A fully parallelized scheme of constructing independent spanning trees on Möbius cubes

Abstract

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.

Independent Spanning Trees in Eisenstein-Jacobi Networks

“…Additionally, The optimal independent spanning trees on Hypercubes is presented in [35]. Further, a fully parallelized construction of ISTs on Mobius cubes has been discussed in [44]. Moreover, An implementation of a fast parallel algorithm for constructing ISTs on Parity Cubes is explained in [4].…”

A Fast Parallel Algorithm for Constructing Independent Spanning Trees on Parity Cubes

“…If F is a subset of V(G), we denote by G − F the graph obtained from G by removing F. A graph G is k-connected if |V(G)| > k and G − F is connected for every subset F⊆V(G) with |F| < k. Zehavi and Itai [20] proposed the following conjecture: If G is a k-connected graph and r ∈ V(G) is an arbitrary node, then G has k ISTs rooted at r. From then on, the conjecture has been confirmed only for k-connected graphs with k ≤ 4 (see [6,7,10,20]), and it is still open for arbitrary k-connected graphs when k ≥ 5. Moreover, by providing construction schemes of ISTs, the conjecture has been proved to be affirmative for several restricted classes of graphs (see recent papers [2][3][4][5][15][16][17][18] and the references quoted therein).…”

The Construction of Multiple Independent Spanning Trees on Burnt Pancake Networks

“…For instance, the construction of ISTS on some variations of hypercubes [3], [22], [32], [33], [40], torus networks [31], recursive circulant graphs [37], [38], and special subclasses of Cayley networks [7], [17], [18], [20], [21]. In particular, special topics related to IST include the research on reducing the height of ISTs [34], [36], [39] and parallel construction of ISTs [4]- [6], [35], [40], [41].…”