Pancyclicity of Möbius cubes with faulty nodes

Xiaofan Yang , Graham M. Megson , David J. Evans Microprocessors and Microsystems 30 (2006) 165–172

指導老師 : 洪春男 教授報告學生 : 林雨淳

Outline

Introduction Notations and terminologies Some interesting properties of small-sized M

öbius cubes Main result Summary

Introduction

The Möbius cube MQn is a variant of the hypercube Qn and has better properties than Qn with the same number of links and processors.

An interconnection network with n nodes is four-pancyclic if it contains a cycle of length l for each integer l with 4 l 2≦ ≦ n.

Introduction

A vertex X = xn-1xn-2 ···x0 ,xn {0,1}, connects ∈to n neighbors Y1,Y2, . . . , Yn, where each Yi satisfies one of the following rules:

Yi = xn-1 xn-2 ···x0 if xn = 0, (1)

Yi = xn-1 xn-2 ···x0 if xn = 1, (2)

If we assume xn = 0, we call the network a “0-Möbius cube”, denoted by MQn

0 ; and if we assume xn =1, we call the network a “1-Möbius cube”, denoted by MQn

1.

000

101100

110 111

011010

001

MQ30

000

110111

101 100

011010

001

MQ31

X=000

Y1 = x2x1x0=100,Y2=x2x1x0=010,Y3=x2x1x0=001 if x0 = 0﹣ ﹣﹣

Introduction

In this paper, we show that an n-dimensional Möbius cube is four-pancyclic in the presence of up to n-2 faulty nodes.

The obtained result is optimal in that, if n-1 nodes are removed, the surviving network may not be four-pancyclic

Notations and terminologies

Property 2.1. [2]. For n 1, MQ≧ 0n(resp. MQ1

n) can be recursively constructed by adding a perfect matching between the nodes of 0MQ0

n － 1 and the nodes of 1MQ1

n － 1.

Property 2.2. Let (u, v) be a 0-edge of MQ0n(re

sp. MQ1n). Let u’ and v’ be the respective k-nei

ghbors of u and v. Then (u’, v’) is also a 0-edge of MQ0

n(resp. MQ1n).

Notations and terminologies

Property 2.3. [4]. For n 2, MQ≧ n is four-pancyclic. For n 3, MQ≧ n is Hamiltonian connected.

Property 2.4. [7] For n 2, MQ≧ n is (n-2)-hybrid-fault-tolerant Hamiltonian. For n 3, MQ≧ n is (n-3) -hybrid-fault-tolerant Hamiltonian connected.

Notations and terminologies

Property 2.5. For n 2, MQ≧ n is (n-1)-hybrid-fault-tolerant Hamiltonian-path.

Property 2.6. [6]. For n 2, MQn is (n-2)-edg≧e-fault-tolerant four-pancyclic.

Some interesting properties of small-sized Möbius cubes Lemma 3.1. MQ3 is 1-node-fault-tolerant fou

r-pancyclic. Proof of Lemma 3.1: we can assume that the

node 000 is faulty and all the remaining nodes are fault-free. For each integer l with 4 1≦ ≦7, Table 1 gives a fault-free cycle of length l withinMQ0

3.

Some interesting properties of small-sized Möbius cubes Lemma 3.2. MQ4 is 2-node-fault-tolerant fou

r-pancyclic.

Lemma 3.5. MQ3 is sub-Hamiltonian connected. Moreover, if (u,v) is a 0-edge of MQ3, then there is a sub-Hamiltonian path P[u,v] that contains two 0-edges.

Proof. As before, we prove the lemma only for MQ03.

In view of the symmetry of MQ03 , we may focus our

attention on the four pairs of nodes: <000,100>, <000, 111>, <000, 011>, <000, 001>. The desired sub-Hamiltonian paths are given in Table 2.

Lemma 3.6. Suppose there is a single faulty node within MQ3. Let (u, v) be a 0-edge with u and v being fault-free. (i) If either u or v is adjacent to the faulty node, the

n MQ3 contains a fault-free path P[u, v] of length l for each integer l{3, 4, 5, 6}.

(ii) If neither u nor v is adjacent to the faulty node, then MQ3 contains a fault-free path P[u, v] of length l for each integer l{3, 4, 6}.

Proof. Again we prove the lemma only for MQ0

3. In view of the symmetry of MQ03 , we may

assume that 000 is the faulty node and (u, v) {(100, 101), (111, 110)}. The required paths are given in Table 3.

Some interesting properties of small-sized Möbius cubes Lemma 3.3. Suppose there are two or three f

aulty nodes within MQ5. If either 0MQ4 or 1MQ4 contains a single faulty node, then MQ5 contains a fault-free cycle of length 16.

Some interesting properties of small-sized Möbius cubes Lemma 3.4. Suppose there are exactly four f

aulty nodes within MQ6 in such a way that there are exactly two faulty nodes within 0MQ5, then MQ6 contains a fault-free cycle of length 31.

Main result

Theorem 4.1. For n 2, MQ≧ n is (n-2)-node-fault-tolerant 4-pancyclic.

Main result

Proof. We argue by induction on n. The theorem is trivial for n 2. In the case n≧ {3, 4}, the correctness of the theorem is ensured by Lemma 3.1 and Lemma 3.2. Suppose the theorem holds for n=m-1 (m 5). Now assume ther≧e are at most (m-2) faulty nodes within MQm. Let F be the set of all the faulty nodes of MQm. Further, let

Case 1. 4 l 2≦ ≦ m-1-f1. Note that f1 (m-2)/2≦ ≦m-3, it follows from the inductive hypothesis that 1MQm-1 and, hence, MQm contains a fault-free cycle of length l.

Main result

Case 2. 2m-1+2-f1 l 2≦ ≦ m-f

Case 3. l=2m-1+1-f1. Clearly, 0MQm-1 contains 2m-2 0-edges. Since 2m-2 ＞ m-2 f, it follows t≧hat 0MQm-1 contains a 0-edge (u, v) so that u and v are fault-free, and u’ and v’ (the respective (m-1)-neighbors of u and v) are fault-free.

Summary

Our result can be viewed as a supplement of a result in [6], which states that an n-dimensional Möbius cube with up to n-2 faulty edges is four-pancyclic.

In view of that hypercube networks are not four-pancyclic, Möbius cubes are superior to hypercubes in terms of the pancyclicity and fault-tolerant pancyclicity.