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( = 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 M Q n . Recently, for constructing n ISTs rooted at an arbitrary node of M Q n , Cheng et al. (Comput J 56(11):1347(Comput J 56(11): -1362(Comput J 56(11): , 2013 and (J Supercomput 65 (3):1279-1301, 2013), respectively, proposed a sequential algorithm to run in O(N log N ) time and a parallel algorithm that takes 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 nonrecursive and fully parallelized approach to construct n ISTs rooted at an arbitrary node of M Q n in O(log N ) time using N nodes of M Q 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.
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