Pancake problems with restricted prefix reversals and some corresponding Cayley networks

The pancake problem concerns the number of prefix reversals (“flips”) needed to sort the elements of an arbitrary permutation, which is the diameter of the n-dimensional pancake network. We restrict the problem by allowing only a few of the possible n-1 flips. Let f_i denote a flip of size i. We consider sets with either O(1) flips or log₂ n flips, and explore their corresponding Cayley networks, such as: The Subcube_n network, for n=2^k, defined by the log₂ n flips {f₂ ,f₄,f₈...f_n}. Subcube_n is isomorphic to a network obtained from an (n-1) dimensional hypercube, Q_n-1, by deleting all but log₂ n of the edges incident to each of its nodes, has diameter (3n/2)-2 (we give an optimum routing algorithm), and hosts Q_n-1 with nearly optimum dilation. The Triad_n network where n is odd and [n/2] mod 4≠0, defined by the set of flips {f[n/2] f[n/2] f_n}. Triad _n has n! nodes and diameter Θ(n log₂ n). Both the n-dimensional shuffle-exchange and shuffle-exchange permutation networks can be emulated by Triad_n with constant slowdown