Algorithmic construction of hamiltonians in pyramids
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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

Algorithmic construction of Hamiltonians in pyramids

H. Sarbazi-Azad, M. Ould-Khaoua, L.M. Mackenzie, IPL, 80, 75-79(2001)


Previous work

Previous work

  • F. Cao, D. F. Hsu, “Fault Tolerance Properties of Pyramid Networks”, IEEE Trans. Comput. 48 (1999) 88-93.

  • Connectivity, fault diameter, container


Meshs

Meshs


Pyramid

Pyramid


Pyramid p n is not regular

Pyramid Pn is not regular

  • (P1)=3, ∆(P1)=4

  • (P2)=3, ∆(P2)=7

  • (Pn)=3, ∆(Pn)=9, for n>=3


Result

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}.


Algorithmic construction of hamiltonians in pyramids

P1


Induction

Induction


Induction cont

Induction (cont.)


Result cont

Result(cont.)

  • Theorem 2. A pyramid of level n, Pn, is Hamiltonian.


Algorithm

algorithm


In fact p n is hamiltonian connected

In fact, Pn is hamiltonian connected


Algorithmic construction of hamiltonians in pyramids

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


Hamiltonian property of m m n

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)


P n is hamiltonian connected

Pn is hamiltonian connected

  • Proof:


Algorithmic construction of hamiltonians in pyramids

P1

  • 剛剛看過了


Induction1

Induction

  • Case 1. x, y 都在上面 n-1層


Algorithmic construction of hamiltonians in pyramids

  • Case 2. x 在上面 n-1 層, y 在第 n層


Algorithmic construction of hamiltonians in pyramids

  • Case 3. x, y 都在第n層


P n is pancyclic

Pn is pancyclic

  • By induction


Algorithmic construction of hamiltonians in pyramids

P1


Induction2

Induction

  • (1) 3~L

  • (2)L+2

  • (3)L+3~L+4

  • (4)L+5~|V(Pn)|

  • (5)L+1


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