Pancyclicity of M ö bius cubes with faulty nodes

1. 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

2. Outline • Introduction • Notations and terminologies • Some interesting properties of small-sized Möbius cubes • Main result • Summary

3. Introduction • The Möbius cube MQnis a variant of the hypercube Qnand has better properties than Qnwith 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≦2n.

4. Introduction • A vertex X = xn-1xn-2 ···x0 ,xn ∈ {0,1}, connects to n neighbors Y1,Y2, . . . , Yn, where each Yisatisfies one of the following rules: Yi= xn-1xn-2 ···x0if xn = 0, (1) Yi= xn-1xn-2 ···x0 if xn = 1, (2) If we assume xn = 0, we call the network a “0-Möbius cube”, denoted by MQn0; and if we assume xn =1, we call the network a “1-Möbius cube”, denoted by MQn1.

5. X=000 Y1= x2x1x0=100,Y2=x2x1x0=010,Y3=x2x1x0=001 if x0 = 0 ﹣ ﹣ ﹣ 000 001 000 001 111 110 100 101 101 100 110 111 010 010 011 011 MQ31 MQ30

8. 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

9. Notations and terminologies • Property 2.1. [2]. For n≧1, MQ0n(resp. MQ1n) can be recursively constructed by adding a perfect matching between the nodes of 0MQ0n－1 and the nodes of 1MQ1n－1. • Property 2.2. Let (u, v) be a 0-edge of MQ0n(resp. MQ1n). Let u’ and v’ be the respective k-neighbors of u and v. Then (u’, v’) is also a 0-edge of MQ0n(resp. MQ1n).

10. Notations and terminologies • Property 2.3. [4]. For n≧2, MQn is four-pancyclic. For n≧3, MQn is Hamiltonian connected. • Property 2.4. [7] For n≧2, MQn is (n-2)-hybrid-fault-tolerant Hamiltonian. For n≧3, MQn is (n-3) -hybrid-fault-tolerant Hamiltonian connected.

11. Notations and terminologies • Property 2.5. For n≧2, MQn is (n-1)-hybrid-fault-tolerant Hamiltonian-path. • Property 2.6. [6]. For n≧2, MQn is (n-2)-edge-fault-tolerant four-pancyclic.

12. Some interesting properties of small-sized Möbius cubes • Lemma 3.1. MQ3 is 1-node-fault-tolerant four-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 withinMQ03.

14. Some interesting properties of small-sized Möbius cubes • Lemma 3.2. MQ4 is 2-node-fault-tolerant four-pancyclic.

15. 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.

17. 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, then 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}.

18. Proof. Again we prove the lemma only for MQ03. 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.

20. Some interesting properties of small-sized Möbius cubes • Lemma 3.3. Suppose there are two or three faulty nodes within MQ5. If either 0MQ4 or 1MQ4 contains a single faulty node, then MQ5 contains a fault-free cycle of length 16.

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

26. Main result • Theorem 4.1. For n≧2, MQn is (n-2)-node-fault-tolerant 4-pancyclic.

27. 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 there are at most (m-2) faulty nodes within MQm. Let F be the set of all the faulty nodes of MQm. Further, let

28. Case 1. 4≦l≦2m-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.

29. Main result • Case 2. 2m-1+2-f1≦l≦2m-f • Case 3. l=2m-1+1-f1. Clearly, 0MQm-1 contains 2m-2 0-edges. Since 2m-2＞ m-2≧f, it follows that 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.

30. 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.