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The spined cube: A new hypercube variant with smaller diameter
Information Processing Letters 111 (2011) 561–567
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
The spined cube: A new hypercube variant with smaller diameter Wujun Zhou a , Jianxi Fan a,∗ , Xiaohua Jia b , Shukui Zhang a a b
School of Computer Science and Technology, Soochow University, Suzhou 215006, China Department of Computer Science, City University of Hong Kong, Hong Kong
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 31 December 2010 Received in revised form 11 March 2011 Accepted 14 March 2011 Available online 17 March 2011 Communicated by A.A. Bertossi Keywords: Interconnection network BC graph Spined cube Diameter
Bijective connection graphs (in brief, BC graphs) are a family of hypercube variants, which contains hypercubes, twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes, etc. It was proved that the smallest diameter of all the known n-dimensional bijective 1 , given a fixed dimension n. An important question connection graphs (BC graphs) is n+ 2 about the smallest diameter among all the BC graphs is: Does there exist a BC graph whose diameter is less than the known BC graphs such as crossed cubes, twisted cubes, Möbius cubes, etc., with the same dimension? This paper answers this question. In this paper, we find that there exists a kind of BC graphs called spined cubes and we prove that the n-dimensional spined cube has the diameter n/3 + 3 for any integer n with n 14. It is the first time in literature that a hypercube variant with such a small diameter is presented. © 2011 Elsevier B.V. All rights reserved.
1. Introduction A multiprocessor interconnection network (in brief, interconnection network) can be represented by a graph G = ( V , E ), where V represents the node set and E represents the edge set, while each node represents a processor and each edge represent a hard link between some two processors. In this paper, we use graphs and interconnection networks interchangeably. Interconnection networks take important roles in parallel computing systems, which decide the performance of them at a large scale. Communication efficiency is a critical metric in a parallel computing systems. And the diameter of an interconnection network is an important metric for communication efficiency. Given an interconnection network (or graph) G and any two nodes u and v, the distance between u and v, denoted as dist(u , v ), is defined as the length of a shortest path between u and v. The diameter of graph G, denoted as diam(G ), is defined as the maximal value of the distances between all the pairs of nodes in G, i.e. diam(G ) = max{dist(u , v )|u , v ∈ V (G )}. Since a
*
Corresponding author. E-mail addresses: [email protected] (W. Zhou), [email protected] (J. Fan), [email protected] (X. Jia), [email protected] (S. Zhang). 0020-0190/$ – see front matter doi:10.1016/j.ipl.2011.03.011
© 2011
Elsevier B.V. All rights reserved.
delay will occur whenever a packet passes through a node, the smaller the diameter, the shorter the delay transferring a packet from the source to the destination under the worst case, the higher the communication efficiency is. As a result, given a fixed number of resources (nodes and edges), whether we can construct an interconnection network with a diameter as small as possible is a significant issue in the interconnection network design. So far, many interconnection networks have been proposed. Among all the interconnection networks, hypercubes have been shown to be popular ones due to their advantageous properties. Recently, Lee et al. proved that the hypercube is appropriate for computer design in nanospace [1]. Hypercubes have a superior nature: the n-dimensional hypercube Q n has diameter n, which is logarithm-level with respect to the node number of Q n . However, by changing some links, various hypercube variants are proposed. Twisted cubes are the first kind of hypercube variants [2]. The n-dimensional twisted cube 1 TQ n has diameter n+ , about a half that of Q n , for any 2 odd positive integer n. Crossed cubes are the second kind of hypercube variants [3]. The n-dimensional crossed cube 1 CQ n has diameter n+ for any positive integer n. Möbius 2 cubes are the third kind of hypercube variants [4]. The n-dimensional Möbius cube MQ n has two types—one is
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0-type and the other is 1-type. The 0-type n-dimensional 2 for any integer n 4 Möbius cube has diameter n+ 2 and the 1-type n-dimensional Möbius cube has diameter n+2 1 for any positive integer n 1. Besides, the expected distance of Möbius cube is lower. Hypercubes and the above three kinds of hypercube variants have two common characters: recursive constructability and bijective connection. That is, for any X n ∈ { Q n , TQ n , CQ n , MQ n }, there are always X n−1 , Y n−1 ∈ { Q n−1 , TQ n−1 , CQ n−1 , MQ n−1 } and a bijection f from V ( X n−1 ) to V (Y n−1 ), such that X n is a graph obtained by connecting each node x in X n−1 with f (x) in Y n−1 . Based on the two characters, Fan and He [5] proposed the definition of n-dimensional bijective connection graphs (in brief, BC graphs). It was proved that hypercubes and the above three kinds of hypercube variants are all BC graphs. It is convenient that once a property was proved on BC graphs, the individual kind of BC graphs will inherit that property. Fan and He [5] studied the general properties of BC graphs. Fan and Jia [6] studied edge-pancyclicity and pathembeddability of a subclass of BC graphs. Besides, Fan and Lin [7] studied t/k-diagnosability of BC graphs. And, Yang et al. [8,9] studied the minimum number of nodes adjacent to a set of nodes in BC graphs. In [10,11], the fourth kind of hypercube variants— locally twisted cubes was proposed. The n-dimensional 1 locally twisted cube LTQ n has diameter n+ for any in2
3 for any positive integer teger n with 1 n 4 and n+ 2 n 5. In fact, locally twisted cubes are also BC graphs. It was proved that the largest diameter of all the n-dimensional BC graphs is n [5]. And, it is easy to find a fact that the smallest diameter of all the above known 1 n-dimensional BC graphs (TQ n , CQ n , MQ n , LTQ n ) is n+ , 2 given a fixed dimension n. An important question about the smallest diameter among all the BC graphs is: Does there exist a BC graph whose diameter is less than the known BC graphs such as crossed cubes, twisted cubes, Möbius cubes, etc., with the same dimension? This paper answers this question. That is, the answer of the question is YES. In detail, we propose a new kind of n-dimensional BC graph called spined cube SQ n and prove that SQ n has the diameter n3 + 3 for any positive integer n with n 14. Obviously, when n is sufficiently big, the diameter of SQ n is about one third of Q n ’s and less than n+2 1 . It is the first time in literature that we find spined cubes have the smallest diameter. The rest of this paper is organized as follows: Section 2 provides the preliminaries. Section 3 gives the definition of spined cubes and several properties of them. Section 4 gives a proof of the diameter of SQ n . In Section 5, we conclude the paper.
2. Preliminaries In this section, we will give some definitions and terminologies used in this paper. Given graph G, let u and v be two nodes in graph G. A walk from u to v is defined as a sequence of nodes W (u , v ): u = u (0) , u (1) , . . . , u (k) = v, such that (u (i ) , u (i +1) ) is an edge of G for any 0 i k − 1. The length of walk W (u , v ) is the number of edges in it, denoted as | W (u , v )|.
A path is a walk without repeated nodes. It is obvious that if there is a walk between nodes u and v, there always exists a path with length no more than | W (u , v )| between u and v. For any integer n 1, a binary string u of length n is denoted by u 1 u 2 · · · un , where u i ∈ {0, 1} for any integer i = {1, 2, . . . , n}. Besides, u 1 is called the first bit of u and un is called the last bit of u. If x = x1 x2 · · · xn is a binary string of length n and s = s1 s2 · · · sm is a binary string of length m for any integer m 1, we use sx to denote the binary string s1 s2 · · · sm x1 x2 · · · xn of length m + n; Furthermore, let U ⊆ {0, 1}n , i.e., U is a set of some binary strings of length n. Then we use sU to denote the set {su |u ∈ U }. If E ⊆ E (G ), we use sE to denote the edges set {(sx, sy)|(x, y ) ∈ E } and use sG to denote (sV , sE ). For the walk W (x, y ): x = u (0) , u (1) , . . . , u (k) = y in G, since for any integer 0 i k − 1, (u (k) , u (k+1) ) ∈ E (G ), we know (su(k) , su(k+1) ) ∈ E (sG ). Thus sx = su(0) , su(1) , . . . , su(k) = sy is a walk in sG. We use sW (x, y ) to denote this walk. Given two graphs G and G , if there exists a bijection ψ from V (G ) to V (G ) such that (u , v ) ∈ E (G ) if and only if (ψ(u ), ψ( v )) ∈ E (G ) for any two nodes u , v ∈ V (G ), then we say that G is isomorphic to G and ψ is an isomorphic mapping from G to G . If the graphs G and G are two isomorphic graphs, we write G ∼ = G . The isomorphic graphs can be regarded as the identical graph. Before introducing the definition of BC graphs, we first give the definition of bijective connection in the following [5]: Definition 1. Let G be a graph. If V (G ) = V 1 ∪ V 2 , V 1 = φ , V 2 = φ , and V 1 ∩ V 2 = φ . We say that there exists a bijective connection between the subsets V 1 and V 2 in G, G
denoted by V 1 ←→ V 2 , if G satisfies the two following conditions: (1) For every u ∈ V 1 , there exists a unique v ∈ V 2 such that {u , v } ∈ E (G ); and (2) For every u ∈ V 2 , there exists a unique v ∈ V 1 such that {u , v } ∈ E (G ). The definition of bijective connection graphs with labels in their nodes was given in [12]. In this paper, we use that definition. An n-dimensional bijective connection graph (in brief, BC graph), denoted by X n , is an n-regular graph with 2n nodes. We identify each node of X n by a unique binary string of length n. The set of all the n-dimensional BC graphs is called the family of the n-dimensional BC graphs, denoted by Ln . Xn and Ln may be recursively defined as below [12]. Definition 2. The 1-dimensional BC graph X 1 is a complete graph on two nodes 0 and 1. The family of the 1-dimensional BC graph is defined as L1 = { X 1 }. Let G be a graph. G is an n-dimensional BC graph, denoted by X n , if there exist V 0 , V 1 ⊂ V (G ) such that the following three conditions hold: (1) V 0 = 0V 0 and V 1 = 0V 1 , where V 0 = V 1 = {0, 1}n−1 ; (2) V (G ) = V 0 ∪ V 1 , V 0 = ∅, V 1 = ∅, and V 0 ∩ V 1 = ∅; and
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Definition 4. For a given integer n with n 2 and any ij ij ij i , j ∈ {0, 1}, the adjacent-decided matrix B n = (bn,n , bn,n−1 , ij
. . . , bn,1 ) for SQ n can be recursively defined as follows: 1. For n = 2, let
01 10 11 B 00 2 = B2 = B2 = B2 = Fig. 1. The 2-dimensional spined cube SQ 2 and 3-dimensional spined cube SQ 3 .
2. For n = 3 and n = 4, let
G
The family of the n-dimensional BC graphs is defined as Ln = {G |G is an n-dimensional BC graph}.
⎤
⎣. .
⎦
1
.
0
⎤
1 0 ··· 0 ⎥ ⎢1 ⎥ ⎢ B n01 = B n11 = ⎢ 0. B 01 ⎥ . n −1 ⎦ ⎣.
.
0 3. For n > 4, let
⎤
⎡
1 0 ··· 0 ⎥ ⎢0 ⎥ ⎢0 ⎥ ⎢ 00 Bn = ⎢ ⎥, ⎢ 0. B n00−1 ⎥ ⎦ ⎣.
.
Definition 3. Let n 1 be an integer. The n-dimensional spined cube, SQ n , is defined recursively as follows:
Fig. 1 demonstrates SQ 2 and SQ 3 . SQ n is obviously an n-regular graph with 2n nodes and n−1 n2 edges, and the labels of any two adjacent nodes of SQ n differ in at most three successive bits. For convenience of presentation, we use node u and the label of u interchangeably. Clearly, 0SQ n−1 ∼ = 1SQ n−1 ∼ = SQ n−1 . Hence, we say that SQ n consists of two (n − 1)-dimensional subspined cubes. In [13], Larson et al. provided a methodology to denote a hypercube variants using matrix. In this paper, for convenience to discussion, we give an equivalent definition of spined cubes using a matrix representation, by the similar method to [13]. We call such a matrix an adjacent-decided matrix. For any integer n 2 and a node x = x1 x2 · · · xn ∈ V (SQ n ). We use an n-dimensional column vector [x1 , x2 , . . . , xn ] T to denote the node x.
⎡
⎡
3. The definition of spined cubes
(1) The 1-dimensional spined cube is a complete graph on two nodes 0 and 1. (2) SQ 2 is a graph consisting of four nodes labeled with 00, 01, 10, and 11, connected by four edges (00, 01), (00, 10), (01, 11), and (10, 11). (3) For n = 3 and n = 4, V (SQ n ) = V (0SQ n−1 )∪ V (1SQ n−1 ), E = E (0SQ n−1 )∪ E (1SQ n−1 )∪ E , where E = {(0x2 x3 · · · xn , 1(x2 ⊕ xn )x3 · · · xn )|xi ∈ {0, 1}, i = 2, 3, . . . , n}. (4) For n > 4, V (SQ n ) = V (0SQ n−1 ) ∪ V (1SQ n−1 ), E = E (0SQ n−1 ) ∪ E (1SQ n−1 ) ∪ E , where E = {(0x2 x3 · · · xn , 1(x2 ⊕ xn−1 )(x3 ⊕ xn )x4 · · · xn )|xi ∈ {0, 1}, i = 2, 3, . . . , n}.
0
1 0 ··· 0 ⎢0 ⎥ ⎢ ⎥ B n00 = B n10 = ⎢ 0. B 00 ⎥ , n −1
(3) V 0 ←→ V 1 , G [ V 0 ] ∈ Ln−1 , and G [ V 1 ] ∈ Ln−1 .
In this section, we will define a new kind of BC graphs called spined cubes and give some properties of them. We first introduce the ⊕ operation. We use ⊕ to denote (modulo 2) sum of two bits a and b (a, b ∈ {0, 1}). And for vectors, letting C = [c 1 , c 2 , . . . , cn ] T and D = [d1 , d2 , . . . , dn ] T be two column vectors, we use C ⊕ D to denote [c 1 ⊕ d1 , c 2 ⊕ d2 , . . . , cn ⊕ dn ] T .
1 0
⎤
⎡
1 0 ··· 0 ⎥ ⎢0 ⎥ ⎢1 ⎥ ⎢ 01 Bn = ⎢ 0 ⎥, ⎢ . B n01−1 ⎥ ⎦ ⎣.
.
0
0
⎤
⎡
1 0 ··· 0 ⎥ ⎢1 ⎥ ⎢0 ⎥ ⎢ B n10 = ⎢ 0 ⎥, ⎢ . B n10−1 ⎥ ⎦ ⎣.
.
0
⎤
⎡
1 0 ··· 0 ⎥ ⎢1 ⎥ ⎢1 ⎥ ⎢ B n11 = ⎢ 0 ⎥. ⎢ . B n11−1 ⎥ ⎦ ⎣.
.
0
The adjacent-decided matrix denotes the neighbor relaij tionship of a node. Let bn,k denote the k-th column vector ij
of B n , 1 k n. Then, for any x = x1 x2 · · · xn ∈ V (SQ n ), x x y = [x1 , x2 , . . . , xn ] T ⊕ bnn,k−1 n is a neighbor of x. We call y the k-dimensional neighbor of x. If we say the node x = x1 x2 x3 · · · xn−1 xn to be of xn−1 xn type. We can divide the nodes of SQ n into four types: 00type, 01-type, 10-type, and 11-type. For example, for n = 5 and a node x = 01011. Because the last two bits xn−1 xn of 01011 is 11, it is a node of 11-type, and B 11 5 represents the neighbor relationship of 11-type nodes. By definition 4, we have
⎡
B 11 5
1
0
0
0
0
⎤
⎢1 1 0 0 0⎥ ⎥ ⎢ ⎥ ⎢ = ⎢1 1 1 0 0⎥. ⎥ ⎢ ⎣0 0 1 1 0⎦ 0
0
0
0
1
Then, the 1-dimensional and 5-dimensional neighbors of node 01011 are respectively
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⎡ ⎤ 0
⎡ ⎤ 0
⎡ ⎤ 0
⎢1⎥ ⎢0⎥ ⎢1⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T 11 [0, 1, 0, 1, 1] ⊕ b5,1 = ⎢ 0 ⎥ ⊕ ⎢ 0 ⎥ = ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣1⎦ ⎣0⎦ ⎣1⎦ 1
⎡ ⎤ 0
1
⎡ ⎤ 1
and
0
⎡ ⎤ 1
⎢1⎥ ⎢1⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T 11 [0, 1, 0, 1, 1] ⊕ b5,5 = ⎢ 0 ⎥ ⊕ ⎢ 1 ⎥ = ⎢ 1 ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣1⎦ ⎣0⎦ ⎣1⎦ 1
0
1
That is, the 1-dimensional and 5-dimensional neighbors of node 01011 are nodes 01010 and 10111, respectively. Definition 4 is obviously equivalent to Definition 3. From Definition 4, we can easily verify the following two properties for any integers i , j ∈ {0, 1}. Proposition 5. For any integers k and n with n 2 and any node x ∈ V (SQ n ), if k 4, then the last two components of ij ij bn,k are always 0, and moreover, for 1 k 3, bn,k have 1’s only in the last three components, i.e. if k 4 the type of the k-dimensional neighbor of x would be the same as x and for 1 k 3, the k-dimensional neighbor of x differs from x only in the last three bits. ij
Proposition 6. The properties of each B n are listed as following: 1. B n00 : each column vector of B n00 contains exact one component of 1 and the other n − 1 components are 0, i.e. each neighbor of a 00-type node differs from that 00-type node in exact one bit. 2. B n01 and B n10 : each column vector of B n01 (or B n10 ) contains at most two components of 1, i.e. each neighbor of a 01type (or 10-type) node differs from that node in at most two bits. 3. B n11 : each column vector of B n11 contains at most three components of 1. Besides, there is not any column with the last two components both 1, i.e. each neighbor of a 11-type node differs from that node in at most three bits and the neighbor node cannot be a node of 00-type. By Definition 4, it is easy to verify these properties. For example, for i j = 11, By Definition 4, we have:
⎡1 ··· ⎢1 ··· ⎢ ⎢1 ··· ⎢ ⎢. ⎢ . .. . ⎢. ⎢ ⎢ 0 ··· B n11 = ⎢ ⎢ ⎢0 ··· ⎢ ⎢0 ··· ⎢ ⎢ ⎢0 ··· ⎢ ⎣0 ··· 0
0
0
0
0
0
0⎤
0
0
0
0
0
0⎥
0
0
0
0
0
.. .
.. .
.. .
.. .
.. .
1
0
0
0
0
1
1
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
0
1
1
⎥
0⎥ ⎥
.. ⎥ ⎥ .⎥ ⎥ 0⎥ ⎥. ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎦
··· 0 0 0 0 0 1
For any integers k, if k 4, the last two components of bn11,k are 00, and moreover, for 1 k 3, bn11,k have 1’s only in the last three components. And it is obvious that each column vector of B n11 contains at most three components of 1. Besides, there is not any column vector with the last two components both 1. 4. The diameter of spined cube In this section, we will study the diameter of spined cubes. We first define ruled walks. Definition 7. If a walk in a spined cube contains the nodes of 00-type, 01-type, 10-type and 11-type, then we call this walk a ruled walk. The following lemma determines the upper bound of the diameter of SQ n for n 2. Lemma 8. For any integer n with n 2, diam(SQ n ) n/3 + 3. Proof. We prove this lemma by proving that there always exists a ruled walk between any two nodes x and y in V (SQ n ) with length at most n/3+ 3 and thus dist(x, y ) n/3 + 3. For any x = x1 x2 · · · xn , y = y 1 y 2 · · · yn ∈ V (SQ n ), the desired ruled walk can be got by induction on n. Base step: For n = 2, we can enumerate ruled walks between any two nodes as follows:
RW (00, 00): RW (00, 01): RW (00, 10): RW (00, 11): RW (01, 01): RW (01, 10): RW (01, 11): RW (10, 10): RW (10, 11): RW (11, 11):
00, 01, 11, 10, 00 00, 10, 11, 01 00, 01, 11, 10 00, 01, 11, 10, 11 01, 11, 10, 00, 01 01, 11, 10, 00, 10 01, 00, 10, 11 10, 00, 01, 11, 10 10, 00, 01, 11 11, 10, 00, 01, 11
The rest can be got by reversing the above ruled walks. The length of these ruled walks is at most 4, which satisfies 2/3 + 3. That is, the lemma holds for n = 2. Fig. 2 illustrates ruled walks starting from node 00. For n = 3, if x1 = y 1 and RW (x2 x3 , y 2 y 3 ) is a ruled walk of n = 2, then x1 RW (x2 x3 , y 2 y 3 ) is a ruled walk from node x1 x2 x3 to node y 1 y 2 y 3 . For instance, if RW (00, 00): 00, 01, 11, 10, 00 is a ruled walk in SQ 2 , then 1RW (00, 00): 100, 101, 111, 110, 100 is a ruled work in SQ 3 . Thus, if x1 = y 1 , there is a ruled walk between nodes x and y with length at most 4. Otherwise, x1 = y¯ 1 . We can enumerate ruled walks between 0x2 x3 and 1 y 2 y 3 as follows:
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Fig. 2. Ruled walks staring from node 00 in SQ 2 , where a bold line segment represents an edge in a ruled walk and the number beside an edge represents the order that a ruled walk passed through this edge.
RW (000, 100): RW (000, 101): RW (000, 110): RW (000, 111): RW (001, 100): RW (001, 101): RW (001, 110): RW (001, 111): RW (010, 100): RW (010, 101): RW (010, 110): RW (010, 111): RW (011, 100): RW (011, 101): RW (011, 110): RW (011, 111):
000, 001, 111, 110, 100 000, 010, 011, 101 000, 001, 111, 110 000, 010, 011, 101, 111 001, 011, 010, 110, 100 001, 000, 010, 011, 101 001, 011, 101, 100, 110 001, 000, 010, 110, 111 010, 011, 101, 100 010, 000, 001, 011, 101 010, 011, 101, 100, 110 010, 000, 100, 101, 111 011, 101, 100, 110, 100 011, 010, 000, 100, 101 011, 001, 000, 010, 110 011, 010, 000, 001, 111
We only enumerate x1 = 0 and y 1 = 1, the case of x1 = 1 and y 1 = 0 can be got by reversing the above ruled walks. In this case, the length of the ruled walks is still at most 4 3/3 + 3. Therefore, the lemma holds for n = 3. For n = 4 and any two nodes x1 x2 x3 x4 and y 1 y 2 y 3 y 4 , if x1 = y 1 , then the ruled walks can be derived similar to the case for n = 3. The length of ruled walks is still at most 4. Otherwise, x1 = y¯ 1 . Let z1 z2 z3 z4 be the 4-dimensional neighbor of x1 x2 x3 x4 . By Definition 4, z1 = x¯1 . Thus, z1 = y 1 . As a result, there is a ruled walk W between z1 z2 z3 z4 and y 1 y 2 y 3 y 4 with length at most 4. Therefore, the walk x1 x2 x3 x4 , W is a ruled walk of length at most 5 4/3 + 3. Thus, the lemma holds for n = 4. In summary, we have proven the lemma for n = 2, n = 3, and n = 4. Induction step: Suppose that for any integer n with τ − 3 n τ − 1, there exists a ruled walk with length at most n/3 + 3 between any two nodes in SQ n (τ 5). In the following, we will prove that the lemma holds for n = τ. By definition 3, the SQ τ is composed of two (τ − 1)dimensional sub-spined cubes 0SQ τ −1 and 1SQ τ −1 . Furthermore, for i ∈ {0, 1}, iSQ τ −1 is composed of two (τ − 2)-dimensional sub-spined cubes i0SQ τ −2 and i1SQ τ −2 . Moreover, for j ∈ {0, 1}, i jSQ τ −2 is composed of two (τ − 3)-dimensional sub-spined cubes i j0SQ τ −3 and i j1SQ τ −3 . Thus, SQ τ can be divided into eight disjoint (τ − 3)dimensional sub-spined cubes and the eight sub-spined cubes are isomorphic to one another and all isomorphic to SQ n−3 (see Fig. 3). That is V (SQ n ) = i jk∈{0,1}3 {i jk ×
V (SQ n−3 )}, E (SQ n ) = i jk∈{0,1}3 {i jkE (SQ n−3 )} ∪ E , where E is edges between i jkV (SQ n−3 ).
Fig. 3. Sub-spined cube division of SQ τ .
Fig. 4. The constructing process of a ruled walk.
According to the induction hypothesis, there exists a ruled walk between any two nodes with length at most (τ − 3)/3 + 3 = τ /3 + 2 in (τ − 3)-dimensional SQ τ −3 . Then, for any two nodes x = x4 x5 · · · xτ and y = y 4 y 5 · · · y τ in SQ τ −3 , there exists a ruled walk RW (x , y ) with length at most τ /3 + 2 in SQ τ −3 . Let y = x1 x2 x3 y 4 y 5 · · · y τ and x = y 1 y 2 y 3 x4 x5 · · · xτ . Because for any i jk ∈ {0, 1}3 , i jkE (SQ n−3 ) ⊂ E (SQ n ), RW (x, y ) = x1 x2 x3 RW (x , y ): x = u (0) , u (1) , . . . , u (k) = y and RW (x , y ) = y 1 y 2 y 3 RW (x , y ): x = v (0) , v (1) , . . . , v (k) = y are still ruled walks in SQ τ , where k is the length of the ruled walks. Furthermore, for any integer j with 0 j k, node u ( j ) and node v ( j ) differ only in the first three bits. If there are two nodes u ( j ) and v ( j ) such that (u ( j ) , v ( j ) ) ∈ E (SQ τ ), then we can construct a ruled walk RW (x, y ): x = u (0) , u (1) , . . . , u ( j −1) , u ( j ) , v ( j ) , v ( j +1) , . . . , v (k) = y. This ruled walks has the length |RW (x, y )| k + 1 τ /3 + 2 + 1 = τ /3 + 3 (see Fig. 4).
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Therefore, the key of the problem is to find a node u ( j ) = x1 x2 x3 z4 z5 · · · zτ which has a neighbor v ( j ) = y 1 y 2 y 3 z4 z5 · · · zτ , where zm ∈ {0, 1} for any integer m with 4 m τ. By Definition 7, there must be nodes of 00-type, 01type, 10-type, and 11-type in a ruled walk. Arbitrarily choose exact one node from nodes of each type in the ruled walk x1 x2 x3 RW (x , y ) and write these nodes of all the four types as u (a00 ) , u (a01 ) , u (a10 ) , and u (a11 ) , respectively, where integers a00 , a01 , a10 , a11 ∈ {1, 2, . . . , k} and u (a pq ) is of pq-type for any p , q ∈ {0, 1}. Based on the result of [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T , the desired adjacent nodes pairs u ( j ) and v ( j ) can be got by the following cases except [x1 , x2 , x3 ] T = [ y 1 , y 2 , y 3 ] T :
Proof. We only need to prove that there exist two nodes x and y in SQ n , such that the length of any path P (x, y ) between x and y satisfies | P (x, y )| n/3 + 3. Let, x = xn xn−1 · · · x3 00 and y = x¯ n x¯ n−1 · · · x¯ 3 00 be two nodes in SQ n . Then any path P (x, y ): x = x(0) , x(1) , . . . , x(k) = y between x and y can be divided into the following three cases.
Case 1. [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T = [1, 0, 0] T , [0, 1, 0] T or [0, 0, 1] T . By definition 4, it is easy to verify that the τ -dimensional, (τ − 1)-dimensional, and (τ − 2)-dimensional neighbors of node v (a00 ) = y 1 y 2 y 3 z4 z5 · · · zτ −2 00 are y¯ 1 y 2 y 3 z4 z5 · · · zτ −2 00, y 1 y¯ 2 y 3 z4 z5 · · · zτ −2 00, and y 1 y 2 y¯ 3 z4 z5 · · · zτ −2 00, respectively. Hence, v (a00 ) must be a neighbor of u (a00 ) .
Case 2. The path P (x, y ) contains at least one node of 01type (or 10-type), and does not contain nodes of 11-type. Because there is at least one node of 01-type (or 10-type) and both x and y are 00-type. There must be two edges in the path, one of which is used for routing from node of 00-type to node of 01-type (or 10-type) and the other of which is used for routing back to node of 00-type. By Proposition 5, for any integers i with 4 i n, the type of x’s i-dimensional neighbor would be the same as x. And, if 1 i 3, x’s i-dimensional neighbor must be different from x in the last three bits. Thus, the two end nodes of the edges between nodes of 00-type and nodes of 01-type (or 10-type) are different from each other only in the last three bits. By Proposition 6, for nodes of 00-type, 01-type, and 10-type, each neighbor of the node differs from that node in at most two bits. Therefore, the length of P (x, y ) is at least (n − 3)/2 + 2. And, when n 14, | P (x, y )| (n − 3)/2 + 2 n/3 + 3.
Case 2. [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T = [1, 1, 0] T . For n 5, 10 T the τ -th column vector of B 10 τ is bτ ,τ = [1, 1, 0, . . . , 0] . ( a10 ) ( a10 ) 10 ( a10 ) Thus v =u ⊕ bτ ,τ . That is, v is a neighbor of u (a10 ) . Case 3. [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T = [1, 0, 1] T . Similar to (a01 ) is a neighbor case 2, v (a01 ) = u (a01 ) ⊕ b01 τ ,τ and thus v of u (a01 ) . Case 4. [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T = [1, 1, 1] T . Similar to (a11 ) is a neighbor case 2, v (a11 ) = u (a11 ) ⊕ b11 τ ,τ and thus v of u (a11 ) . Case 5. [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T = [0, 1, 1] T . For n = 5, 01 T the 4-th column vector of B 01 5 is b 5,4 = [0, 1, 1, 0, 0] . Thus v (a01 ) is a neighbor of u (a01 ) . For n 6, the (τ − 1)-th col10 T umn vector of B 10 τ is bτ ,τ −1 = [0, 1, 1, 0, 0, . . . , 0] . Thus ( a10 ) ( a10 ) v is a neighbor of u .
It is clear that for all the cases except [x1 , x2 , x3 ] T = [ y 1 , y 2 , y 3 ] T , there is a u ( j ) which has a neighbor v ( j ) and thus there is a ruled walk with length |RW (x, y )| τ /3 + 3. The case [x1 , x2 , x3 ] T = [ y 1 , y 2 , y 3 ] T is trivial. Because x1 x2 x3 RW (x , y ) is a ruled walk with length |RW (x, y )| τ /3 + 2 between x and y in SQ n . Therefore, the lemma holds for all the cases. That completes the proof. 2 On the other hand, the following lemma will give a lower bound of the diameter of SQ n for any integer n with n 14. Lemma 9. For any integer n with n 14, diam(SQ n )
n/3 + 3.
Case 1. The path P (x, y ) only contains nodes of 00-type. By Proposition 6, for any integer i with 0 i < k, x(i ) differs from x(i +1) in exact one bit. Since there are n − 2 different bits between x and y, the length of this kind of path P (x, y ) is at least n − 2. When n 14, | P (x, y )| n − 2 > n/3 + 3.
Case 3. The path P (x, y ) contains at least one node of 11type. Similar to the above case, because both x and y are 00-type, there must be edges for routing to and back from nodes of 11-type. But by Proposition 6, nodes of 11-type only have neighbors of 01-type or 10-type. Thus, routing from a node of 00-type to a node of 11-type and back to a node of 00-type requires at least four edges, and these edges will not change the first n − 3 bits. Moreover, for each node on the path, each neighbor of the node differs from that node in at most three bits. Therefore, | P (x, y )| (n − 3)/3 + 4 = n/3 + 3. In summary, we have dist(x, y ) n/3 + 3 for any integer with n 14. 2 By Lemmas 8 and 9, we have: Theorem 10. For any integer n with n 14, diam(SQ n ) = n/3 + 3. 5. Conclusion In this paper, we define a new kind of BC graphs. We proved that the diameter of n-dimensional spined cube is n/3 + 3 for any integer with n 14, which is smaller than any known n-dimensional BC graph when n 18.
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Whether is there a kind of BC graphs whose diameter is less than those of spined cube? This question is awaiting our further investigation in the future. Acknowledgement This research is supported by Natural Science Foundation of China (No. 60873047, 60970117, and 61070169), Natural Science Foundation of Jiangsu Province (No. BK2008154), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20103201110018), and Application Foundation Research of Suzhou of China (No. SYG201034) and sponsored by Qing Lan Project. References [1] S.C. Lee, L.R. Hook IV, Logic and computer design in nanospace, IEEE Transactions on Computers 57 (7) (2008) 965–977. [2] S. Abraham, K. Padmanabhan, The twisted cube topology for multiprocessors: a study in network asymmetry, Journal of Parallel and Distributed Computing 13 (1) (1991) 104–110. [3] K. Efe, A variation on the hypercube with lower diameter, IEEE Trans-
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actions on Computers 40 (11) (1991) 1312–1316. [4] S.M. Larson, P. Cull, The Möbius cubes, IEEE Transactions on Computers 44 (5) (1995) 647–659. [5] J. Fan, L. He, BC interconnection networks and their properties, Chinese Journal of Computers 26 (1) (2003) 84–90. [6] J. Fan, X. Jia, Edge-pancyclicity and path-embeddability of bijective connection graphs, Information Sciences 178 (2) (2008) 340–351. [7] J. Fan, X. Lin, The t /k-diagnosability of the BC graphs, IEEE Transactions on Computers 54 (2) (2005) 176–184. [8] X. Yang, J. Cao, G.M. Megson, J. Luo, Minimum neighborhood in a generalized cube, Information Processing Letters 97 (3) (2006) 88– 93. [9] X. Yang, G.M. Megson, J. Cao, J. Luo, A lower bound on the size of k-neighborhood in generalized cubes, Applied Mathematics and Computation 179 (1) (2006) 47–54. [10] X. Yang, D.J. Evans, G.M. Megson, The locally twisted cubes, International Journal of Computer Mathematics 82 (4) (2005) 401–413. [11] X. Yang, G.M. Megson, D.J. Evans, Locally twisted cubes are 4-pancyclic, Applied Mathematics Letters 17 (8) (2004) 919–925. [12] J. Fan, X. Jia, X. Liu, S. Zhang, J. Yu, Efficient unicast in bijective connection networks with the restricted faulty node set, Information Sciences 181 (11) (2011) 2303–2315. [13] P. Cull, S.M. Larson, A linear equation model for twisted cube networks, in: L.M. Ni (Ed.), ICPADS, IEEE Computer Soc. Press, 1994, pp. 709–715.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
The spined cube: A new hypercube variant with smaller diameter Wujun Zhou a , Jianxi Fan a,∗ , Xiaohua Jia b , Shukui Zhang a a b
School of Computer Science and Technology, Soochow University, Suzhou 215006, China Department of Computer Science, City University of Hong Kong, Hong Kong
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 31 December 2010 Received in revised form 11 March 2011 Accepted 14 March 2011 Available online 17 March 2011 Communicated by A.A. Bertossi Keywords: Interconnection network BC graph Spined cube Diameter
Bijective connection graphs (in brief, BC graphs) are a family of hypercube variants, which contains hypercubes, twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes, etc. It was proved that the smallest diameter of all the known n-dimensional bijective 1 , given a fixed dimension n. An important question connection graphs (BC graphs) is n+ 2 about the smallest diameter among all the BC graphs is: Does there exist a BC graph whose diameter is less than the known BC graphs such as crossed cubes, twisted cubes, Möbius cubes, etc., with the same dimension? This paper answers this question. In this paper, we find that there exists a kind of BC graphs called spined cubes and we prove that the n-dimensional spined cube has the diameter n/3 + 3 for any integer n with n 14. It is the first time in literature that a hypercube variant with such a small diameter is presented. © 2011 Elsevier B.V. All rights reserved.
1. Introduction A multiprocessor interconnection network (in brief, interconnection network) can be represented by a graph G = ( V , E ), where V represents the node set and E represents the edge set, while each node represents a processor and each edge represent a hard link between some two processors. In this paper, we use graphs and interconnection networks interchangeably. Interconnection networks take important roles in parallel computing systems, which decide the performance of them at a large scale. Communication efficiency is a critical metric in a parallel computing systems. And the diameter of an interconnection network is an important metric for communication efficiency. Given an interconnection network (or graph) G and any two nodes u and v, the distance between u and v, denoted as dist(u , v ), is defined as the length of a shortest path between u and v. The diameter of graph G, denoted as diam(G ), is defined as the maximal value of the distances between all the pairs of nodes in G, i.e. diam(G ) = max{dist(u , v )|u , v ∈ V (G )}. Since a
*
Corresponding author. E-mail addresses: [email protected] (W. Zhou), [email protected] (J. Fan), [email protected] (X. Jia), [email protected] (S. Zhang). 0020-0190/$ – see front matter doi:10.1016/j.ipl.2011.03.011
© 2011
Elsevier B.V. All rights reserved.
delay will occur whenever a packet passes through a node, the smaller the diameter, the shorter the delay transferring a packet from the source to the destination under the worst case, the higher the communication efficiency is. As a result, given a fixed number of resources (nodes and edges), whether we can construct an interconnection network with a diameter as small as possible is a significant issue in the interconnection network design. So far, many interconnection networks have been proposed. Among all the interconnection networks, hypercubes have been shown to be popular ones due to their advantageous properties. Recently, Lee et al. proved that the hypercube is appropriate for computer design in nanospace [1]. Hypercubes have a superior nature: the n-dimensional hypercube Q n has diameter n, which is logarithm-level with respect to the node number of Q n . However, by changing some links, various hypercube variants are proposed. Twisted cubes are the first kind of hypercube variants [2]. The n-dimensional twisted cube 1 TQ n has diameter n+ , about a half that of Q n , for any 2 odd positive integer n. Crossed cubes are the second kind of hypercube variants [3]. The n-dimensional crossed cube 1 CQ n has diameter n+ for any positive integer n. Möbius 2 cubes are the third kind of hypercube variants [4]. The n-dimensional Möbius cube MQ n has two types—one is
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0-type and the other is 1-type. The 0-type n-dimensional 2 for any integer n 4 Möbius cube has diameter n+ 2 and the 1-type n-dimensional Möbius cube has diameter n+2 1 for any positive integer n 1. Besides, the expected distance of Möbius cube is lower. Hypercubes and the above three kinds of hypercube variants have two common characters: recursive constructability and bijective connection. That is, for any X n ∈ { Q n , TQ n , CQ n , MQ n }, there are always X n−1 , Y n−1 ∈ { Q n−1 , TQ n−1 , CQ n−1 , MQ n−1 } and a bijection f from V ( X n−1 ) to V (Y n−1 ), such that X n is a graph obtained by connecting each node x in X n−1 with f (x) in Y n−1 . Based on the two characters, Fan and He [5] proposed the definition of n-dimensional bijective connection graphs (in brief, BC graphs). It was proved that hypercubes and the above three kinds of hypercube variants are all BC graphs. It is convenient that once a property was proved on BC graphs, the individual kind of BC graphs will inherit that property. Fan and He [5] studied the general properties of BC graphs. Fan and Jia [6] studied edge-pancyclicity and pathembeddability of a subclass of BC graphs. Besides, Fan and Lin [7] studied t/k-diagnosability of BC graphs. And, Yang et al. [8,9] studied the minimum number of nodes adjacent to a set of nodes in BC graphs. In [10,11], the fourth kind of hypercube variants— locally twisted cubes was proposed. The n-dimensional 1 locally twisted cube LTQ n has diameter n+ for any in2
3 for any positive integer teger n with 1 n 4 and n+ 2 n 5. In fact, locally twisted cubes are also BC graphs. It was proved that the largest diameter of all the n-dimensional BC graphs is n [5]. And, it is easy to find a fact that the smallest diameter of all the above known 1 n-dimensional BC graphs (TQ n , CQ n , MQ n , LTQ n ) is n+ , 2 given a fixed dimension n. An important question about the smallest diameter among all the BC graphs is: Does there exist a BC graph whose diameter is less than the known BC graphs such as crossed cubes, twisted cubes, Möbius cubes, etc., with the same dimension? This paper answers this question. That is, the answer of the question is YES. In detail, we propose a new kind of n-dimensional BC graph called spined cube SQ n and prove that SQ n has the diameter n3 + 3 for any positive integer n with n 14. Obviously, when n is sufficiently big, the diameter of SQ n is about one third of Q n ’s and less than n+2 1 . It is the first time in literature that we find spined cubes have the smallest diameter. The rest of this paper is organized as follows: Section 2 provides the preliminaries. Section 3 gives the definition of spined cubes and several properties of them. Section 4 gives a proof of the diameter of SQ n . In Section 5, we conclude the paper.
2. Preliminaries In this section, we will give some definitions and terminologies used in this paper. Given graph G, let u and v be two nodes in graph G. A walk from u to v is defined as a sequence of nodes W (u , v ): u = u (0) , u (1) , . . . , u (k) = v, such that (u (i ) , u (i +1) ) is an edge of G for any 0 i k − 1. The length of walk W (u , v ) is the number of edges in it, denoted as | W (u , v )|.
A path is a walk without repeated nodes. It is obvious that if there is a walk between nodes u and v, there always exists a path with length no more than | W (u , v )| between u and v. For any integer n 1, a binary string u of length n is denoted by u 1 u 2 · · · un , where u i ∈ {0, 1} for any integer i = {1, 2, . . . , n}. Besides, u 1 is called the first bit of u and un is called the last bit of u. If x = x1 x2 · · · xn is a binary string of length n and s = s1 s2 · · · sm is a binary string of length m for any integer m 1, we use sx to denote the binary string s1 s2 · · · sm x1 x2 · · · xn of length m + n; Furthermore, let U ⊆ {0, 1}n , i.e., U is a set of some binary strings of length n. Then we use sU to denote the set {su |u ∈ U }. If E ⊆ E (G ), we use sE to denote the edges set {(sx, sy)|(x, y ) ∈ E } and use sG to denote (sV , sE ). For the walk W (x, y ): x = u (0) , u (1) , . . . , u (k) = y in G, since for any integer 0 i k − 1, (u (k) , u (k+1) ) ∈ E (G ), we know (su(k) , su(k+1) ) ∈ E (sG ). Thus sx = su(0) , su(1) , . . . , su(k) = sy is a walk in sG. We use sW (x, y ) to denote this walk. Given two graphs G and G , if there exists a bijection ψ from V (G ) to V (G ) such that (u , v ) ∈ E (G ) if and only if (ψ(u ), ψ( v )) ∈ E (G ) for any two nodes u , v ∈ V (G ), then we say that G is isomorphic to G and ψ is an isomorphic mapping from G to G . If the graphs G and G are two isomorphic graphs, we write G ∼ = G . The isomorphic graphs can be regarded as the identical graph. Before introducing the definition of BC graphs, we first give the definition of bijective connection in the following [5]: Definition 1. Let G be a graph. If V (G ) = V 1 ∪ V 2 , V 1 = φ , V 2 = φ , and V 1 ∩ V 2 = φ . We say that there exists a bijective connection between the subsets V 1 and V 2 in G, G
denoted by V 1 ←→ V 2 , if G satisfies the two following conditions: (1) For every u ∈ V 1 , there exists a unique v ∈ V 2 such that {u , v } ∈ E (G ); and (2) For every u ∈ V 2 , there exists a unique v ∈ V 1 such that {u , v } ∈ E (G ). The definition of bijective connection graphs with labels in their nodes was given in [12]. In this paper, we use that definition. An n-dimensional bijective connection graph (in brief, BC graph), denoted by X n , is an n-regular graph with 2n nodes. We identify each node of X n by a unique binary string of length n. The set of all the n-dimensional BC graphs is called the family of the n-dimensional BC graphs, denoted by Ln . Xn and Ln may be recursively defined as below [12]. Definition 2. The 1-dimensional BC graph X 1 is a complete graph on two nodes 0 and 1. The family of the 1-dimensional BC graph is defined as L1 = { X 1 }. Let G be a graph. G is an n-dimensional BC graph, denoted by X n , if there exist V 0 , V 1 ⊂ V (G ) such that the following three conditions hold: (1) V 0 = 0V 0 and V 1 = 0V 1 , where V 0 = V 1 = {0, 1}n−1 ; (2) V (G ) = V 0 ∪ V 1 , V 0 = ∅, V 1 = ∅, and V 0 ∩ V 1 = ∅; and
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Definition 4. For a given integer n with n 2 and any ij ij ij i , j ∈ {0, 1}, the adjacent-decided matrix B n = (bn,n , bn,n−1 , ij
. . . , bn,1 ) for SQ n can be recursively defined as follows: 1. For n = 2, let
01 10 11 B 00 2 = B2 = B2 = B2 = Fig. 1. The 2-dimensional spined cube SQ 2 and 3-dimensional spined cube SQ 3 .
2. For n = 3 and n = 4, let
G
The family of the n-dimensional BC graphs is defined as Ln = {G |G is an n-dimensional BC graph}.
⎤
⎣. .
⎦
1
.
0
⎤
1 0 ··· 0 ⎥ ⎢1 ⎥ ⎢ B n01 = B n11 = ⎢ 0. B 01 ⎥ . n −1 ⎦ ⎣.
.
0 3. For n > 4, let
⎤
⎡
1 0 ··· 0 ⎥ ⎢0 ⎥ ⎢0 ⎥ ⎢ 00 Bn = ⎢ ⎥, ⎢ 0. B n00−1 ⎥ ⎦ ⎣.
.
Definition 3. Let n 1 be an integer. The n-dimensional spined cube, SQ n , is defined recursively as follows:
Fig. 1 demonstrates SQ 2 and SQ 3 . SQ n is obviously an n-regular graph with 2n nodes and n−1 n2 edges, and the labels of any two adjacent nodes of SQ n differ in at most three successive bits. For convenience of presentation, we use node u and the label of u interchangeably. Clearly, 0SQ n−1 ∼ = 1SQ n−1 ∼ = SQ n−1 . Hence, we say that SQ n consists of two (n − 1)-dimensional subspined cubes. In [13], Larson et al. provided a methodology to denote a hypercube variants using matrix. In this paper, for convenience to discussion, we give an equivalent definition of spined cubes using a matrix representation, by the similar method to [13]. We call such a matrix an adjacent-decided matrix. For any integer n 2 and a node x = x1 x2 · · · xn ∈ V (SQ n ). We use an n-dimensional column vector [x1 , x2 , . . . , xn ] T to denote the node x.
⎡
⎡
3. The definition of spined cubes
(1) The 1-dimensional spined cube is a complete graph on two nodes 0 and 1. (2) SQ 2 is a graph consisting of four nodes labeled with 00, 01, 10, and 11, connected by four edges (00, 01), (00, 10), (01, 11), and (10, 11). (3) For n = 3 and n = 4, V (SQ n ) = V (0SQ n−1 )∪ V (1SQ n−1 ), E = E (0SQ n−1 )∪ E (1SQ n−1 )∪ E , where E = {(0x2 x3 · · · xn , 1(x2 ⊕ xn )x3 · · · xn )|xi ∈ {0, 1}, i = 2, 3, . . . , n}. (4) For n > 4, V (SQ n ) = V (0SQ n−1 ) ∪ V (1SQ n−1 ), E = E (0SQ n−1 ) ∪ E (1SQ n−1 ) ∪ E , where E = {(0x2 x3 · · · xn , 1(x2 ⊕ xn−1 )(x3 ⊕ xn )x4 · · · xn )|xi ∈ {0, 1}, i = 2, 3, . . . , n}.
0
1 0 ··· 0 ⎢0 ⎥ ⎢ ⎥ B n00 = B n10 = ⎢ 0. B 00 ⎥ , n −1
(3) V 0 ←→ V 1 , G [ V 0 ] ∈ Ln−1 , and G [ V 1 ] ∈ Ln−1 .
In this section, we will define a new kind of BC graphs called spined cubes and give some properties of them. We first introduce the ⊕ operation. We use ⊕ to denote (modulo 2) sum of two bits a and b (a, b ∈ {0, 1}). And for vectors, letting C = [c 1 , c 2 , . . . , cn ] T and D = [d1 , d2 , . . . , dn ] T be two column vectors, we use C ⊕ D to denote [c 1 ⊕ d1 , c 2 ⊕ d2 , . . . , cn ⊕ dn ] T .
1 0
⎤
⎡
1 0 ··· 0 ⎥ ⎢0 ⎥ ⎢1 ⎥ ⎢ 01 Bn = ⎢ 0 ⎥, ⎢ . B n01−1 ⎥ ⎦ ⎣.
.
0
0
⎤
⎡
1 0 ··· 0 ⎥ ⎢1 ⎥ ⎢0 ⎥ ⎢ B n10 = ⎢ 0 ⎥, ⎢ . B n10−1 ⎥ ⎦ ⎣.
.
0
⎤
⎡
1 0 ··· 0 ⎥ ⎢1 ⎥ ⎢1 ⎥ ⎢ B n11 = ⎢ 0 ⎥. ⎢ . B n11−1 ⎥ ⎦ ⎣.
.
0
The adjacent-decided matrix denotes the neighbor relaij tionship of a node. Let bn,k denote the k-th column vector ij
of B n , 1 k n. Then, for any x = x1 x2 · · · xn ∈ V (SQ n ), x x y = [x1 , x2 , . . . , xn ] T ⊕ bnn,k−1 n is a neighbor of x. We call y the k-dimensional neighbor of x. If we say the node x = x1 x2 x3 · · · xn−1 xn to be of xn−1 xn type. We can divide the nodes of SQ n into four types: 00type, 01-type, 10-type, and 11-type. For example, for n = 5 and a node x = 01011. Because the last two bits xn−1 xn of 01011 is 11, it is a node of 11-type, and B 11 5 represents the neighbor relationship of 11-type nodes. By definition 4, we have
⎡
B 11 5
1
0
0
0
0
⎤
⎢1 1 0 0 0⎥ ⎥ ⎢ ⎥ ⎢ = ⎢1 1 1 0 0⎥. ⎥ ⎢ ⎣0 0 1 1 0⎦ 0
0
0
0
1
Then, the 1-dimensional and 5-dimensional neighbors of node 01011 are respectively
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⎡ ⎤ 0
⎡ ⎤ 0
⎡ ⎤ 0
⎢1⎥ ⎢0⎥ ⎢1⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T 11 [0, 1, 0, 1, 1] ⊕ b5,1 = ⎢ 0 ⎥ ⊕ ⎢ 0 ⎥ = ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣1⎦ ⎣0⎦ ⎣1⎦ 1
⎡ ⎤ 0
1
⎡ ⎤ 1
and
0
⎡ ⎤ 1
⎢1⎥ ⎢1⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T 11 [0, 1, 0, 1, 1] ⊕ b5,5 = ⎢ 0 ⎥ ⊕ ⎢ 1 ⎥ = ⎢ 1 ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣1⎦ ⎣0⎦ ⎣1⎦ 1
0
1
That is, the 1-dimensional and 5-dimensional neighbors of node 01011 are nodes 01010 and 10111, respectively. Definition 4 is obviously equivalent to Definition 3. From Definition 4, we can easily verify the following two properties for any integers i , j ∈ {0, 1}. Proposition 5. For any integers k and n with n 2 and any node x ∈ V (SQ n ), if k 4, then the last two components of ij ij bn,k are always 0, and moreover, for 1 k 3, bn,k have 1’s only in the last three components, i.e. if k 4 the type of the k-dimensional neighbor of x would be the same as x and for 1 k 3, the k-dimensional neighbor of x differs from x only in the last three bits. ij
Proposition 6. The properties of each B n are listed as following: 1. B n00 : each column vector of B n00 contains exact one component of 1 and the other n − 1 components are 0, i.e. each neighbor of a 00-type node differs from that 00-type node in exact one bit. 2. B n01 and B n10 : each column vector of B n01 (or B n10 ) contains at most two components of 1, i.e. each neighbor of a 01type (or 10-type) node differs from that node in at most two bits. 3. B n11 : each column vector of B n11 contains at most three components of 1. Besides, there is not any column with the last two components both 1, i.e. each neighbor of a 11-type node differs from that node in at most three bits and the neighbor node cannot be a node of 00-type. By Definition 4, it is easy to verify these properties. For example, for i j = 11, By Definition 4, we have:
⎡1 ··· ⎢1 ··· ⎢ ⎢1 ··· ⎢ ⎢. ⎢ . .. . ⎢. ⎢ ⎢ 0 ··· B n11 = ⎢ ⎢ ⎢0 ··· ⎢ ⎢0 ··· ⎢ ⎢ ⎢0 ··· ⎢ ⎣0 ··· 0
0
0
0
0
0
0⎤
0
0
0
0
0
0⎥
0
0
0
0
0
.. .
.. .
.. .
.. .
.. .
1
0
0
0
0
1
1
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
0
1
1
⎥
0⎥ ⎥
.. ⎥ ⎥ .⎥ ⎥ 0⎥ ⎥. ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎦
··· 0 0 0 0 0 1
For any integers k, if k 4, the last two components of bn11,k are 00, and moreover, for 1 k 3, bn11,k have 1’s only in the last three components. And it is obvious that each column vector of B n11 contains at most three components of 1. Besides, there is not any column vector with the last two components both 1. 4. The diameter of spined cube In this section, we will study the diameter of spined cubes. We first define ruled walks. Definition 7. If a walk in a spined cube contains the nodes of 00-type, 01-type, 10-type and 11-type, then we call this walk a ruled walk. The following lemma determines the upper bound of the diameter of SQ n for n 2. Lemma 8. For any integer n with n 2, diam(SQ n ) n/3 + 3. Proof. We prove this lemma by proving that there always exists a ruled walk between any two nodes x and y in V (SQ n ) with length at most n/3+ 3 and thus dist(x, y ) n/3 + 3. For any x = x1 x2 · · · xn , y = y 1 y 2 · · · yn ∈ V (SQ n ), the desired ruled walk can be got by induction on n. Base step: For n = 2, we can enumerate ruled walks between any two nodes as follows:
RW (00, 00): RW (00, 01): RW (00, 10): RW (00, 11): RW (01, 01): RW (01, 10): RW (01, 11): RW (10, 10): RW (10, 11): RW (11, 11):
00, 01, 11, 10, 00 00, 10, 11, 01 00, 01, 11, 10 00, 01, 11, 10, 11 01, 11, 10, 00, 01 01, 11, 10, 00, 10 01, 00, 10, 11 10, 00, 01, 11, 10 10, 00, 01, 11 11, 10, 00, 01, 11
The rest can be got by reversing the above ruled walks. The length of these ruled walks is at most 4, which satisfies 2/3 + 3. That is, the lemma holds for n = 2. Fig. 2 illustrates ruled walks starting from node 00. For n = 3, if x1 = y 1 and RW (x2 x3 , y 2 y 3 ) is a ruled walk of n = 2, then x1 RW (x2 x3 , y 2 y 3 ) is a ruled walk from node x1 x2 x3 to node y 1 y 2 y 3 . For instance, if RW (00, 00): 00, 01, 11, 10, 00 is a ruled walk in SQ 2 , then 1RW (00, 00): 100, 101, 111, 110, 100 is a ruled work in SQ 3 . Thus, if x1 = y 1 , there is a ruled walk between nodes x and y with length at most 4. Otherwise, x1 = y¯ 1 . We can enumerate ruled walks between 0x2 x3 and 1 y 2 y 3 as follows:
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Fig. 2. Ruled walks staring from node 00 in SQ 2 , where a bold line segment represents an edge in a ruled walk and the number beside an edge represents the order that a ruled walk passed through this edge.
RW (000, 100): RW (000, 101): RW (000, 110): RW (000, 111): RW (001, 100): RW (001, 101): RW (001, 110): RW (001, 111): RW (010, 100): RW (010, 101): RW (010, 110): RW (010, 111): RW (011, 100): RW (011, 101): RW (011, 110): RW (011, 111):
000, 001, 111, 110, 100 000, 010, 011, 101 000, 001, 111, 110 000, 010, 011, 101, 111 001, 011, 010, 110, 100 001, 000, 010, 011, 101 001, 011, 101, 100, 110 001, 000, 010, 110, 111 010, 011, 101, 100 010, 000, 001, 011, 101 010, 011, 101, 100, 110 010, 000, 100, 101, 111 011, 101, 100, 110, 100 011, 010, 000, 100, 101 011, 001, 000, 010, 110 011, 010, 000, 001, 111
We only enumerate x1 = 0 and y 1 = 1, the case of x1 = 1 and y 1 = 0 can be got by reversing the above ruled walks. In this case, the length of the ruled walks is still at most 4 3/3 + 3. Therefore, the lemma holds for n = 3. For n = 4 and any two nodes x1 x2 x3 x4 and y 1 y 2 y 3 y 4 , if x1 = y 1 , then the ruled walks can be derived similar to the case for n = 3. The length of ruled walks is still at most 4. Otherwise, x1 = y¯ 1 . Let z1 z2 z3 z4 be the 4-dimensional neighbor of x1 x2 x3 x4 . By Definition 4, z1 = x¯1 . Thus, z1 = y 1 . As a result, there is a ruled walk W between z1 z2 z3 z4 and y 1 y 2 y 3 y 4 with length at most 4. Therefore, the walk x1 x2 x3 x4 , W is a ruled walk of length at most 5 4/3 + 3. Thus, the lemma holds for n = 4. In summary, we have proven the lemma for n = 2, n = 3, and n = 4. Induction step: Suppose that for any integer n with τ − 3 n τ − 1, there exists a ruled walk with length at most n/3 + 3 between any two nodes in SQ n (τ 5). In the following, we will prove that the lemma holds for n = τ. By definition 3, the SQ τ is composed of two (τ − 1)dimensional sub-spined cubes 0SQ τ −1 and 1SQ τ −1 . Furthermore, for i ∈ {0, 1}, iSQ τ −1 is composed of two (τ − 2)-dimensional sub-spined cubes i0SQ τ −2 and i1SQ τ −2 . Moreover, for j ∈ {0, 1}, i jSQ τ −2 is composed of two (τ − 3)-dimensional sub-spined cubes i j0SQ τ −3 and i j1SQ τ −3 . Thus, SQ τ can be divided into eight disjoint (τ − 3)dimensional sub-spined cubes and the eight sub-spined cubes are isomorphic to one another and all isomorphic to SQ n−3 (see Fig. 3). That is V (SQ n ) = i jk∈{0,1}3 {i jk ×
V (SQ n−3 )}, E (SQ n ) = i jk∈{0,1}3 {i jkE (SQ n−3 )} ∪ E , where E is edges between i jkV (SQ n−3 ).
Fig. 3. Sub-spined cube division of SQ τ .
Fig. 4. The constructing process of a ruled walk.
According to the induction hypothesis, there exists a ruled walk between any two nodes with length at most (τ − 3)/3 + 3 = τ /3 + 2 in (τ − 3)-dimensional SQ τ −3 . Then, for any two nodes x = x4 x5 · · · xτ and y = y 4 y 5 · · · y τ in SQ τ −3 , there exists a ruled walk RW (x , y ) with length at most τ /3 + 2 in SQ τ −3 . Let y = x1 x2 x3 y 4 y 5 · · · y τ and x = y 1 y 2 y 3 x4 x5 · · · xτ . Because for any i jk ∈ {0, 1}3 , i jkE (SQ n−3 ) ⊂ E (SQ n ), RW (x, y ) = x1 x2 x3 RW (x , y ): x = u (0) , u (1) , . . . , u (k) = y and RW (x , y ) = y 1 y 2 y 3 RW (x , y ): x = v (0) , v (1) , . . . , v (k) = y are still ruled walks in SQ τ , where k is the length of the ruled walks. Furthermore, for any integer j with 0 j k, node u ( j ) and node v ( j ) differ only in the first three bits. If there are two nodes u ( j ) and v ( j ) such that (u ( j ) , v ( j ) ) ∈ E (SQ τ ), then we can construct a ruled walk RW (x, y ): x = u (0) , u (1) , . . . , u ( j −1) , u ( j ) , v ( j ) , v ( j +1) , . . . , v (k) = y. This ruled walks has the length |RW (x, y )| k + 1 τ /3 + 2 + 1 = τ /3 + 3 (see Fig. 4).
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Therefore, the key of the problem is to find a node u ( j ) = x1 x2 x3 z4 z5 · · · zτ which has a neighbor v ( j ) = y 1 y 2 y 3 z4 z5 · · · zτ , where zm ∈ {0, 1} for any integer m with 4 m τ. By Definition 7, there must be nodes of 00-type, 01type, 10-type, and 11-type in a ruled walk. Arbitrarily choose exact one node from nodes of each type in the ruled walk x1 x2 x3 RW (x , y ) and write these nodes of all the four types as u (a00 ) , u (a01 ) , u (a10 ) , and u (a11 ) , respectively, where integers a00 , a01 , a10 , a11 ∈ {1, 2, . . . , k} and u (a pq ) is of pq-type for any p , q ∈ {0, 1}. Based on the result of [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T , the desired adjacent nodes pairs u ( j ) and v ( j ) can be got by the following cases except [x1 , x2 , x3 ] T = [ y 1 , y 2 , y 3 ] T :
Proof. We only need to prove that there exist two nodes x and y in SQ n , such that the length of any path P (x, y ) between x and y satisfies | P (x, y )| n/3 + 3. Let, x = xn xn−1 · · · x3 00 and y = x¯ n x¯ n−1 · · · x¯ 3 00 be two nodes in SQ n . Then any path P (x, y ): x = x(0) , x(1) , . . . , x(k) = y between x and y can be divided into the following three cases.
Case 1. [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T = [1, 0, 0] T , [0, 1, 0] T or [0, 0, 1] T . By definition 4, it is easy to verify that the τ -dimensional, (τ − 1)-dimensional, and (τ − 2)-dimensional neighbors of node v (a00 ) = y 1 y 2 y 3 z4 z5 · · · zτ −2 00 are y¯ 1 y 2 y 3 z4 z5 · · · zτ −2 00, y 1 y¯ 2 y 3 z4 z5 · · · zτ −2 00, and y 1 y 2 y¯ 3 z4 z5 · · · zτ −2 00, respectively. Hence, v (a00 ) must be a neighbor of u (a00 ) .
Case 2. The path P (x, y ) contains at least one node of 01type (or 10-type), and does not contain nodes of 11-type. Because there is at least one node of 01-type (or 10-type) and both x and y are 00-type. There must be two edges in the path, one of which is used for routing from node of 00-type to node of 01-type (or 10-type) and the other of which is used for routing back to node of 00-type. By Proposition 5, for any integers i with 4 i n, the type of x’s i-dimensional neighbor would be the same as x. And, if 1 i 3, x’s i-dimensional neighbor must be different from x in the last three bits. Thus, the two end nodes of the edges between nodes of 00-type and nodes of 01-type (or 10-type) are different from each other only in the last three bits. By Proposition 6, for nodes of 00-type, 01-type, and 10-type, each neighbor of the node differs from that node in at most two bits. Therefore, the length of P (x, y ) is at least (n − 3)/2 + 2. And, when n 14, | P (x, y )| (n − 3)/2 + 2 n/3 + 3.
Case 2. [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T = [1, 1, 0] T . For n 5, 10 T the τ -th column vector of B 10 τ is bτ ,τ = [1, 1, 0, . . . , 0] . ( a10 ) ( a10 ) 10 ( a10 ) Thus v =u ⊕ bτ ,τ . That is, v is a neighbor of u (a10 ) . Case 3. [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T = [1, 0, 1] T . Similar to (a01 ) is a neighbor case 2, v (a01 ) = u (a01 ) ⊕ b01 τ ,τ and thus v of u (a01 ) . Case 4. [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T = [1, 1, 1] T . Similar to (a11 ) is a neighbor case 2, v (a11 ) = u (a11 ) ⊕ b11 τ ,τ and thus v of u (a11 ) . Case 5. [x1 , x2 , x3 ] T ⊕ [ y 1 , y 2 , y 3 ] T = [0, 1, 1] T . For n = 5, 01 T the 4-th column vector of B 01 5 is b 5,4 = [0, 1, 1, 0, 0] . Thus v (a01 ) is a neighbor of u (a01 ) . For n 6, the (τ − 1)-th col10 T umn vector of B 10 τ is bτ ,τ −1 = [0, 1, 1, 0, 0, . . . , 0] . Thus ( a10 ) ( a10 ) v is a neighbor of u .
It is clear that for all the cases except [x1 , x2 , x3 ] T = [ y 1 , y 2 , y 3 ] T , there is a u ( j ) which has a neighbor v ( j ) and thus there is a ruled walk with length |RW (x, y )| τ /3 + 3. The case [x1 , x2 , x3 ] T = [ y 1 , y 2 , y 3 ] T is trivial. Because x1 x2 x3 RW (x , y ) is a ruled walk with length |RW (x, y )| τ /3 + 2 between x and y in SQ n . Therefore, the lemma holds for all the cases. That completes the proof. 2 On the other hand, the following lemma will give a lower bound of the diameter of SQ n for any integer n with n 14. Lemma 9. For any integer n with n 14, diam(SQ n )
n/3 + 3.
Case 1. The path P (x, y ) only contains nodes of 00-type. By Proposition 6, for any integer i with 0 i < k, x(i ) differs from x(i +1) in exact one bit. Since there are n − 2 different bits between x and y, the length of this kind of path P (x, y ) is at least n − 2. When n 14, | P (x, y )| n − 2 > n/3 + 3.
Case 3. The path P (x, y ) contains at least one node of 11type. Similar to the above case, because both x and y are 00-type, there must be edges for routing to and back from nodes of 11-type. But by Proposition 6, nodes of 11-type only have neighbors of 01-type or 10-type. Thus, routing from a node of 00-type to a node of 11-type and back to a node of 00-type requires at least four edges, and these edges will not change the first n − 3 bits. Moreover, for each node on the path, each neighbor of the node differs from that node in at most three bits. Therefore, | P (x, y )| (n − 3)/3 + 4 = n/3 + 3. In summary, we have dist(x, y ) n/3 + 3 for any integer with n 14. 2 By Lemmas 8 and 9, we have: Theorem 10. For any integer n with n 14, diam(SQ n ) = n/3 + 3. 5. Conclusion In this paper, we define a new kind of BC graphs. We proved that the diameter of n-dimensional spined cube is n/3 + 3 for any integer with n 14, which is smaller than any known n-dimensional BC graph when n 18.
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Whether is there a kind of BC graphs whose diameter is less than those of spined cube? This question is awaiting our further investigation in the future. Acknowledgement This research is supported by Natural Science Foundation of China (No. 60873047, 60970117, and 61070169), Natural Science Foundation of Jiangsu Province (No. BK2008154), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20103201110018), and Application Foundation Research of Suzhou of China (No. SYG201034) and sponsored by Qing Lan Project. References [1] S.C. Lee, L.R. Hook IV, Logic and computer design in nanospace, IEEE Transactions on Computers 57 (7) (2008) 965–977. [2] S. Abraham, K. Padmanabhan, The twisted cube topology for multiprocessors: a study in network asymmetry, Journal of Parallel and Distributed Computing 13 (1) (1991) 104–110. [3] K. Efe, A variation on the hypercube with lower diameter, IEEE Trans-
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