Construction independent spanning trees on locally twisted cubes in parallel

Abstract

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.