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Algorithmic construction of Hamiltonians in pyramids

Algorithmic construction of Hamiltonians in pyramids. H. Sarbazi-Azad, M. Ould-Khaoua, L.M. Mackenzie, IPL, 80, 75-79(2001). Previous work. F. Cao, D. F. Hsu, “ Fault Tolerance Properties of Pyramid Networks ”, IEEE Trans. Comput. 48 (1999) 88-93. Connectivity, fault diameter, container.

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Algorithmic construction of Hamiltonians in pyramids

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  1. Algorithmic construction of Hamiltonians in pyramids H. Sarbazi-Azad, M. Ould-Khaoua, L.M. Mackenzie, IPL, 80, 75-79(2001)

  2. Previous work • F. Cao, D. F. Hsu, “Fault Tolerance Properties of Pyramid Networks”, IEEE Trans. Comput. 48 (1999) 88-93. • Connectivity, fault diameter, container

  3. Meshs

  4. Pyramid

  5. Pyramid Pn is not regular • (P1)=3, ∆(P1)=4 • (P2)=3, ∆(P2)=7 • (Pn)=3, ∆(Pn)=9, for n>=3

  6. result • Theorem 1. A Pn contains Hamiltonian paths starting with any node x  P = { Pn▲, Pn◤, Pn◣, Pn◥, Pn◢ } and lasting at any node y  P – {x}.

  7. P1

  8. Induction

  9. Induction (cont.)

  10. Result(cont.) • Theorem 2. A pyramid of level n, Pn, is Hamiltonian.

  11. algorithm

  12. In fact, Pn is hamiltonian connected

  13. A. Itai, C. Papadimitriou, J. Szwarcfiter, “Hamilton Paths in grid graphs”, SIAM Journal on Computing, 11 (4) (1982) 676-686.

  14. Hamiltonian property of M(m, n) • In fact, M(m, n) is bipartite. • M(m,n) is even-size if m*n is even. • Roughly speaking, for a even-sized M(m, n), there exists a hamiltonian path between any two nodes x, y iff x and y belong to a same partite set. • There are a few exceptions. (detail)

  15. Pn is hamiltonian connected • Proof:

  16. P1 • 剛剛看過了

  17. Induction • Case 1. x, y 都在上面 n-1層

  18. Case 2. x 在上面 n-1 層, y 在第 n層

  19. Case 3. x, y 都在第n層

  20. Pn is pancyclic • By induction

  21. P1

  22. Induction • (1) 3~L • (2)L+2 • (3)L+3~L+4 • (4)L+5~|V(Pn)| • (5)L+1

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