algorithmic construction of hamiltonians in pyramids h. sarbazi-azad, m. ould-khaoua, l.m....
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![Page 1: Algorithmic construction of Hamiltonians in pyramids H. Sarbazi-Azad, M. Ould-Khaoua, L.M. Mackenzie, IPL, 80, 75- 79(2001)](https://reader035.vdocuments.mx/reader035/viewer/2022062423/56649ee55503460f94bf563f/html5/thumbnails/1.jpg)
Algorithmic construction of Hamiltonians in pyramids
H. Sarbazi-Azad, M. Ould-Khaoua, L.M. Mackenzie, IPL, 80, 75-79(2
001)
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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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Meshs
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Pyramid
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Pyramid Pn is not regular
(P1)=3, ∆(P1)=4(P2)=3, ∆(P2)=7(Pn)=3, ∆(Pn)=9, for n>=3
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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}.
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P1
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Induction
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Induction (cont.)
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Result(cont.)
• Theorem 2. A pyramid of level n, Pn, is Hamiltonian.
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algorithm
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In fact, Pn is hamiltonian connected
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• A. Itai, C. Papadimitriou, J. Szwarcfiter, “Hamilton Paths in grid graphs”, SIAM Journal on Computing, 11 (4) (1982) 676-686.
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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)
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Pn is hamiltonian connected
• Proof:
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P1
• 剛剛看過了
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Induction
• Case 1. x, y 都在上面 n-1 層
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• Case 2. x 在上面 n-1 層 , y 在第 n 層
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• Case 3. x, y 都在第 n 層
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Pn is pancyclic
• By induction
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P1
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Induction
• (1) 3~L
• (2)L+2
• (3)L+3~L+4
• (4)L+5~|V(Pn)|
• (5)L+1