Pancake Problems with Restricted Prefix Reversals and some Corresponding Cayley Networks

The pancake problem, which has attracted considerable attention [10, 12, 9, 15], concerns the number of prefix reversals or flips needed to sort the elements of an arbitrary permutation. The number of prefix reversals to sort permutations is also the diameter of the often We consider restricted pancake problems, for example when only 3 of the possible n-1 flips are allowed. Let f i denote a flip of size i. Each flip is itself a permutation. For example, a flip of size 4, i. e., f 4 , on eight symbols has the effect of changing, say, 3 5 1 2 4 6 8 7 into 2 1 5 3 4 6 8 7. f 4 is the permutation 4 3 2 1 5 6 7 8. We investigate sets of permutations corresponding to flips as generators of the symmetric group S n. Let n be the number of symbols in a permutation. We consider sets with either a constant number of generators (i. e., flips) or with log 2 n generators. In special interesting cases, the corresponding Cayley networks, defined by a given set of generators and a given group of permutations, are explored. Specifically, we investigate two special families of networks: 1) The Subcube n network, for n = 2 k , defined by the log 2 n generators in the set {f 2 , f 4 , f 8 … f n }. We prove that: • Subcube n is isomorphic to a network obtained from an (n-1) dimensional hypercube, Q n-1 , by deleting all but log 2 n of the edges incident to each of its nodes.