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Algorithmic construction of Hamiltonians in pyramidsPowerPoint Presentation

Algorithmic construction of Hamiltonians in pyramids

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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)

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

- (P1)=3, ∆(P1)=4
- (P2)=3, ∆(P2)=7
- (Pn)=3, ∆(Pn)=9, for n>=3

- 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}.

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

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

- 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)

- Proof:

- 剛剛看過了

- Case 1. x, y 都在上面 n-1層

- Case 2. x 在上面 n-1 層, y 在第 n層

- Case 3. x, y 都在第n層

- By induction

- (1) 3~L
- (2)L+2
- (3)L+3~L+4
- (4)L+5~|V(Pn)|
- (5)L+1