ON PANCAKE NETWORKS Marissa P. Justan A symmetric interconnection network (SIM) has the property that the network viewed from any vertex of the network looks the same. In such a network, congestion problems are minimized since the load will be distributed uniformly through all the vertices. In designing SIMs, the overall objective has been to construct large vertex symmetric graphs with small degree and diameter, high connectivity, and offering simple routing algorithms. (We will use the term graph interchangeably with network). The diameter corresponds to the worst communication delay for broadcasting messages in the network. Thus, the smaller the diameter, the better for a network. One class of SIMs, namely the pancake network, Pn is especially attractive for distributed processing. Dweighter [2] described the pancake network by the following combinatorial problem: Given any stack of n pancakes with unique sizes, what is the maximum number of flips as a function f(n), to sort them so that the smallest winds up on top, and so on, down to the largest at the bottom by grabbing several from the top and flipping them over? Gates and Papadimitriou [3] introduced the burnt pancake flipping problem. Here, one side of each pancake is burnt, and the pancakes must be sorted with the burnt side down. We shall refer to this problem as the two-sided pancake flipping problem and the network that it describes as the two-sided pancake graph, 2Pn. In the generalized pancake flipping problem [Justan, 5], a stack of n pancakes all come out in different sizes, and each pancake has m-sides. 23
The problem is getting the maximum number of flips that will have to sort any arrangement of pancakes so that one particular side is up. This problem is referred as the m-sided pancake flipping problem, and its network is described as the msided pancake graph, mPn. Tables 1 and 2 summarized the results of the study No. of vertices Degree
Lower bound of diameter
Upper Bound of diameter
3Pn
n!3n
2n
n+ 1
2n-1
mPn
n!mn
2n
n+ 1
m/2 n
Table 1. Summary of order, degree and known best bounds.
The exact value of f (13)=15 was derived by Heydari and Sudborough [4]. Previous computer search becomes impractical for n>13. Pn
f(13) = 15
2Pn
g(10) = 18
mPn
h(7) = 9 Table 2. Last known values of diameters f(n), g(n), h(n).
The current work here is (1) the improvement of the bounds for the diameters and (2) the computation of the diameters by applying high performance computing techniques. BIBLIOGRAPHY [1] Sheldon Akers and Balakrishnan Krishnamurthy, A Group-Theoretic Model For Symmetric Interconnection Networks, IEEE Transactions on Computers, Vol. 38, No. 4. Pp. 555-566. April 1989. [2] Harry Dweighter, American Mathematical Monthly, Vol. 82, No. 1, p.1010, 1975. [3] William H. Gates and Christos H. Papadimitriou, Bounds For Sorting By Prefix Reversal, Discrete Mathematics 27, pp. 47-57, 1979 [4] Mohammad H. Heydari and I. Hal Sudburough, On the Diameter of the Pancake Network, Vol. 25, No. 1, pp.67-94, Oct 1997. [5] Marissa P. Justan, On the Generalization of the Pancake Flipping Problem, Ph.D. Dissertation, Mathematics Department, Ateneo de Manila University, 1999.
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[1,2,3] [3,2,1]
[2,1,3]
[2,3,1]
[3,1,2] [1,3,2]
Figure 1: Graph of P
[4,3,2,1]
[1,2,3,4] [3,2,1,4]
3
[2,1,3,4] [3,4,2,1]
[2,3, 4,1]
[3,1,2,4] [2,4,3,1]
[3,2,4,1]
[2,3,1,4] [1,3,2,4]
[4,2,3,1] [2,4,1,3]
[3,1,4,2] [4,1,3,2] [1,4,3,2]
[1,3,4,2] [4,2,1,3]
[1,4,2,3]
[4,3,1,2] [1,2,4,3]
[4,1,2,3] [2,1,4,3]
[3,4,1,2]
Figure 2: Graph of P4 [(0,1),(0,2)]
[(1,1),(0,2)]
[(1,2),(1,1)]
[(1,2),(0,1)]
[(0,2),(1,1)]
[(0,2),(0,1)]
[(1,1),(1,2)]
[(0,1),(1,2)]
Figure 3: Graph of 2P 2
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