Constructing dual-CISTs of folded divide-and-swap cubes

Abstract

Spanning trees T1,T2,…,Tk (k⩾2) in a network are called completely independent spanning trees (CISTs for short) if they are pairwise edge-disjoint and inner-node-disjoint. Particularly, if k=2, the two CISTs are called a dual-CIST. Hasunuma (2002) [8] proved that the problem of determining whether there exists a dual-CIST in a network is NP-complete. Tapolcai (2013) [20] showed that a dual-CIST has an application on configuring a protection routing in intra-domain IP networks. The adoption of protection routing guarantees that there is a loop-free alternate path for packet forwarding when a single link or node failure occurs. Pai et al. (2020) [17] demonstrated that protection routing is also suitable for relatively large (static) network topologies with scalability, such as interconnection networks with recursive structure and data center networks (DCNs). Recently, Kim et al. (2019) [10] newly proposed a hypercube-variant network called folded divide-and-swap cube, denoted as FDSC(n), which is obtained from the divide-and-swap cube DSC(n) by adding an extra edge to each node. They also showed that DSC(n) and FDSC(n) have a better network performance than that of hypercubes, where the performance is measured by the product of degree and diameter. In this paper, we first point out that FDSC(n) is suitable as a candidate topology for DCNs. Then, we investigate the construction of a dual-CIST {T1,T2} in FDSC(n). In particular, the diameters of Ti for i=1,2 we constructed arediam(Ti)={9if n=4;478n−15−23(4⌊d−42⌋−1)(dmod2+1)if n=2d for d⩾3.