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Some Interesting Properties of Interconnection Networks

This paper explores fault-tolerant Hamiltonicity in various interconnection networks such as twisted-cubes, crossed-cubes, and Möbius cubes. It includes definitions of these networks and their fault-tolerant properties. The paper also presents optimal fault-tolerant designs for meshes and ladders.

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Some Interesting Properties of Interconnection Networks

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  1. Some Interesting Properties of Interconnection Networks Lih-Hsing Hsu

  2. R.S. Chou and L.H. Hsu (1994), "1-Edge Fault Tolerant Design for Meshes," Parallel Processing Letters, Vol. 4, pp. 385-389.

  3. Y.C. Chuang, C.H. Chang, and L.H. Hsu (2002), "Optimal 1-Edge Fault-Tolerant Designs for Ladders," Information Processing Letters, Vol. 84, pp. 87-92.

  4. Good Graph Theorist Good-Graph Theorist

  5. A k-regular Hamiltonian and Hamiltonian connected graph G is super fault-tolerant Hamiltonian if G remains Hamiltonian after removing at most k-2 nodes and/or edges and remains Hamiltonian connected after removing at most k-3 nodes and/or edges.

  6. Introduction • The fault-tolerant Hamiltonicity, Hf(G), is defined to be the maximum integer l such that G-F remains Hamiltonian for every F  V(G)  E(G) with |F| l if G is Hamiltonian, and undefined if otherwise. • Obviously, Hf(G) (G)-2, where (G) = min{deg(v) | v  V(G)}. • A regular graph G is optimal fault-tolerant Hamiltonian if Hf(G) = (G)-2.

  7. Introduction • The fault-tolerant Hamiltonian connectivity, Hkf(G), is defined to be the maximum integer l such that G-F remains Hamiltonian connected for every F  V(G)  E(G) with |F| l if G is Hamiltonian connected, and undefined if otherwise. • Obviously, Hf(G) (G)-3, where (G) = min{deg(v) | v V(G)}. • A regular graph G is optimal fault-tolerant Hamiltonian connected if Hf(G) = (G)-3.

  8. Preliminaries Preliminaries • The hypercube is a popular network because of its attractive properties, including regularity, symmetry, small diameter, strong connectivity, recursive construction, partitionability, and relatively low link complexity.

  9. Preliminaries 000 010 001 011 110 100 101 111 Hypercube Q3 Hypercube Q4

  10. Preliminaries • There are some variations of the hypercube appearing in literature; such as Twisted-cubes, Crossed-cubes, Möbius cubes, and so on. These variations preserve most of the good topological properties of the hypercube, and even better. • We generalize these cubes and maintain its fault tolerance.

  11. Twisted-cube Definition of Twisted-cube TQn • The Twisted n-cube TQn is defined for odd values of n. The vertex set of the twisted n-cube TQn is the set of all binary strings of length n. Let u = un-1un-2… u1u0 be any vertex in TQn. • A twisted 1-cube, TQ1, is a complete graph with two vertices 0 and 1.

  12. Twisted-cube • For 0 in-1, let the i-th parity function be Pi(u) = uiui-1 … u0, where  is the exclusive-or operation. • Suppose that n  3. We can decompose the vertices of TQn into four sets, TQn-2, TQn-2, TQn-2, and TQn-2 where TQn-2 consists of those vertices u with un-1 = i and un-2 = j. • For each (i,j) {(0,0), (0,1), (1,0), (1,1)}, the induced subgraph of TQn-2 in TQn is isomorphic to TQn-2.

  13. Twisted-cube • The edges that connect these four subtwisted cubes can be described as follows: Any vertex un-1un-2… u1u0 with Pn-3(u) = 0 is connected to ūn-1ūn-2… u1u0 and ūn-1un-2… u1u0; and to un-1ūn-2… u1u0 and ūn-1un-2… u1u0 if Pn-3(u) = 1.

  14. Twisted-cube

  15. Twisted-cube

  16. W.T. Huang, J.M. Tan, C.N. Hung, and L.H. Hsu (2002), "Fault-Tolerant Hamiltonicity of Twisted Cubes," Journal of Parallel and Distributed Computing, Vo.l 62, pp. 591-604.

  17. Crossed-cube Definition of Crossed-cube CQn • Two two-digit binary strings x = x1x0 and y = y1y0 are pair related, denoted by x ~ y, if and only if (x,y)  {(00,00), (10,10), (01,11), (11,01)}. • CQ1 is a complete graph with two vertices labeled by 0 and 1. • CQn consists of two identical (n-1)-dimension crossed cubes, CQn-10 and CQn-11.

  18. Crossed-cube • The vertex u = 0un-2…u0V(CQn-10) and vertex v = 1vn-2…v0V(CQn-11) are adjacent in CQn if and only if (1) un-2 = vn-2 if n is even; and (2) for 0 i < (n-1)/2, u2i+1u2i ~ v2i+1v2i.

  19. Crossed-cube

  20. W.T. Huang, Y.C. Chuang, L.H. Hsu, and J.M. Tan (2002), "On the Fault-Tolerant Hamiltonicity of Crossed Cubes," IEICE Transaction on Fundamentals, Vol. E85-A, pp. 1359-1371.

  21. Möbius cube Definition of Möbius cube MQn • The Möbius cube, MQn = (V,E), has 2n vertices. • Each vertex is labeled by a unique n-bit binary string as its address and has connections to n other distinct vertices. • The vertex with address X = xn-1xn-2…x0 connects to n other vertices Yi, 0 in-1, where the address of Yi satisfies (1) Yi = (xn-1…xi+1xi…x0) if xi+1=0; or (2) Yi = (xn-1…xi+1xi…x0) if xi+1=1.

  22. Möbius cube • X connects to Yi by complementing the bit Xi if xi+1=0, or by complementing all bits of xi...x0 if xi+1=1. • For the connection between X and Yn-1, we can assume that the unspecified xn is either 0 or 1, which gives slightly different topologies. • If xn is 0, we call the network generated the ``0-möbius cube", denoted by 0-MQn; and if xn is 1, we call the network generated the ``1-möbius cube", denoted by 1-MQn.

  23. Möbius cube 0-MQ4 1-MQ4

  24. W.T. Huang, Y.C. Chuang, L.H. Hsu, and J.M. Tan (2000), "Fault-Free Hamiltonian Cycle in Faulty Mobius Cube," J. Computing and Sys, Vol. 4, pp.106-114.

  25. construction schemes One construction scheme • Graph G(G1,G2 ; M) G G a perfect 1 2 matching M

  26. Y. C. Chen, C. H. Tsai, L. H. Hsu, and Jimmy J. M. Tan (2004), "On Some Super Fault-Tolerant Hamiltonian Graphs," Applied Mathematics and Computation, Vol. 148, pp. 729-741.

  27. Theorem • Assume that G1 and G2 are k-regular super fault-tolerant Hamiltonian where k 5 and |V(G1)| = |V(G2)|. Then G(G1, G2 ; M) is (k+1)-regular super fault-tolerant Hamiltonian.

  28. Y.C. Chen, L.H. Hsu, and Jimmy J.M. Tan (2006), "A Recursively Construction Scheme for Super Fault-Tolerant Hamiltonian Graphs," Applied Mathematics and Computation, Vol. 177, pp. 465-481. (k5) • T.L. Kueng, C.K. Lin, T. Liang, J.J.M Tan, and L.H. Hsu (2008), "Fault-tolerant Hamiltonian Connectedness of Cycle Composition Networks," Applied Mathematics and Computation, Vol. 196 pp. 245-256. (4)

  29. C.K. Lin, T.Y. Ho, J.M. Tan, and L.H. Hsu ``Fault-Tolerant Hamiltonicity and Fault-Tolerant Hamiltonian Connectivity of the Folded Petersen Cube Networks", accepted by International Journal of Computer Mathematics. PkQn

  30. Preliminaries • Let G1 and G2 be two k-regular super fault-tolerant Hamiltonian graphs with the same number of nodes, and let M be an arbitrary perfect matching. Then G(G1,G2 ; M) is (k+1)-regular. We expect that its fault-tolerant Hamiltonicity Hf(G) and fault-tolerant Hamiltonian connectivity Hf(G) are also increased by 1.

  31. Lemma 1 • Let G be a k-Hamiltonian graph, FG be a set of faults in G with |FG| k, and u be a healthy node in G. Then there are at least k- fG +2 edges incident to node u, such that each one of them is on some Hamiltonian cycle in G - FG.

  32. We know that G is k-Hamiltonian, and • There are fG faults in G. • Suppose fG < k. Let HC be a Hamiltonian cycle in G-FG, and let e be an edge on HC and incident to node u. • Deleting edge e, G-FG-{e} still contains a Hamiltonian cycle. • Hence, G-FG is still Hamiltonian even if we add k-fG more faults to G-FG. • Repeating this process k-fG times, we find k-fG+2 edges incident to node u and each one of them is on some Hamiltonian cycle in G-FG. u fG faults e e Graph G

  33. Lemma 2 • Let G be a k-Hamiltonian connected graph, FG be a set of faults in G with |FG| k, and {x,y,u} be three distinct health nodes in G. Then there are at least k-fG+2 edges incident to node u, such that each one of them is on some x,y-Hamiltonian path in G-FG.

  34. It is known that G is k-Hamiltonian connected, and there are fG faults in G. • Thus, G-FG is still Hamiltonian connected even if we add fG more faults to G-FG. • Suppose fG < k. Let HP be an x,y-Hamiltonian path in G-FG, and let e be an edge on HP and incident to node u. • Deleting edge e, G-FG-{e} still contains an x,y-Hamiltonian path. • Repeating this process k-fG times, we find k-FG+2 edges incident to node u and each one of them is on some x,y-Hamiltonian path in G-FG. x u fG faults e y Graph G

  35. Lemma 3 • Let Gr and Gs be two k-regular graphs with the same number of nodes. If the total number of faults in G(Gr ,Gs;M) is no greater than k, there exists at least one healthy matching edge between Gr and Gs. k+1 vertices

  36. Lemma 4 • Let Gr and Gs be two k-regular graphs with the same number of nodes, and let x and y be two healthy nodes in G(Gr, Gs;M). If the total number of faults in G(Gr,Gs;M) is no greater than k-2, there exists at least one healthy matching edge between Gr and Gs whose endpoints are neither x nor y. y k+1 vertices x

  37. Main Result Theorem 1 • Assume k 4. Let G1 and G2 be two k-regular super fault-tolerant Hamiltonian graphs and |V(G1)| = |V(G2)|. Then graph G(G1, G2 ; M) is (k-1)-Hamiltonian.

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