Bubble-sort Networks Research Articles

Double declined subnetwork reliability analysis in bubble-sort networks under node fault model

The n-dimensional bubble-sort network Bn is a competitive interconnection network that could be used to construct large-scale multiprocessor computer systems. Let p be the reliability of each node in Bn and let Rn,n−2(p) be the Bn−2-subnetwork reliability (i.e., double declined subnetwork reliability) in Bn under the equiprobability node fault model. In this paper, the bounds on Rn,n−2(p) are derived for n≥4, and an algorithm is given to further verify the accuracy of the lower and upper bounds. The MTTFs that maintain the healthy state of different numbers of disjoint Bn−2's in Bn under different partition patterns are calculated. Simulations are also carried out to verify the MTTFs. Both the theoretical result and the simulating result show that using the flexible partition pattern will make the network more reliable in the sense of double declined subnetwork availability.

On strong Menger connectivity of [formula omitted]-bubble-sort networks

Let t≥0, r≥0 be two integers and G be a connected graph. Let S‾⊆V(G) and E‾⊆E(G) be any sets satisfying |S‾|≤t and |E‾|≤t. For any vertices u1, u2 of G−S‾ (resp. G−E‾) with u1≠u2, if G−S‾ (resp. G−E‾) has min{dG−S‾(u1),dG−S‾(u2)} (resp. min{dG−E‾(u1),dG−E‾(u2)}) vertex (resp. edge) disjoint paths connecting u1 and u2, then G is t-strongly Menger vertex (resp. edge) connected or briefly t-SMVC (resp. t-SMEC). If G is t-SMVC (resp. t-SMEC) for any set S‾ (resp. E‾) satisfying |S‾|≤t with δ(G−S‾)≥r (resp. |E‾|≤t with δ(G−E‾)≥r), then G is t-SMVC (resp. t-SMEC) of order r. We show that (n,k)-bubble-sort network Bn,k is (k−2)-SMVC, (n−3)-SMEC of order 1, (2n−8)-SMEC of order 2 and (3n−15)-SMEC of order 3. Moreover, we give the conditions when the four bounds are sharp respectively.

A parallel algorithm for constructing multiple independent spanning trees in bubble-sort networks

The use of multiple independent spanning trees (ISTs) for data broadcasting in networks provides a number of advantages, including the increase of fault-tolerance and secure message distribution. Thus, the designs of multiple ISTs on several classes of networks have been widely investigated. Kao et al. (2019) [18] proposed an algorithm to construct independent spanning trees in bubble-sort networks. The algorithm is executed in a recursive function and thus is hard to parallelize. In this paper, we focus on the problem of constructing ISTs in bubble-sort networks Bn and present a non-recursive algorithm. Our approach can be fully parallelized, i.e., every vertex can determine its parent in each spanning tree in constant time. This solves the open problem from the paper by Kao et al. Furthermore, we show that the total time complexity O(n⋅n!) of our algorithm is asymptotically optimal, where n is the dimension of Bn and n! is the number of vertices of the network.

Hyper star fault tolerance of bubble sort networks

Let H and M be two connected subgraphs of an interconnection network G. If the removal of any minimum H-structure-cut (resp. minimum H-substructure-cut) splits G into exactly two components, one of which is isomorphic to M, then G is said to be hyperH|M-connected (resp. hyper sub-H|M-connected). The bubble sort graph Bn is one of alternative interconnection networks for multiprocessor systems. Let n≥7 and 2≤r≤4. Denote by K1 the complete graph on one vertex and denote by K1,r a star on 1+r vertices. In this paper, we prove that (i) both the K1,r-structure connectivity and the sub-K1,r-structure connectivity of Bn are ⌈n−12⌉; and (ii)Bn is hyper K1,r|K1-connected and hyper sub-K1,r|K1-connected.

Extra (component) connectivity and diagnosability of bubble sort networks

The generalizations of traditional connectivity and diagnosability, h-component connectivity and diagnosability along with h-extra connectivity and diagnosability are significant parameters to characterize invulnerability of multiprocessor systems. In this paper, we determine that h-extra connectivity of bubble sort networks Bn is κh(Bn)=(h+1)(n−1)−h(h+3)2 for 0≤h≤n2. Afterwards, we show that h-extra diagnosability of Bn under PMC model and MM⁎ model is th(Bn)=(h+1)(n−1)−h(h+3)2+h. In addition, we prove that (h+1)-component connectivity of Bn is cκh+1(Bn)=h(n−1)−h(h+1)2+1, and h-component diagnosability of Bn under PMC model and MM⁎ model is cth(Bn)=h(n−1)−h(h+1)2+1 for 1≤h≤n2.

The (strong) structure connectivity and (strong) substructure connectivity of the [formula omitted]-bubble-sort network

A network is always viewed as a simple connected graph. The connectivity of a network can evaluate its fault tolerance and reliability. Since the faulty elements of a graph may have special structures, the (strong) structure connectivity and (strong) substructure connectivity were proposed as generalizations of connectivity. We give the (strong) structure connectivity and (strong) substructure connectivity of the (n,k)-bubble-sort network Bn,k in this paper, where the special structures are complete graphs.

Reliability assessment for modified bubble-sort networks

In large-scale multiprocessor systems, it is significant to estimate the reliability of the interconnection networks. The κk and λk are two generalized measurements of reliability for networks. The κk (λk) of G is the minimum cardinality of a set of vertices (edges) of G, if any, whose deletion disconnects G, and every remaining component has minimum degree at least k. For the modified bubble-sort networks MBn, this paper proves κk(MBn)=λk(MBn)=2k(n−k) for k≤n/2. The conclusions extend some known results in the literature, imply that to make a modified bubble-sort network MBn disconnected with all remaining vertices have at least k neighbors, at least 2k(n−k) vertices or edges have to be faulty.

Vertex-pancyclicity of the (n,k)-bubble-sort networks

The cycle embedding is an important problem of networks, which can determine the fault tolerance of the networks. A network can be viewed as a graph. Let m be an integer with m≥4, G be a graph and w∈V(G) be an arbitrary vertex. The graph G is vertex-pancyclic if G has a cycle Cl of length l with w∈V(Cl) for every l∈{3,4,⋯,|V(G)|} and G is m-weak-vertex-pancyclic if G has a cycle Cl of length l with w∈V(Cl) for every l∈{m,m+1,⋯,|V(G)|}. Let G′ be a bipartite graph and w′∈V(G′) be an arbitrary vertex. The graph G′ is vertex-bipancyclic if G′ has a cycle Ch of length h with w′∈V(Ch) for any even integer h with 4≤h≤|V(G′)|. In this paper, we study the cycle embedding in the (n,k)-bubble-sort network Bn,k. We obtain that (1) Bn,1 is vertex-pancyclic for n≥3. (2) Bn,n−1 is vertex-bipancyclic for n≥4. (3) B4,2 and B5,2 are 6-weak-vertex-pancyclic and B5,3 is vertex-pancyclic. (4) Bn,k is vertex-pancyclic for n≥6 with 2≤k≤n−2 and every constructed cycle of Bn,k contains a residual edge for n≥4 with 2≤k≤n−2.

The fault tolerance of [formula omitted]-bubble-sort networks

The h-good neighbor connectivity κh(G) is an important parameter to evaluate the fault tolerance of an interconnection network G. So far, almost all known results of κh(G) are about small h′s except the hypercubes, the star graphs, the k-ary n-cubes and hierarchical hypercube network HHCn et al. In this paper, we focus on κh(Bn,k) for the (n,k)-bubble-sort network Bn,k, which is a generalization of the bubble-sort graph Bn as it is a special case of (n,k)-bubble-sort networks for k=n−1. We show that κh(Bn,k)=n+h(k−2)−1 for 2≤k≤⌈n+32⌉ and 0≤h≤⌊n−32⌋, which implies that at least n+h(k−2)−1 vertices of Bn,k have to be deleted to get a disconnected graph without vertices of degree less than h.

Generalized measures of fault tolerance for bubble sort networks

The κk and λk are two generalized measurements for fault tolerance of large-scale processing system. For the bubble sort networks Bn, this paper determines κk(Bn)=λk(Bn)=2k(n−k−1) for k ≤ n/2. The results show that to disconnect Bn with each vertex in resulting graph has at least k fault-free neighbors, at least 2k(n−k−1) faulty vertices or faulty edges have to occur. In particular, the results also settle affirmatively a conjecture proposed by Shi and Wu (Acta Math. Appl. Sin-E., 33 (4)(2017), 933–944).

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.

Dual-CISTs: Configuring a Protection Routing on Some Cayley Networks

A set of $k~(\geqslant 2)$ spanning trees in the underlying graph of a network topology is 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. However, it has been proved that determining if there exists a dual-CIST in a graph is an NP-hard problem. Kwong et al. [IEEE/ACM Transactions Networking 19(5) 1543–1556, 2011] defined that a routing is protected, if there is an alternate with loop-free forwarding, when a single link or node failure occurs. Shortly afterward, Tapolcai [Optim. Lett. 7(4) 723–730, 2013] showed that a network possessing a dual-CIST suffices to establish a protection routing. It is well-known that Cayley graphs have a large number of desirable properties of interconnection networks. Although many results of constructing dual-CISTs on interconnection networks have been proposed in the literature, so far, the work has not been dealt with on Cayley graphs due to that their expansions are in exponential scalability. In this paper, we try to make a breakthrough of this work on some famous subclasses of Cayley graphs, including alternating group networks, bubble-sort network, and star networks. We first propose tree searching algorithms for helping the construction of dual-CISTs on low-dimensional networks. Then, by inductive construction, we show that dual-CISTs on high-dimensional networks can also be constructed agreeably. As a result, we can configure protection routings by using the constructed dual-CISTs. In addition, we complement some analysis with a simulation study of the proposed construction to evaluate the corresponding performance.

Estimating the subsystem reliability of bubblesort networks

The exact reliability of a complicated network system is usually difficult to determine, and numerical approximations may play a crucial role in indicating the reliable probability that a system is still operational under a specified suite of conditions. In this paper, we establish upper and lower bounds on the first-order subsystem reliability of bubblesort networks using the probabilistic fault model. Numerical results show that the curves of upper- and lower-bounded reliability are in good agreement, especially when the node reliability is at a low level.

Embedded connectivity of recursive networks

Let Gn be an n-dimensional recursive network. The h-embedded connectivity ζh(Gn) (resp. edge-connectivity ηh(Gn)) of Gn is the minimum number of vertices (resp. edges) whose removal results in disconnected and each vertex is contained in an h-dimensional subnetwork Gh. This paper determines ζh and ηh for the hypercube Qn and the star graph Sn, and η3 for the bubble-sort network Bn.

Subnetwork preclusion for bubble-sort networks

For two positive integers n and m with n>m, the Gn−m preclusion node (resp. link) number Fm(Gn) (resp. fm(Gn)) of an n-dimensional interconnection network Gn is the minimum number of nodes (resp. links), if any, whose deletion results in a network with no subnetwork isomorphic to Gn−m. The n-dimensional bubble-sort network Bn is one of the most attractive interconnection networks for multiprocessor systems. In this paper, we prove that F2(Bn)=f2(Bn)=n(n−1) for n≥6.

Conditional connectivity of recursive interconnection networks respect to embedding restriction

Large-scale multiprocessor systems always take some recursive interconnection networks as underlying topologies. Let Gn be an n-dimensional recursive interconnection network. The m-embedding-restricted connectivity ζm(Gn) (resp. the m-embedding-restricted edge connectivity ηm(Gn)) of Gn is the cardinality of a minimum subset of nodes (resp. edges), if any, whose deletion disconnects Gn and each node of the remaining components lies in an undamaged m-dimensional sub-network Gm. In this paper, we present some relationships between the proposed indices and other conditional connectivity indices in general recursive interconnection networks. We give some bounds on these two indices in k-ary n-cubes and bubble-sort networks. In addition, we determine these two indices in k-ary n-cubes and bubble-sort networks in some cases.

Transposition networks as a class of fault-tolerant robust networks

The paper proposes designs of interconnection networks (graphs) which can tolerate link failures. The networks under study belong to a subclass of Cayley graphs whose generators are subsets of all possible transpositions. We specifically focus on star and bubble sort networks. Our approach is to augment existing dimensions (or generators) with one or more dimensions. If the added dimension is capable of replacing any arbitrary failed dimension, it is called a wildcard dimension. It is shown that, up to isomorphism among digits used in labeling the vertices, the generators of the star graph are unique. The minimum number of extra dimensions needed to acquire i wildcard dimensions is derived for the star and bubble sort networks. Interestingly, the optimally augmented star network coincides with the Transposition network, T/sub n/. Transposition networks are studied rigorously. These networks are shown to be optimally fault tolerant. T/sub n/ is also shown to possess wide containers with short length. Fault diameter of T/sub n/ is shown to be n. While the T can efficiently embed star and bubble sort graphs, it can also lend itself to an efficient embedding of meshes and hypercubes.