A parallel algorithm for constructing independent spanning trees in twisted cubes

Abstract

A long-standing conjecture mentions that a $k$-connected graph $G$ admits $k$ independent spanning trees (ISTs for short) rooted at an arbitrary node of $G$. An $n$-dimensional twisted cube, denoted by $T{Q}_n$, is a variation of hypercube with connectivity $n$ and has many features superior to those of hypercube. Yang (2010) first proposed an algorithm to construct $n$ edge-disjoint spanning trees in $T{Q}_n$ for any odd integer $n⩾3$ and showed that half of them are ISTs. At a later stage, Wang et al. (2012) inferred that the above conjecture in affirmative for $T{Q}_n$ by providing an $\mathcal{O}(NlogN)$ time algorithm to construct $n$ ISTs, where $N=2^n$ is the number of nodes in $T{Q}_n$. However, this algorithm is executed in a recursive fashion and thus is hard to be parallelized. In this paper, we revisit the problem of constructing ISTs in twisted cubes and present a non-recursive algorithm. Our approach can be fully parallelized to make the use of all nodes of $T{Q}_n$ as processors for computation in such a way that each node can determine its parent in all spanning trees directly by referring its address and tree indices in $\mathcal{O}(logN)$ time.