The shuffle-cubes and their generalization
✩Tseng-Kuei Li
a, Jimmy J.M. Tan
a, Lih-Hsing Hsu
a,∗, Ting-Yi Sung
baDepartment of Computer and Information Science, National Chiao Tung University, 1001 Ta Hsueh Road Hsinchu, Taiwan 30050 bInstitute of Information Science, Academia Sinica, Taipei, Taiwan 115
Received 23 February 2000; received in revised form 23 June 2000 Communicated by F.Y.L. Chin
Abstract
In this paper, we first present a new variation of hypercubes, denoted by SQn. SQnis obtained from Qnby changing some links. SQnis also an n-regular n-connected graph but of diameter about n/4. Then, we present a generalization of SQn. For any positive integer g, we can construct an n-dimensional generalized shuffle-cube with 2n vertices which is n-regular and n-connected. However its diameter can be about n/g if we consider g as a constant.2001 Elsevier Science B.V. All rights reserved.
Keywords: Hypercubes; Diameter; Connectivity; Interconnection network
1. Introduction
The topology of any interconnection network for parallel and distributed systems can be represented by an undirected graph. For the graph theoretic def-initions and notations we follows Harary’s book [7].
G= (V, E) is a graph if V is a finite set and E is a
subset of{(a, b) | (a, b) is an unordered pair of V }. We say that V is the vertex set and E is the edge set. The
degree of a vertex x, denoted by deg(x), is the number
of edges incident with x. A k-regular graph is a graph with deg(x)= k for any vertex x ∈ V . A sequence of vertices P= hx0, x1, . . . , xki is a path from x0to xk if (xi−1, xi)∈ E for 1 6 i 6 k and xi6= xj if i6= j. The
length of P is k. Let u and v be two vertices of G. The distance between u and v, denoted by d(u, v), is the ✩This work was supported in part by the National Science Council of the Republic of China under Contract NSC 89-2213-E-009-013.
∗Corresponding author.
E-mail address: lhhsu@cc.nctu.edu.tw (L.-H. Hsu).
length of the shortest path from u to v. The diameter of
G, denoted by D(G), is max{d(u, v) | u, v ∈ V }. The
connectivity of G, denoted by κ(G), is the minimum
number of vertices whose removal leaves the remain-ing graph disconnected or trivial.
Network topology is a crucial factor for intercon-nection networks since it determines the performance of a network. However, designing an interconnection network is a multiple-objective optimization problem. Usually, we want to minimize the diameter and to maximize the connectivity. There are a lot of inter-connection network topologies proposed in literature. Among these topologies, the n-dimensional hyper-cube, denoted by Qn, is one of many popular
topolo-gies. It is known that D(Qn)= n and κ(Qn)= n [11].
However, a hypercube does not make the best use of its hardware. It is possible to fashion networks with lower diameters than that of Qnand with the same
connec-tivity. For example, the cross cubes [4–6,9], twisted cubes [1,8], and Möbius cubes [3] are derived from Qn
by changing the connection of some hypercube links.
0020-0190/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 0 - 0 1 9 0 ( 0 0 ) 0 0 1 4 7 - 2
All of these topologies have connectivity n and have diameter around n/2. Thus, this is an improvement of approximately a factor of 2. A natural question raised is: if there is another way to change the connection of some hypercube links to lower the diameter.
In this paper, we first present a variant of hyper-cubes, called the shuffle-hyper-cubes, SQn. SQn is obtained
from Qnby changing some links of Qn. It has
connec-tivity n and has diameter around n/4. Then we present a generalization of shuffle-cubes. For any positive in-teger g, we can construct an n-dimensional general-ized shuffle-cube with 2n vertices which is n-regular and n-connected. Its diameter can be about n/g if we consider g as a constant.
2. Shuffle-cubes
We use n-bit binary strings to represent vertices, for example, u= un−1un−2. . . u1u0 for ui ∈ {0, 1} and
06 i 6 n − 1. We use pj(u) to denote the j -prefix
of u, i.e., pj(u)= un−1un−2. . . un−j, and si(u) the
i-suffix of u, i.e., si(u)= ui−1ui−2. . . u1u0. Let u and v
be two vertices. The number of bits that are differing in
u and v is called the Hamming distance between u and v, denoted by h(u, v). The n-dimensional hypercube, Qn, consists of all of the n-bit binary strings as its
vertices and two vertices are adjacent if and only if
h(u, v)= 1. It is known that Qn can be recursively
constructed from two copies of Qn−1. For this reason, Q0 is the complete graph K1 as the basis of the
hypercubes. We will use ⊕ to denote addition with modulo 2.
To construct shuffle-cubes, we define the following four sets:
V00= {1111, 0001, 0010, 0011}, V01= {0100, 0101, 0110, 0111}, V10= {1000, 1001, 1010, 1011}, V11= {1100, 1101, 1110, 1111}.
For ease of exposition, we limit our discussion to
n= 4k + 2 for k > 0.
Definition 1. The n-dimensional shuffle-cube, SQn,
is recursively defined as follows: SQ2 is Q2. For n> 3, SQn consists of 16 subcubes SQi1i2i3i4
n−4 , where ij∈ {0, 1} for 1 6 j 6 4 and p4(u)= i1i2i3i4for all
Fig. 1. SQ6.
vertices u in SQi1i2i3n−4 i4. The vertices u= un−1un−2. . . u1u0and v= vn−1vn−2. . . v1v0in different subcubes
of dimension n− 4 are adjacent in SQnif and only if (1) sn−4(u)= sn−4(v), and
(2) p4(u)⊕ p4(v)∈ Vs2(u).
For example, the vertex 111101 in SQ6is linked to the following vertices in different subcubes of dimen-sion 2: 101101, 101001, 100101 and 100001. We il-lustrate SQ6in Fig. 1 showing only edges incident at
vertices in SQ00002 and omitting others. Obviously, the degree of each vertex of SQnis n and the number of vertices (edges, respectively) is the same as that of Qn.
For 16 j 6 k, the jth 4-bit of u, denoted by uj4, is defined as uj4= u4j+1u4ju4j−1u4j−2. In particular,
the 0th 4-bit of u, u04, is defined as u04= u1u0. u j 4= v
j 4
if and only if u4j+i = v4j+i for−2 6 i 6 1. Thus,
similar to Hamming distance, we define 4-bit
Ham-ming distance between u and v, denoted by h4(u, v),
as the number of 4-bits uj4 with 06 j 6 k such that
uj46= vj4, i.e.,
h4(u, v)= {j| uj46= vj4for 06 j 6 k} .
Using the notion of h4(u, v), we can redefine SQn
as follows: The vertex u and the vertex v are linked by an edge if and only if one of the following conditions holds:
(1) uj4∗⊕ v4j∗∈ Vu0
4 for exactly one j
∗satisfying 16
j∗6 k and uj4= v4j for all 06 j 6= j∗6 k. (2) u04⊕ v40∈ {01, 10} and uj4= vj4for all 16 j 6 k.
For example, the ten neighbors of 1011000010 in
0001000010, 0000000010, 1011100010, 1011100110, 1011101010, 1011101110, 1011000000, and 1011000011. In other words, the vertex u is adjacent to the vertex v only if h4(u, v)= 1. The converse is
not necessarily true. For example, u= 0000000000 is not adjacent to v= 0000000011 though h4(u, v)= 1.
Thus, d(u, v)> h4(u, v) for any two vertices u, v
of SQn.
3. Properties of shuffle-cubes
In this paper, we only discuss on the connectivity and the diameter of SQn.
Theorem 1. SQnis n-connected.
Proof. We prove this theorem by induction. Since
SQ2 = Q2, SQ2 is 2-connected. Since SQn is
n-regular, it suffices to show that after removing arbitrary
f vertices from SQnfor 16 f 6 n − 1, the remaining
graph is still connected. Let F be an arbitrary set of f vertices.
Now consider n= 6. By definition, SQ6 consists of 16 SQ2 subcubes. We decompose SQ6 into two subgraphs H1and H2, where H1consists of those SQ2
subcubes containing vertices in F , and H2 consists
of the remaining SQ2 subcubes. It is observed that
H2is connected, and that H1− F is not necessarily
connected. We distinguish the following two cases:
Case 1.1. Each SQ2subcube has at most one vertex
in F . It follows that each subcube of SQ6− F is still connected and has at least three vertices. Furthermore,
H1contains at most five subcubes since|F | 6 5. Let Q0be a subcube in H1− F . Since Q0contains three
vertices, it has twelve edges connected with eleven or twelve other subcubes in SQ6− F . Since there are
at most five subcubes in H1− F , Q0 is connected to
some subcubes in H2. Since H2is connected and each
subcube in H1− F is also connected to H2, SQ6− F
is connected.
Case 1.2. There is a subcube containing at least two
vertices of F . Let v be a vertex in H1− F . Then v
is connected to four other subcubes. Since |F | 6 5,
H1contains at most four subcubes and therefore, v is
connected to a subcube in H2. Since H2is connected,
it follows that each vertex in H1− F is connected to
some vertices in H2. Therefore, SQ6− F is connected.
Hence, SQ6is 6-connected.
We assume that SQ4k−2 is (4k− 2)-connected for
k> 2. Now consider SQn for n= 4k + 2 and k > 2,
and there are at most 4k+ 1 vertices in F . Each subcube of SQ4k+2 is an SQ4k−2. We distinguish the
following two cases for F :
Case 2.1. Each subcube contains at most 4k− 3
vertices in F . By the induction hypothesis, each subcube is still connected. Consider two arbitrary subcubes SQi1i2i3i4
4k−2 and SQ j1j2j3j4
4k−2 . The edges (u, v)
between SQi14ki2i3i4−2 and SQj1j2j3j44k−2 satisfy s4k−2(u)= s4k−2(v), and p4(u)⊕ p4(v)= i1i2i3i4⊕ j1j2j3j4∈ Vs2(u). Therefore, the number of edges in SQnbetween
SQi1i2i3i4
4k−2 and SQ j1j2j3j4
4k−2 is 24k−4 which is greater
than |F |. Consequently, each subcube SQi14ki2i3i4−2 −
F is connected to every subcube SQj1j2j3j4 4k−2 − F .
Furthermore, it follows from the induction hypothesis that each subcube SQi1i2i34k−2i4 − F is connected. Hence
SQ4k+2− F is connected.
Case 2.2. There is a subcube containing at least
4k− 2 vertices in F . It follows that H1 contains at most four subcubes. The proof is similar to Case 1.2 for SQ6.
Hence, SQ4k+2 is 4k + 2 connected. And the theorem follows. 2
Lemma 1. D(SQn)> dn/4e + 3 if n = 4k + 2 with k> 4 .
Proof. Let P be any path of SQnfrom u to v. We can view P as a sequence of 4-bits changing from u to v. Let u= un−1un−2. . . u1u0= uk4uk4−1. . . u04with u04=
00, u14= 1100, u24= 1000, u34= 0100, and uj4= 0001 if 46 j 6 k. Let v = vn−1vn−2. . . v1v0 with vj = 0
for 06 j < n.
Note that 0001, 0100, 1000, and 1100 are only in
V00, V01, V10, and V11, respectively. We can change
any 4-bit 0001, 0100, 1000, or 1100 into 0000 in one step only if the 0th 4-bit is 00, 01, 10, or 11, respec-tively. Thus, d(u, v)> dn/4e + 3. Hence D(SQn)> dn/4e + 3. 2
Next, we propose a routing algorithm on SQn. Let u and v be two vertices of SQn. We use h∗4(u, v)
to denote the number of uj4 for 16 j 6 k such that
uj46= v4j.
Route1(u, v)
(1) If u= v, then accept the message.
(2) Find a neighbor w of u such that h∗4(w, v)= h∗4(u, v)− 1 if w exists. Then route into w.
(3) If there is no neighbor w of u such that h∗4(w, v)= h∗4(u, v)− 1, then route into the neighbor w of u
that changes u1u0in a cyclic manner with respect
to 00, 01, 11, 10. For example, w= pn−2(u)00 if u1u0= 10.
Example 1. Let u= 0001000101001000110000 and
v = 0000000000000000000011 be two vertices of
SQ20. The path obtained from Route1(u, v) is
0001000101001000110000, 0000000101001000110000, 0000000001001000110000, 0000000001001000110001, 0000000000001000110001, 0000000000001000110011, 0000000000001000000011, 0000000000001000000010, 0000000000000000000010, 0000000000000000000000, 0000000000000000000001, 0000000000000000000011.
We note that this path is not the shortest path.
Applying the above algorithm to any two vertices u and v on SQn, it is observed that we may apply step (3) at most three times to obtain a vertex w such that
h∗4(w, v)= 0. Hence the algorithm will find a path,
not necessarily the shortest path, of length at most
h∗4(u, v)+ 6 that joins u to v. Therefore, D(SQn)6 dn/4e + 5. We will discuss the exact value of D(SQn)
after we introduce the concept of generalized shuffle-cubes.
4. Generalized shuffle-cubes
In this section, we generalize the shuffle-cubes into
generalized shuffle-cubes. For any positive integer l,
we use S(l) to denote the set of all binary strings of length l and we use S∗(l) to denote S(l)− {00 · · ·0| {z }
l }.
Let b and g be any positive integers satisfying 2b>
(2g− 1)/g. For each i1i2· · ·ib∈ S(b), we associate
it with a subset Ai1i2···ib of S∗(g) with the following
properties:
(1) |Ai1i2···ib| = g, and
(2) Si1i2···i
b∈S(b)Ai1i2···ib= S∗(g).
We say the family A= {Ai1i2···ib | i1i2· · · ib∈ S(b)}
with the above properties is a normal (g, b) family. For example,{A00, A01, A10, A11} is the normal (4, 2)
family where A00, A01, A10, A11 are defined in
Sec-tion 2.
Definition 2. Let B be any b-regular graph with
vertex set S(b) and A be any normal (g, b) family. Then we can recursively define the n-dimensional generalized shuffle-cube GSQ(n, A, B) for any n=
kg+ b for k > 0 with its vertex set to be S(n) as
follows:
(1) If n= b, GSQ(n, A, B) is B.
(2) If n= kg + b for k > 1, any two vertices u and v in GSQ(n, A, B) are adjacent if and only if (a) sn−g(u) and sn−g(v) are adjacent in GSQ(n−
g, A, B), and pg(u)= pg(v); or
(b) sn−g(u) = sn−g(v) and pg(u) ⊕ pg(v) ∈ Aub−1ub−2···u0.
For example, Qn is the GSQ(n, A, B) where A= {A0} is a normal (1, 0) family with A0= {1} and B= Q0; and SQn is the GSQ(n, A, B) where A=
{A00, A01, A10, A11} is a normal (4, 2) family and B
is Q2.
Assume that GSQ(n, A, B) be a generalized shuffle-cube. Obviously, GSQ(n, A, B) is an n-regular graph with 2n vertices. Let u and v be vertices in GSQ(n,
A, B). For 16 j 6 k, the jth g-bit of u, denoted by ujg, is ujg= ugj+b−1ugj+b−2· · ·ugj+b−g. In particular,
the 0th g-bit of u is u0g = ub−1ub−2· · · u0. The
g-bit Hamming distance between u and v, denoted
by hg(u, v), is the number of g-bits ujg with 0 6 j 6 k such that ujg 6= vjg, i.e., hg(u, v)= |{j | ujg 6=
vgj for 06 j 6 k}|. We also use h∗g(u, v) to denote the
number of ujgfor 16 j 6 k such that ujg6= vjg.
Applying similar arguments to Theorem 1, we have the following theorem.
Theorem 2. κ(GSQ(n, A, B))= n if κ(B) = b.
To discuss the diameter of a generalized shuffle-cube GSQ(n, A, B), we assume that B has some Hamiltonian properties. Let G be a graph. A sequence of vertices C= hx0, x1, . . . , xki in a graph G is a cycle
if k> 3, (xi−1, xi)∈ E for 1 6 i 6 k, x0= xk, and xi6= xjfor 06 i 6= j < k. A Hamiltonian path (cycle)
is a path (cycle) that spans all the vertices of G. We say that G is Hamiltonian if G has a Hamiltonian cycle.
A graph G is Hamiltonian connected if there exists a Hamiltonian path from u to v for any two different vertices u and v in G. However, it is known that any bipartite graph with at least three vertices is not Hamiltonian connected. A bipartite graph with bipartition (X, Y ) is Hamiltonian laceable if there exists a Hamiltonian path from u to v for any two different vertices u and v that are in different parts, i.e., one in X and one in Y . For example, Qn is
Hamiltonian laceable [10].
Suppose that B is Hamiltonian. Let C= hx0, x1, . . . , xk= x0i is a Hamiltonian cycle of B. The cycle h00,
01, 11, 10, 00i, for example, is a Hamiltonian cycle of
Q2. We generalize the routing algorithm Route1(u, v)
for GSQ(n, A, B) as follows:
Route2(u, v)
(1) If u= v, then accept the message.
(2) Find a neighbor w of u such that h∗g(w, v)= h∗g(u, v)− 1 if w exists. Then route into w.
(3) If h∗g(u, v) > 0 and there is no neighbor w of u
such that h∗g(w, v)= h∗g(u, v)− 1, then route into
the neighbor w of u that changes u0g in a cyclic manner with respect to C.
(4) If h∗g(u, v)= 0, find a neighbor z of sb(u) in B
such that the distance between z and sb(v) is the
distance between sb(u) and sb(v) minus one. Then
route into pn−b(u)z.
So we have the following theorem.
Theorem 3. D(GSQ(n, A, B))6 (n − b)/g + 2b−
1+ D(B) if B is Hamiltonian.
The upper bound for the D(GSQ(n, A, B)) can be further reduced if B is Hamiltonian connected or Hamiltonian laceable. Assume that B is Hamiltonian laceable. To route u to v, we first compute a vertex sequence Z(u, v) of S(b) as follows: If sb(u) and sb(v) are in different parts, set Z(u, v) to be any
Hamiltonian path from sb(u) to sb(v). If sb(u) and sb(v) are in the same part, find a neighborhood sb(z)
of sb(v) in B, let P be a Hamiltonian path from sb(u)
to sb(z), and set Z(u, v) to be the vertex sequence hP, sb(v)i. Then the path of GSQ(n, A, B) from u to v can be determined by the following algorithm:
Route3(u, v)
(1) If u= v, then accept the message.
(2) Find a neighbor w of u such that h∗g(w, v) = h∗g(u, v)− 1 if w exists. Then route into w.
(3) If there is no neighbor w of u such that h∗g(w, v)= h∗g(u, v)− 1, then route into the neighbor w of u
that changes u0gin the order of Z(u, v).
Example 2. As we point out before, SQn is a gener-alized shuffle-cube GSQ(n, A, B) with B= Q2. It is known that Q2is Hamiltonian laceable. Let
u= 0001000101001000110000
and
v= 0000000000000000000011
be two vertices of SQ20. Obviously, 00 and 11 are in the same part and 10 is a neighbor of 11. Hence
h00, 01, 11, 10i is a Hamiltonian path from 00 to 10 in Q2. Thus, we can set Z(u, v) ash00, 01, 11, 10, 11i.
The path obtained from Route3(u, v) is 0001000101001000110000, 0000000101001000110000, 00000000 01001000110000, 0000000001001000110001, 0000000000001000110001, 0000000000001000110011, 0000000000001000000011, 0000000000001000000010, 0000000000000000000010, 0000000000000000000011.
We note that this path is shorter than the path obtained in Example 1.
It is observed that we should apply step (3) exactly 2b−1 times to obtain a vertex w such that either w = v or w is a neighbor of v. Thus, we have the following theorem.
Theorem 4. (n− b)/g 6 D(GSQ(n, A, B)) 6 (n −
b)/g+ 2bif B is Hamiltonian laceable.
Assume that B is Hamiltonian connected. To avoid trivial case, we assume that b> 1. To route from u to v in GSQ(n, A, B), we compute a vertex sequence
Z(u, v) of S(b) as follows: If sb(u) 6= sb(v), set Z(u, v) to be any Hamiltonian path from sb(u) to sb(v). If sb(u)= sb(v), find a neighborhood sb(z) of sb(v) in B, let P be a Hamiltonian path from sb(u)
to sb(z), and set Z(u, v) to be the vertex sequence hP, sb(v)i. We can also apply Route3(u, v) to obtain
a path from u to v. So we have the following theorem.
Theorem 5. D(GSQ(n, A, B))6 (n − b)/g + 2bif B is Hamiltonian connected.
Now, we can determine the diameter of SQn for
n= 4k + 2.
Theorem 6. Assume n= 4k + 2. Then D(SQn) is 2 if n= 2; 4 if n = 6; or dn/4e + 3 if n > 10.
Proof. Using breadth first search, we can easily
de-termine D(SQ2)= 2, D(SQ6)= 4, D(SQ10)= 6, and D(SQ14)= 7. Combining Lemma 1 and Theorem 4,
we can conclude that D(SQn)= dn/4e + 3 if n > 18.
The theorem is proved. 2
5. Conclusion
In this paper, we present a variation of hypercubes, called shuffle-cubes, and their generalization. All the present known variations of hypercubes of dimension
n are n regular graphs with connectivity n and of
diameter around n/2. The shuffle-cube of dimension
n, SQn, has the same parameters of these topological
properties except the diameter is around n/4. Further-more, for any positive integer g, we can construct an
n-dimensional generalized shuffle-cube with 2n ver-tices which is n-regular and n-connected. Its diameter can be about n/g if we consider g as a constant.
We can also choose g as log n and choose b as log n− loglog n. Obviously, 2b > (2g − 1)/g. So we can find a (g, b) family ˆA and construct an
n-dimensional generalized shuffle-cube GSQ(n, ˆA, Qb).
By Theorem 4,
D GSQ(n, ˆA, Qb)
6 2 n
log n.
Let N= 2nbe the number of nodes of GSQ(n, ˆA, Qb).
Thus, the diameter of GSQ(n, ˆA, Qb) is
O log N log log N .
The star graphs, Sn, is another family of famous
interconnection networks [2]. Snis an (n− 1)-regular
graphs with N= n! vertices and of diameter b3(n − 1)
/2c. Thus, the diameter of Snis also O(log log Nlog N ) which
is of the same order as that of GSQ(n, ˆA, Qb).
Acknowledgements
The authors are very grateful to the anonymous referees for their thorough review of the paper and concrete and helpful suggestions.
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