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Some New Directions about Interconnection Networks

This talk delves into the fascinating world of interconnection networks, focusing on hypercubes and their various structures. We discuss the Hamming distance, adjacency in n-dimensional hypercubes, and introduce different graph families including twisted cubes, star graphs, and more. The session also shares research strategies for students and professors, highlighting fault tolerance in graph theory and insights from pertinent literature. Dive into the complexity of network design and understand how these principles apply in real-world computing and communication systems.

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Some New Directions about Interconnection Networks

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Presentation Transcript

  1. Some New Directions about Interconnection Networks

  2. Purpose of this talk • A story of research • How to do research as a student • How to do research as a professor

  3. Interconnection Networks

  4. Hypercubes 100 110 0 00 10 000 010 101 111 1 01 11 001 011 Q1 Q2 Q3

  5. Let u = un−1un−2. . .u1u0 and v = vn−1vn−2. . .v1v0 be two n-bit binary strings. • The Hamming distanceh(u, v) between two vertices u and v is the number of different bits in the corresponding strings of both vertices. • The n-dimensional hypercubeconsists of all n-bit binary strings as its vertices and two vertices u and v are adjacent if and only if h(u, v) = 1.

  6. Hypercubic Like graphs • Twisted cubes • Cross cubes • Mobius cubes • Locally twisted cubes n regular graph with 2n vertices

  7. Bypartite Hypercubic Like Graphs

  8. Other Cubic Graphs • Folded hypercubes (bipartite or nonbipartite) • Enhance hypercubes (bipartite or nonbipartite) • Augmented cubes

  9. Other Families • Star graphs (bipartite) • Pancake graphs • (n,k)-star graphs • Arrangement graphs • Butterfly (bipartite or nonbipartite) • Recursive circulant graphs • Cubic family (honeycomb torus, Christmas tree, honeycomb disk, spider web, brother tree) • etc

  10. S. Latifi, S. Zheng, N. Bagherzadeh, Optimal ring embedding in hypercubes with faulty links, in: Fault-Tolerant Computing Symp., 1992, pp. 178–184. • Y.C. Tseng, S.H. Chang, and J.P. Sheu, Fault-tolerant ring embedding in a star graph with both link and node failures, IEEE Trans Parallel Distrib Syst 8 (1997), 1185–1195.

  11. Another story • F. Harary, J.P. Hayes, Edge fault tolerance in graphs, Networks 23 (1993) 135–142. • F. Harary, J.P. Hayes, Node fault tolerance in graphs, Networks 27 (1996) 19–23. Diameter about n/4

  12. K.Mukhopadhyaya, B.P. Sinha, Hamiltonian graphs with minimum number of edges for fault-tolerant topologies, Inform. Process. Lett. 44 (1992) 95–99. Diameter about n/6

  13. Diameter about O(n1/2) Diameter about O(log n)

  14. Fault Hamiltonian and Fault Hamiltonian Connected

  15. Home Home Home

  16. Fault Hamiltonian and Fault Hamiltonian Connected • n-2 fault tolerant hamiltonian and n-3 fault tolerant hamiltonian connected • d-2 fault tolerant hamiltonian and d-3 fault tolerant hamiltonian connected

  17. General Rules • 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.

  18. Other families of Interconnection Networks • C.H. Tsai, J.M. Tan, Y.C. Chen, and L.H. Hsu, (2002) "Hamiltonian Properties of Faulty Recursive Circulant Graphs," Journal of Interconnection Networks, Vol 3, Nos, 3&4, pp. 273-289. • C.N. Hung, H. C. Hsu, K. Y. Liang, and L. H. Hsu, (2003) "Ring Embedding in Faulty Pancake Graphs," Information Processing Letters, Vol 86, pp. 271-275.

  19. H.C. Hsu, Y.L. Hsieh, J.M. Tan, and L.H. Hsu, (2003) " Fault Hamiltonicity and Fault Hamiltonian Connectivity of the (n,k)-star Graphs," Networks, Vol 42(4), pp. 189-201. • H.C. Hsu, T.K. Li, J.M. Tan, and L.H. Hsu (2004). "Fault Hamiltonicity and Fault Hamiltonian Connectivity of the Arrangement Graphs," IEEE Trans. on Computers, Vol. 53 (1), pp. 39-53.

  20. C.H. Tsai, J.M. Tan, T. Liang, and L.H. Hsu (2002), ``Fault-Tolerant Hamiltonian Laceability of Hypercubes", Information Processing Letters, Vol. 83, pp. 301-306. • H.C. Hsu, L.C.Chiang, Jimmy J.M. Tan, L.H. Hsu (2005), `` Fault hamiltonicity of augmented cubes", Parallel Computing, Vol. 31, pp.131-145. • Y.H. Teng, Jimmy J.M. Tan, L.H. Hsu (2005), ``Honeycomb rectangular disks", Parallel Computing, Vol. 31, pp.371-388.

  21. How about bipartite graphs Hamiltonian laceable

  22. Edge fault tolerance hamiltonian laceable

  23. Edge fault tolerance strong hamiltonian laceable

  24. Hyper hamiltonian laceable

  25. pancyclic

  26. pancyclic

  27. Panconnected • A lot of people work on pancyclic and panconnected recently.

  28. Globally 3*-connected Graphs • M. Albert, E.R.L. Aldred, D. Holton, and J. Sheehan, On globally 3*-connected graphs, Australasian Journal of Combinatorics, 24, 2001, 193-207.

  29. Global 3*-connected graph

  30. Mutually independent hamiltonian cycles and hamiltonian paths

  31. 出入口

  32. 出入口

  33. Mutually independent hamiltonian cycles and hamiltonian paths

  34. Star and cycle (independent hamiltonian cycles) • (n,k)-star graph (independent hamiltonian paths) • Folded hypercubes (KBJ) • Other families of graphs and math works • Independent paths and cycles

  35. 0 出入口 1 1 2 2 2 2 3 2 1 2 1 Panpositionable Hamiltonian

  36. 出入口 出入口

  37. Panpositionable Hamiltonian

  38. Diagnosability

  39. Thanks

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