2 tl+2 . . . tnn t11 Proof. For an arbitrary vertex x = tl+1
j
135
l+1 , and (φ(x1 ), φ(x2 )) = (hl, (j1 − tnn t11 . . . tl l tl+1
References [1] F. Annexstein, M. Baumslag, A.L. Rosenberg, Group action graphs and parallel architectures, SIAM J. Comput. 19 (3) (1990) 544–569. [2] G. Chen, F.C.M. Lau, Comments on “A new family of Cayley graph interconnection networks of constant degree four”, IEEE Trans. Parallel Distributed Systems 8 (12) (1997) 1299–1300. [3] A.W. Fu, S.C. Chau, Cyclic-cubes: A new family of interconnection networks of even fixed-degrees, IEEE Trans. Parallel Distributed Systems 9 (12) (1998) 1253–1268. [4] S.C. Hwang, G.H. Chen, Geodesics of butterfly graphs, Submitted. [5] P. Vadapalli, P.K. Srimani, A new family of Cayley graph interconnection networks of constant degree four, IEEE Trans. Parallel Distributed Systems 7 (1) (1996) 26–32.