Bipancyclicity of Hierarchical Hypercube Networks
Ruei-Yu Wu1, 4, ∗, Gen-Huey Chen1, and Jung-Sheng Fu2, and Gerard J. Chang3
1 Dept. of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan 2 Dept. of Electronic Engineering, National United University, Miaoli, Taiwan
3 Dept. of Mathematics, National Taiwan University, Taipei, Taiwan
4 Department of Management Information Systems, Hwa Hsia Institute of Technology, Taiwan
∗
Ruei-Yu Wu is the speaker. E-mail: fish@inrg.csie.ntu.edu.tw
Abstract - The hierarchical hypercube network is suitable
for massively parallel systems. The number of links in the hierarchical hypercube network forms a compromise between those of the hypercube and the cube-connected cycles. Recently, some interesting properties of the hierarchical hypercube network were investigated. Since the hierarchical hypercube is bipartite. A bipartite graph is bipancyclic if it contains cycles of every even length from 4 to |V(G)| inclusively. In this paper, we show that the hierarchical hypercube network is bipancyclic.
Keywords: Bipancyclic, bipartite graph, embedding, Gray
code, Hamiltonian cycle, hierarchical hypercube networks, hypercube.
1 Introduction
In recent decades, many interconnection network topologies have been proposed in the literature (see [12], [16]) for purpose of connecting hundreds or thousands of processing elements. Among these topologies, the hypercube network is a popular interconnection network with many attractive properties such as regularity, symmetry, small diameter, strong connectivity, recursive construction, partition ability, and relatively low link complexity [21].
Malluhi and Bayoumi were proposed the hierarchical hypercube network [19], which is an alternative to the hypercube. It owns many favorable topological properties for building massively parallel systems. An appealing property of this network is the low number of connections per processor which enhances the VLSI design and fabrication of the system. Other alluring features include regularity, symmetry and logarithmic diameter which imply easy and fast algorithms for communication. Besides, it can perform one-to-one communication, one-to-all communication and divide-and-conquer algorithms efficiently [17]-[19]. Moreover, the one-to-one disjoint paths algorithm was investigated in [24].
On the other hand, linear arrays and rings, which are two of the most fundamental networks for parallel and
distributed computation, are suitable for developing simple algorithms with low communication costs. Many efficient algorithms designed on linear arrays and rings for solving a variety of algebraic problems and graph problems can be found in previous works [16]. The pancyclicity of a network represents its power of embedding cycles of all possible lengths. An n-node network (graph) is pancyclic if it contains all cycles of lengths from 3 to n [3]. It can embed rings of all possible lengths with dilation 1, congestion 1, load 1, and expansion 1. The pancycle
problem on a network W asks, for every integer 3≤l≤ |W|,
whether or not W contains a cycle of length l, where |W| is the number of nodes contained in W. Obviously, a pancyclic network is Hamiltonian because a cycle of length
n corresponds to a Hamiltonian cycle. The pancycle
problem was solved on many networks, e.g., the twisted cube [5], the butterfly graph [13], the arrangement graph [7], the hypercomplete network [6], the alternating group graph [15] the CCC network [9], and the hierarchical cubic network [8].
The hypercube network [16] and the hierarchical hypercube network [19] are bipartite graphs. Bipancyclicity is essentially a restriction of the concept of pancyclicity to bipartite graphs whose cycles are necessarily of even length. A bipartite graph is bipancyclic if it contains a cycle of every even length from 4 to the number of its vertices. In this paper we solve the pancycle problem on the hierarchical hypercube network, that is, we show that the hierarchical hypercube network is bipancyclic.
The rest of this paper is organized as follows: In the next section, the structure of the hierarchical hypercube network is first reviewed. And the cycle embedding problem in the network is solved in Section 3. Finally, this paper concludes in Section 4.
2 Preliminaries
A network is conveniently represented as an undirected graph whose vertices represent the nodes (i.e., processors) of the network and whose edges represent the communication links of the network. Throughout this paper,
for the graph theoretical definitions and notations we follow [22].
Let G=(V, E) be a connected graph, where the set of
vertices V(G) represent processors, and the set of edges
E(G) represent links between processors. We use network
and graph, node and link (vertex and edge) interchangeably. A graph G=(V0∪V1, E) is bipartite if V(G) is the union of
two disjoint sets V0 and V1, such that every edge joins V0
with V1. Two vertices, u and v, have the same color if and
only if u and v are in the same partite set. If e1 and e2 are
distinct edges that are incident to a common vertex, then e1
and e2 are adjacent edges.
The degree of a vertex in G is the number of edges
incident to it. If all vertices have the same degree d, then G is called regular or d-regular. The distance between two vertices u and v, denoted by d(u, v), is the length of the shortest path between u and v.
An n-dimensional hypercube, denoted by Qn, is one
of the most popular networks. There are 2n nodes contained
in a Qn network, each is uniquely represented by a binary
sequence bn−1bn−2…b0 of length n. Two nodes in a Qn
network are adjacent if and only if they differ at exactly one bit position. An edge of Qn network is dimension k
(0≤k≤n−1) if its two end vertices differ at bk. The
hypercube network suffers from a practical limitation when it is used as the topology of a multiprocessor system. As n increases, it becomes more difficult to design and fabricate the nodes of Qn because of the large fanout.
To remove the limitation, the cube-connected cycles (CCC) network [20] was designed as a substitute for the hypercube. The node degree of CCC is restricted to three. However, this restriction degrades the performance of CCC at the same time. For example, CCC has a larger diameter
than the hypercube. Taking both the practical limitation and the performance into account, the hierarchical hypercube (HHC) network [19] was proposed as a compromise between the hypercube and CCC. An HHC network, which has a two-level structure, takes hypercube as basic modules and connects them in a hypercube manner. An HHC network has a logarithmic diameter, which is the same as a hypercube network. Since the topology of an HHC is closely related to the topology of a hypercube network, it inherits some favorable properties from the latter.
Recall that a CCC network can be obtained by replacing each node of a Qk network with a cycle of k
nodes so that these k nodes are connected to the k neighbors of the original node in the Qk network. Actually,
an HHC network is a modification of a CCC network in which the k−node cycle is replaced with a hypercube. Assume k=2m. An HHC network can be constructed as
follows: start with Q2m network and replace each node of it
with a Qm network.
Since there are a total of 2m
2 ×2m=22m+m nodes in the
HHC network, each node of the HHC network can be uniquely represented by a binary sequence bn−1bn−2…b0,
where n=2m+m. Refer to Figure 1, where an example with
m=2 is shown. For convenience, bn−1bn−2…b0 is expressed
as a two-tuple (S, P), where S=bn−1bn−2…bm tells which Qm
network the node is located in and P=bm−1bm−2…b0 gives
the address of the node in the located Qm network.
Let P(l)=bm−1…bl+1blbl−1…b0 (S(m+l)= bn−1…bm+l+1bm+1
bm+l−1…bm), where bl denote the complement of bl. An
HHC network can be defined in terms of graph as follows.
0101 0100 0001 0000 0010 0011 0110 0111 1000 1001 1010 1011 1100 1110 1101 1111 0101 0100 0001 0000 0010 0011 0110 0111 1111 1101 1110 1100 1011 1001 1000 1010 10 11 01 00 00 10 11 01 10 11 01 00 00 10 11 01 01 01 11 00 11 00 10 10 01 01 11 00 11 00 10 10 10 11 01 00 00 10 11 01 10 11 01 00 00 10 11 01 01 01 11 00 00 11 10 10 01 01 11 00 00 11 10 10 Q22 HHC (m=2)
Definition 2.1 The node set of an n-dimensional HHC
(n-HHC for short) is {(S, P)| S = bn−1bn−2…bm, and P =
bm−1bm−2…b0, and bi ∈{0, 1} for all 0≤i≤n−1}, where n=
2m +
m and m≥1. Node adjacency of an n-HHC network is defined as follows: (S, P) is adjacent to (1) (S, P(l)) for all 0
≤ l ≤ m−1 and (2) (S(m+dec(P)), P), where dec(P) is the decimal value of P.
Edges defined by (1) are referred to as internal edges, and those defined by (2) are referred to as external edges. Internal edges are within Qm networks and each of external
edges connects two Qm networks. As Figure 1, node (0000,
01) and node (0000, 11) are connected by an internal edge; node (0000, 01) and node (0010, 01) are connected by an external edge. Notice that an n-HHC network is (m+ 1)-regular, symmetric, and has a diameter of 2m+1 (see [19]).
In subsequent discussion, whenever a node A of an n-HHC network is mentioned, we use AS and AP to denote the S
part and P part of A, respectively. For each bi in S part, we
decrease each index i by m, so that all the index i would follow 0≤i≤2m − 1 in the rest of paper.
In the following, we define Gray codes, which will be used in the next section.
Definition 2.2 [11] An m-bit Gray code, denoted by Gm,
defines an ordering among all the m-bit binary numbers. G1 is defined as (0, 1), and for m > 1, Gm is defined
recursively in terms of Gm−1 as (0Gm−1, 1Gm−1r), where
Gm−1r stands for the reverse ordering of Gm−1 and 0Gm−1
(1Gm−1r) stands for prefixing each binary number in Gm−1
(Gm−1r) with 0 (1).
For example, G2 can be (00, 01, 11, 10) and G3 can be
(000, 001, 011, 010, 110, 111, 101, 100). Notice that every two adjacent binary numbers, including the first one and the last one, in Gm differ in exactly one bit position.
3 Cycles Embedding in HHC Networks
In this section, we embed cycles of all possible lengths into an n-HHC network. Since an n-HHC network is bipartite (see [19]), only cycles of even lengths, ranging from 4 to22m, can be embedded.
We used dH(V0, V1) to denoted the Hamming distance
between V0 and V1, which is the number of different bits
between V0 and V1. A path from V0 to Vm is denoted V0 →
V1 → V2 → …→ Vm. It can be also abbreviated to a V0-Vm
path. A cycle cl is denoted V0 → V1 → V2 → …→ Vm→ V0,
where l is the length of the cycle. Obeying the convention of most graph books, every path (or cycle) in this paper contains no repeated node.
For example, A=(00000000, 000)→ (00000001, 000)
*→ (00000001, 010) → (00000101, 010) *→ (00000101, 000) → (00000100, 000) *→ (00000100, 010) → (00000000, 010)=B expresses a path, denoted by A-B path,
from A = (00000000, 000) to B =(00000000, 010), where *→ denotes a shortest path within a Q3 network. The path
in a 11-HHC contains internal edges and external edges alternately. Each subpath of it within a Q3 network is
maintained shortest. It is easy to obtain a shortest path between any two distinct nodes in a Qm [21]. So, if the
subpaths within Qm networks are ignored, then a path in an
n-HHC network can be simply represented by a sequence
of external edges, called an external edge sequence (EES). In this example, the path contains four external edges that can be represented by their P parts, i.e., 000, 010, 000 and 010 in sequence. Hence, the path can be simply represented by an EES (000, 010, 000, 010).
Lemma 3.1 [8] Suppose dH(X, Y)=d≥1. There are X-Y
path in a Qm whose length are d+2, d+4,…, c, where m≥1,
c=2m−1 if d is odd, and c=2m−2 if d is even.
Let dH(X, Y)=1. By lemma 3.1, there are X-Y path in
Qm whose length is ranging from 3 to 2m−1 connect these
adjacent nodes, X and Y. Then, there are cycles in Qm
whose length is even and ranging from 4 to 2m. In the
other word, we have following corollary.
Corollary 3.2 An m-cube (Qm) is bipancyclic, where m>1.
Lemma 3.3 Suppose 4≤l≤22m+mand cl is a cycle or path
within an n-HHC. Let A and B are two arbitrary adjacent vertices in cl such that no external edges in cl incident to
them. Then, we can replace the link (A, B) by a path which obtained according to the EES (AP, BP, AP, BP). Then, the
cycle (or path) is extended to length l+6.
For example, by corollary 3.2, we can construct cycles c6 = (00000000, 000) → (00000000, 001) →
(00000000, 011) →(00000000, 111) → (00000000, 110) → (00000000, 010) → (00000000, 000) within a Q3 of an
11-HHC (m=3). Arbitrarily select two adjacent nodes A= (00000000, 111) and B= (00000000, 110) from c6. By
lemma 3.3, the link (A, B) can be replaced by the path which obtained according to the EES (111, 110, 111, 110). Then, the cycle is extended to length 12. The extended cycle is described as follow and the path obtained by the EES (111, 110, 111, 110) is underlined. c12: (00000000, 000)→(00000000, 001)→ (00000000, 011)→(00000000, 111)→(10000000, 111)→(10000000, 110)→(11000000, 110)→ (11000000, 111)→(01000000, 111)→(01000000, 110)→(00000000, 110)→(00000000, 010)→ (00000000, 000).
Some observations on above example, the original cycle c6 is located within a Q3 of an 11-HHC. The cycle c12
is the result of extending cycle c6 and the cycle c12 pass
through 4 different Q3’s. Clearly, the result of applying
lemma 3.2 one time can increase 3 Qm’s to the cycle. By
lemma 3.1, we can extend the cycle cl repeatedly until l≤
32 (=4×|V(Q3)|) and l is even. Then, we will describe the
Theorem 3.4 An n-HHC network contains cycles of
lengths ranging from 4 to 2n (= 2m+m
2 ), where m>2.
Proof. In the proof, we assume l is even. To construct a
cycle of length l, two cases should be considered as follows. Case 1. (4≤l≤2m): Without losing generality, we apply
corollary 3.2 in a Qm which is located in
m
2
0 , where 2m 0
represents 2m consecutive 0’s. By corollary 3.2, we can
construct cycles in a Qm of an n-HHC whose length is even
and ranging from 4 to 2m.
Case 2. (2m+2≤
l≤22m+m): Let node X and node Y be two
adjacent vertices in c2m−2 such that no external edges in cl−4 incident to them. Without loss of generality, assume XP =
xm-1xm-2…x10 and YP = XP(1) = xm-1xm-2…x11. There are at
most 2m-1 can be selected. We sort them by Gray code ordering. When l = k2m +2 and 1≤
k≤22m−m, the cycle is
extended by adding new Qm’s. We describe how to add
new Qm’s in two parts: (A) k=1 or k=2t, where 2≤t≤2m−
m; (B) otherwise.
(A) First, we select a new link (X, Y) by Gary code ordering. Then, we replace the link (X, Y) of Qm
m
2 0 by a path which is obtained according to the EES (AP, BP, AP, BP)
by applying lemma 3.3. Clearly, we add three Qm to the
cycle, and therefore the length of the cycle is l+2 (=l−4+
6).
(B) We can find a Qm in cl−4 where Qm's link (X, Y) is not
replaced. And we apply lemma 3.3 to extend the cycle. Clearly, we add three Qm to the cycle and the length of the
cycle is l+2 (=l−4+6).
Then, we can apply corollary 3.2 to extend the cycle
cl, where k2m+4≤l≤(k+3)2m and 1≤k≤
m 2
2 −1. Apply the method describe above repeatedly until all Qm’s of an
n-HHC are added to the cycle. We can extend the cycles with all even length from 4×2m+2 to 2m+m
2 . There are 2m
2 Qm’s
in an n-HHC. Each time we apply lemma 3.3, we can add three Qm’s to the cycle. After (
m 2
2 −1)/3 times, we can add all Qm’s of an n-HHC to the cycle. Note that (
m 2 2 −1)/3 is an integer since 2m 2 −1= 2 12 2 m−× −1=(22m−1)2−1=(22m−1 +1) (22m−1 −1)=3
∏
m=1−1 i ( i 22 +1). As a result, all Qm’s can be
added to the cycle. ■
To consider the m≤2, these cases are special. When
m=1, obviously a 3-HHC is also a cycle with length 8.
When m=2, there are 16 Q2’s in a 6-HHC. Clearly, a Q2 is
also a cycle c4. We use the construction method of theorem
3.4 which repeatedly applies lemma 3.3 five (=( 22
2 −1)/3)
times to add all Q2’s to the cycle. Then, we have cycles cl,
where 10 ≤l ≤26 and l is even. So, a 6-HHC network
contains cycles of all possible even lengths, except 6.
4 Conclusions
The hierarchical hypercube network was originally proposed in [17]-[19] for building massively parallel systems. It uses logarithmic links of a comparable hypercube and owns many favorable topological properties include regularity, symmetry and logarithmic diameter which imply easy and fast algorithms for communication. And an appealing property of this network is the low number of connections per processor which enhances the VLSI design and fabrication of the system. Besides, it can perform one-to-one communication, one-to-all communication and divide-and-conquer algorithms efficiently [17]-[19]. Moreover, the one-to-one disjoint paths algorithm was investigated in [24]. The hierarchical hypercube network was originally proposed in [9-11] for building massively parallel systems. It uses logarithmic links of a comparable hypercube and owns many favorable topological properties include regularity, symmetry and logarithmic diameter which imply easy and fast
In this paper, we solved the cycle embedding problem by showing that there are cycles of all possible even length in the n-dimensional hierarchical hypercube network, where m > 2. Consequently, the hierarchical hypercube network can efficiently execute all algorithms that are executable on linear arrays or rings. Many of such algorithms can be found in [1].
Finally, further research problems on the hierarchical hypercube network are suggested. For instance, Hamiltonian-laceability [10], [14], [23] and conditional faults [2], [4] problems were proposed. It still includes research issues in the hierarchical hypercube network.
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