Algorithmic construction of Hamiltonians in pyramids

H. Sarbazi-Azad, M. Ould-Khaoua, L.M. Mackenzie, IPL, 80, 75-79(2

001)

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

Pyramid

Pyramid Pn is not regular

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

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}.

P1

Induction

Induction (cont.)

Result(cont.)

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

algorithm

In fact, Pn is hamiltonian connected

• 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)

• 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)

Pn is hamiltonian connected

• Proof:

P1

• 剛剛看過了

Induction

• 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

Pn is pancyclic

• By induction

P1

Induction

• (1) 3~L

• (2)L+2

• (3)L+3~L+4

• (4)L+5~|V(Pn)|

• (5)L+1