Hamiltonicity of k-Sided Pancake Networks with Fixed-Spin: Efficient Generation, Ranking, and Optimality

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 $\ell$, which reverses the first $\ell$ 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 $k>2$. In addition to our generation results, we provide ranking and unranking algorithms for our Hamiltion cycle that run in $O(n^2)$ time, and show that the cycle is globally optimal in terms of minimizing the total number of pancakes that are flipped. Finally, we characterize the Hamiltonicity of k-sided pancake networks with any fixed “spin” s.

Appendices

Appendix

An Example Illustrating the Auxiliary Variable for the Loop-Free Generation

The following array (read top to bottom, then left to right) shows how the $c_i$’s and $f_i$’s evolve for each flip generated by FlipSeq for $\sigma _{3, 3}$. Each entry corresponds to the flip in the same position in the array to the right of the vertical bar in Example 2. Each entry is a $2\times 3$ matrix of the form $\begin{pmatrix} c_1 &{} c_2 &{} c_3 \\ f_1 &{} f_2 &{} f_3 \end{pmatrix}$. Note that $c_4=0$ except for in the last entry where $c_4=1$ and $f_4$ is always equal to 4, so we chose to omit these.