Construction independent spanning trees on locally twisted cubes in parallel

Let \(LTQ_n\) be the n-dimensional locally twisted cube. Hsieh and Tu (Theor Comput Sci 410(8–10):926–932, 2009) proposed an algorithm to construct n edge-disjoint spanning trees rooted at a particular vertex 0 in \(LTQ_n\). Later on, Lin et al. (Inf Process Lett 110(10):414–419, 2010) proved that Hsieh and Tu’s spanning trees are indeed independent spanning trees (ISTs for short), i.e., all spanning trees are rooted at the same vertex r and for any other vertex \(v(\ne r)\), the paths from v to r in any two trees are internally vertex-disjoint. Shortly afterwards, Liu et al. (Theor Comput Sci 412(22):2237–2252, 2011) pointed out that \(LTQ_n\) fails to be vertex-transitive for \(n\geqslant 4\) and proposed an algorithm for constructing n ISTs rooted at an arbitrary vertex in \(LTQ_n\). Although this algorithm can simultaneously construct n ISTs, it is hard to be parallelized for the construction of each spanning tree. In this paper, from a modification of Hsieh and Tu’s algorithm, we present a fully parallelized scheme to construct n ISTs rooted at an arbitrary vertex in \(LTQ_n\) in \({\mathcal O}(n)\) time using \(2^n\) vertices of \(LTQ_n\) as processors.