A Hamilton Cycle in the k-Sided Pancake Network

Abstract

We present a Hamilton cycle in the k-sided pancake network and four combinatorial algorithms to traverse the cycle. The network’s vertices are coloured permutations $\pi = p_1p_2\cdots p_n$, where each $p_i$ has an associated colour in $\{0,1,\ldots , k\,-\,1\}$. There is a directed edge $(\pi _1,\pi _2)$ if $\pi _2$ can be obtained from $\pi _1$ by a “flip” of length j, which reverses the first j elements and increments their colour modulo k. Our particular cycle is created using a greedy min-flip strategy, and the average flip length of the edges we use is bounded by a constant. By reinterpreting the order recursively, we can generate successive coloured permutations in O(1)-amortized time, or each successive flip by a loop-free algorithm. We also show how to compute the successor of any coloured permutation in O(n)-time. Our greedy min-flip construction generalizes known Hamilton cycles for the pancake network (where $k=1$) and the burnt pancake network (where $k=2$). Interestingly, a greedy max-flip strategy works on the pancake and burnt pancake networks, but it does not work on the k-sided network when $k>2$.