Search

All-to-all broadcasting means to distribute the exclusive message of each node in the network to all other nodes. It can be handled by rings, and a Hamiltonian cycle is a ring that visits each vertex exactly once. Multiple edge-disjoint Hamiltonian cycles, abbreviated as EDHCs, have two application advantages: (1) parallel data broadcast and (2) edge fault-tolerance in network communications. There are three edge-disjoint Hamiltonian cycles on n-dimensional locally twisted cubes and n-dimensional crossed cubes while n ≥ 6, respectively. Locally twisted cubes, crossed cubes, folded locally twisted cubes (denoted as FLTQn), and folded crossed cubes (denoted as FCQn) are among the hypercube-variant network. The topology of hypercube-variant network has more wealth than normal hypercubes in network properties. Then, the following results are presented in this paper: (1) Using the technique of edge exchange, three EDHCs are constructed in FLTQ5 and FCQ5, respectively. (2) According to the recursive structure of FLTQn and FCQn, there are three EDHCs in FLTQn and FCQn while n ≥ 6. (3) Considering that multiple faulty edges will occur randomly, the data broadcast performance of three EDHCs in FLTQn and FCQn is evaluated by simulation when 5 ≤ n ≤ 9. [ABSTRACT FROM AUTHOR]

Multiple edge-disjoint Hamiltonian cycles (EDHCs) provide the advantages of data broadcast in parallel and edge fault-tolerance in network communications. This paper investigates how to construct more EDHCs in a hypercube-variant network called crossed cube, denoted as C Q n . The topology of C Q n has more wealth than normal hypercubes in network properties, e.g., it has about half of the diameter of a hypercube with the same dimension. Then, we obtain the following results in this paper: (1) We first provide the construction of three EDHCs in C Q 6 . (2) According to the recursive structure of C Q n , we prove by induction that there exist also three EDHCs in C Q n for n ⩾ 7 . (3) Finally, we evaluate the performance of data broadcasting by simulation through three EDHCs and compare it against the best previous result in [18] using two EDHCs. In particular, our findings significantly improved the average success rate in edge fault-tolerant data broadcasting and two specific metrics concerning the broadcasting delivery time (latency). [ABSTRACT FROM AUTHOR]

• We point out that FDSC(n) is suitable as a candidate topology for DCNs. • We propose a recursive construction of a dual-CIST of FDSC(n). • The diameter of constructed CISTs is no more than 47 8 n. • As a by-product, we can use the dual-CIST to configure a protection routing. Spanning trees T 1 , T 2 , ... , T k (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 { T 1 , T 2 } in FDSC(n). In particular, the diameters of T i for i = 1 , 2 we constructed are diam (T i) = { 9 if n = 4 ; 47 8 n − 15 − 2 3 (4 ⌊ d − 4 2 ⌋ − 1) (d mod 2 + 1) if n = 2 d for d ⩾ 3. [ABSTRACT FROM AUTHOR]

The protection routing uses the multi-paths technique for integrating route discovery and route maintenance mechanisms in a network, and thus it can tolerate the failure of one component (including a node or a link). Tapolcai (2013) proposed a method showing that a network possessing two completely independent spanning trees (CISTs for short) suffices to configure a protection routing. However, it is well-known that the problem of determining whether there exist two CISTs in a graph (or network) is NP-complete. In this paper, we extend Tapolcai's method such that the protection routing is configured by a combination of multiple CISTs. The first application of such an extension is that it can be used to deal with the problem of security in transmission, which we call the secure-protection routing scheme (SPR-scheme for short). Thus, a network transmission using the SPR-scheme ensures that no node other than the destination can receive the complete message. Moreover, we show that if a message M is transmitted in a network G using the SPR-scheme configured by a combination of n CISTs, then each intermediate node of G can receive a maximum of 2 ∕ n ratio of M. From a similar idea of the extension, another application is the so-called multiple-protection routing scheme (MPR-scheme for short) which can be used to increase the capability of fault-tolerance. For assessing the performance of routing using MPR-scheme, we first propose a construction of three CISTs in the two types of Möbius cubes, which are hypercube-variant networks and are superior to hypercubes due to having a smaller diameter. For the n -dimensional Möbius cube, the diameters of CISTs we constructed are at most 14 when n = 6 and at most 2 n + 1 when n ⩾ 7. So, we configure the desired protection routing in the Möbius cubes via the three CISTs. Then, we provide simulation results to experimentally evaluate the performance of the newly proposed MPR-scheme in the n -dimensional Möbius cubes for 6 ⩽ n ⩽ 10. As an important point, our results show that the adoption of routing using MPR-scheme will result in a significant slowdown in the transmission failure rate. • We propose a construction of three CISTs in a Möbius cube M Q n . • For n ⩾ 7 , the diameters of three CISTs of M Q n we constructed are at most 2n+1. • Protection routing employing MPR-scheme can increase the capability of fault-tolerance. • Protection routing using SPR-scheme ensures that no node other than the destination can receive the complete message. [ABSTRACT FROM AUTHOR]

For the underlying graph G of a network, k (⩾ 2) spanning trees of G are called completely independent spanning trees (CISTs for short) if they are mutually inner-node-disjoint. It is well-known that determining the existence of k CISTs in a graph is an NP-hard problem, even for k = 2. Also, there exists a k -connected graph which does not possess two CISTs for any k ⩾ 2. Accordingly, researches focused on the problem of constructing multiple CISTs in some particular famous networks. Pai and Chang (2016) proposed a unified approach to recursively construct two CISTs with diameter 2 n − 1 in several n -dimensional hypercube-variant networks for n ⩾ 4 , including locally twisted cubes L T Q n and crossed cubes C Q n. Later on, new constructions showed that the diameter of two CISTs can be reduced to 2 n − 2 if n = 4 and 2 n − 3 if n ⩾ 5 for L T Q n , and to 2 n − 2 if n = 4 , 5 and 2 n − 3 if n ⩾ 6 for C Q n. In this paper, we intend to construct more CISTs for those networks that are paid attention. We develop a novel tree searching algorithm, called two-stages tree-searching algorithm (abbreviated as TS 2), which can be used to construct three CISTs of a 6-regular graph if the scale of the graph is not too large and there exist three CISTs for certain. In particular, we demonstrate that three CISTs of L T Q 6 and C Q 6 can be acquired by using TS 2. Moreover, for high-dimensional L T Q n and C Q n with n ⩾ 7 , we show that their three CISTs can be constructed by recursion. Consequently, we obtain the following results: (i) For L T Q n , the diameters of three CISTs we constructed are 11, 12 and 12 when n = 6 , and all are 2 n − 1 when n ⩾ 7 ; (ii) For C Q n , all diameters of three CISTs we constructed are 12 when n = 6 , and 2 n + 1 when n ⩾ 7. [ABSTRACT FROM AUTHOR]

Given a graph G , a set of spanning trees of G are completely independent spanning trees (CISTs for short) if for any two vertices x , y ∈ V ( G ) , the paths joining x and y on these trees have neither vertex nor edge in common, except x and y . Hasunuma [9,10] first introduced the concept of CISTs and conjectured that there are k CISTs in any 2 k -connected graph. Unfortunately, Péterfalvi [16] disproved this conjecture by constructing a k -connected graph, for each k ⩾ 2 , which does not have two CISTs. In this paper, we provide a unified approach for constructing two CISTs in several hypercube-variant networks, including hypercubes, locally twisted cubes, crossed cubes, parity cubes, and Möbius cubes. In particular, for an n -dimensional hypercube-variant network, the diameters of the constructed CISTs are 2 n − 1 . [ABSTRACT FROM AUTHOR]