Amortized efficiency of constructing multiple independent spanning trees on bubble-sort networks

A set of spanning trees in a graph G is called independent spanning trees (ISTs for short) if they are rooted at the same vertex, say r, and for each vertex \(v(\ne r)\) in G, the two paths from v to r in any two trees share no common edge and no common vertex except for v and r. Constructing ISTs has applications on fault-tolerant broadcasting and secure message distribution in reliable communication networks. Since Cayley graphs have been used extensively to design the topologies of interconnection networks, construction of ISTs on Cayley graphs is significative. It is well-known that star networks \(S_n\) and bubble-sort network \(B_n\) are two of the most attractive subclasses of Cayley graphs. Although it has been dealt with about two decades for the construction of ISTs on \(S_n\) (which has been pointed out that there is a flaw and has been corrected recently), so far the problem of constructing ISTs on \(B_n\) is not dealt with yet. In this paper, we present an algorithm to construct \(n-1\) ISTs of \(B_n\). Moreover, we show that our algorithm has amortized efficiency for multiple trees construction. In particular, every vertex can determine its parent in each spanning tree in a constant amortized time. Accordingly, except for the star networks, it seems that our work is the latest breakthrough on the problem of ISTs for all subfamilies of Cayley graphs.