A fully parallelized scheme of constructing independent spanning trees on Möbius cubes

A set of spanning trees in a graph is said to be independent (ISTs for short) if all the trees are rooted at the same node \(r\) and for any other node \(v(\ne r)\), the paths from \(v\) to \(r\) in any two trees are node-disjoint except the two end nodes \(v\) and \(r\). It was conjectured that for any \(n\)-connected graph there exist \(n\) ISTs rooted at an arbitrary node. Let \(N=2^n\) be the number of nodes in the \(n\)-dimensional Möbius cube \(MQ_n\). Recently, for constructing \(n\) ISTs rooted at an arbitrary node of \(MQ_n\), Cheng et al. (Comput J 56(11):1347–1362, 2013) and (J Supercomput 65(3):1279–1301, 2013), respectively, proposed a sequential algorithm to run in \({\mathcal O}(N\log N)\) time and a parallel algorithm that takes \({\mathcal O}(N)\) time using \(\log N\) processors. However, the former algorithm is executed in a recursive fashion and thus is hard to be parallelized. Although the latter algorithm can simultaneously construct \(n\) ISTs, it is not fully parallelized for the construction of each spanning tree. In this paper, we present a non-recursive and fully parallelized approach to construct \(n\) ISTs rooted at an arbitrary node of \(MQ_n\) in \({\mathcal O}(\log N)\) time using \(N\) nodes of \(MQ_n\) as processors. In particular, we derive useful properties from the description of paths in ISTs, which make the proof of independency to become easier than ever before.