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

Abstract

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.