Models For Interconnection Networks Research Articles

Boundary enumerator polynomial of hypercubes in Fibonacci cubes

Hypercubes and their special subgraphs, Fibonacci cubes, have been proposed as basic models for interconnection networks. By the recursive nature of Fibonacci cubes, they contain many smaller dimensional hypercubes as subgraphs. In this work, we consider the boundary enumerator polynomial of the k-dimensional hypercubes in Fibonacci cubes of dimension n. We obtain recursive relations satisfied by these polynomials.

From Subkautz Digraphs to Cyclic Kautz Digraphs

The Kautz digraphs K(d, ℓ) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related to these, the cyclic Kautz digraphs CK(d, ℓ) were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed. In this paper we propose a new approach to the cyclic Kautz digraphs by introducing the family of the subKautz digraphs sK(d, ℓ), from where the cyclic Kautz digraphs can be obtained as line digraphs. This allows us to give exact formulas for the distance between any two vertices of both sK(d, ℓ) and CK(d, ℓ). Moreover, we compute the diameter and the semigirth of both families, also providing efficient routing algorithms to find the shortest path between any pair of vertices. Using these parameters, we also prove that sK(d, ℓ) and CK(d, ℓ) are maximally vertex-connected and super-edge-connected. Whereas K(d, ℓ) are optimal with respect to the diameter, we show that sK(d, ℓ) and CK(d, ℓ) are optimal with respect to the mean distance, whose exact values are given for both families when ℓ = 3. Finally, we provide a lower bound on the girth of CK(d, ℓ) and sK(d, ℓ).

$q$-counting hypercubes in Lucas cubes

Lucas and Fibonacci cubes are special subgraphs of the binary hypercubes that have been proposed as models of interconnection networks. Since these families are closely related to hypercubes, it is natural to consider the nature of the hypercubes they contain. Here we study a generalization of the enumerator polynomial of the hypercubes in Lucas cubes, which $q$-counts them by their distance to the all 0 vertex. Thus, our bivariate polynomials refine the count of the number of hypercubes of a given dimension in Lucas cubes and for $q=1$ they specialize to the cube polynomials of Klavžar and Mollard. We obtain many properties of these polynomials as well as the $q$-cube polynomials of Fibonacci cubes themselves. These new properties include divisibility, positivity, and functional identities for both families.

Rapid Calculation of Max-Min Fair Rates for Multi-Commodity Flows in Fat-Tree Networks

Max-min fairness is often used in the performance modeling of interconnection networks. Existing methods to compute max-min fair rates for multi-commodity flows have high complexity and are computationally infeasible for large networks. In this work, we show that by considering topological features, this problem can be solved efficiently for the fat-tree topology that is widely used in data centers and high performance compute clusters. Several efficient new algorithms are developed for this problem, including a parallel algorithm that can take advantage of multi-core and shared-memory architectures. Using these algorithms, we demonstrate that it is possible to find the max-min fair rate allocation for multi-commodity flows in fat-tree networks that support tens of thousands of nodes. We evaluate the run-time performance of the proposed algorithms and show improvement in orders of magnitude over the previously best known method. We further demonstrate a new application of max-min fair rate allocation that is only computationally feasible using our new algorithms.

Cube-Connected Circulants as Efficient Models for Interconnection Networks

We introduce cube-connected circulants as efficient models for communication networks. We give an algorithm for computing a shortest path between any pair of vertices in a cube-connected circulant. We give formulas for the diameter of a cube-connected circulant and the distance between any pair of vertices in such a graph. Then we give an embedding of cube-connected circulants into hypercubes, and an embedding of hypercubes into cube-connected circulants. We show cube-connected circulants outperform a few well-known network structures in several invariants.

Upper Bounds on the Domination and Total Domination Number of Fibonacci Cubes

One of the basic model for interconnection networks is the $n$-dimensional hypercube graph $Q_n$ and the vertices of $Q_n$ are represented by all binary strings of length $n$. The Fibonacci cube $\Gamma_n$ of dimension $n$ is a subgraph of $Q_n$, where the vertices correspond to those without two consecutive 1s in their string representation. In this paper, we deal with the domination number and the total domination number of Fibonacci cubes. First we obtain upper bounds on the domination number of $\Gamma_n$ for $n\ge 13$. Then using these result we obtain upper bounds on the total domination number of $\Gamma_n$ for $n\ge 14$ and we see that these upper bounds improve the bounds given in [1].

The spectra of subKautz and cyclic Kautz digraphs

Kautz digraphs K(d,ℓ) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz CK(d,ℓ) and the subKautz sK(d,2) digraphs were recently introduced by Böhmová, Huemer and the author.In this paper we propose a new method to obtain the complete spectra of subKautz sK(d,2) and cyclic Kautz CK(d,3) digraphs, for all d≥3, through the Hoffman–McAndrew polynomial and regular partitions. This approach can be useful to find the spectra of other families of digraphs with high regularity.

The wide-diameter of [formula omitted

The l-wide-diameter of a graph G is the minimum integer d for which there exist at least l internally disjoint paths of length at most d between any two distinct vertices in G. This parameter measures the fault tolerance and transmission delay in communication networks modelled by graphs. Hypercube-like graphs are widely used as graph models for interconnection networks. In this paper, we study the wide-diameters of a special type of hypercube-like graphs: the Z-cubes Zn,k. It is known that Zn,k have diameter at most ⌈nk+1⌉+2k in Zhu (2015). We show that the k-wide-diameter of Zn,k is at most ⌈nk+1⌉+2k+k+4. In particular, if k=⌈log2n−2log2log2n⌉, then the k-wide-diameter of Zn,k is (1+o(1))nlog2n, which is asymptotically optimal.

Recursive cubes of rings as models for interconnection networks

We study recursive cubes of rings as models for interconnection networks. We first redefine each of them as a Cayley graph on the semidirect product of an elementary abelian group by a cyclic group in order to facilitate the study of them by using algebraic tools. We give an algorithm for computing shortest paths and the distance between any two vertices in recursive cubes of rings, and obtain the exact value of their diameters. We obtain sharp bounds on the Wiener index, vertex-forwarding index, edge-forwarding index and bisection width of recursive cubes of rings. The cube-connected cycles and cube-of-rings are special recursive cubes of rings, and hence all results obtained in the paper apply to these well-known networks.

Some new models for multiprocessor interconnection networks

A multiprocessor system can be modeled by a graph G. The vertices of G correspond to processors while edges represent links between processors. To find suitable models for multiprocessor interconnection networks (briefly MINs), one can apply tools and techniques of spectral graph theory. In this paper, we extend some of the existing results and present several graphs which could serve as models for efficient MINs based on the small values of the previously introduced graph tightness. These examples of possible MINs arise as a result of some well-known and widely used graph operations. We also examine the suitability of strongly regular graphs (briefly SRGs) to model MINs, and prove the uniqueness of some of them.

The restricted edge-connectivity and restricted connectivity of augmented k-ary n-cubes

Augmented k-ary n-cube is proposed as a new interconnection network model by Xiang and Steward [Augmented k-ary n-cubes, Inform. Sci. 181(1) (2011), pp. 239–256]. For a connected graph G, an edge-cut (vertex-cut) S is called a restricted edge-cut (restricted vertex-cut) if G–S contains no isolated vertices. The restricted edge-connectivity (restricted connectivity) of G, denoted by (), is the minimum cardinality over all restricted edge-cuts (vertex-cuts) of G. In this paper, we completely determine the restricted (edge-)connectivity of . Precisely, for ; for , for , for and , but does not have restricted vertex-cut.

Cascading failures in interconnected networks with dynamical redistribution of loads

Cascading failures of loads in isolated networks and coupled networks have been studied in the past few years. In most of the corresponding results, the topologies of the networks are destroyed. Here, we present an interconnected network model considering cascading failures based on the dynamic redistribution of flow in the networks. Compared with the results of single scale-free networks, we find that interconnected scale-free networks have higher vulnerability. Additionally, the network heterogeneity plays an important role in the robustness of interconnected networks under intentional attacks. Considering the effects of various coupling preferences, the results show that there are almost no differences. Finally, the application of our model to the Beijing interconnected traffic network, which consists of a subway network and a bus network, shows that the subway network suffers more damage under the attack. Moreover, the interconnected traffic network may be more exposed to damage after initial attacks on the bus network. These discussions are important for the design and optimization of interconnected networks.

Conditional diagnosability and strong diagnosability of Split-Star Networks under the PMC model

An interconnection network's diagnosability is an important measure of its self-diagnostic capability. The classical problems of fault diagnosis are explored widely. The conditional diagnosability is proposed by Lai et al. as a new measure of diagnosability, which can better measure the diagnosability of regular interconnection networks. The conditional diagnosability is an important indicator of the robustness of a multiprocessor system in presence of failed processors. Furthermore, a multiprocessor system is strongly t-diagnos-able, if it is t-diagnosable and can achieve diagnosability t+1 except for the case where a node's neighbors are all faulty. The conditional diagnosability and strong diagnosability were proposed later to better reflect the networks' self-diagnostic capability under more realistic assumptions. In this paper, we determine the conditional diagnosability of an n-dimensional Split-Star Network (denoted as Sn2), a well-known interconnection network model for multiprocessor systems, under the PMC (Preparata, Metze, and Chien) model. We show that the conditional diagnosability of Sn2(n≥4) is 8n−23, which is about four times of its traditional diagnosability. As a byproduct, the strong diagnosability of Sn2 is also obtained.

Conditional diagnosability of arrangement graphs under the PMC model

Processor fault diagnosis has played an important role in measuring the reliability of a multiprocessor system, and the diagnosabilities of many well-known multiprocessor systems have been investigated. The conditional diagnosability has been widely accepted as a measure of diagnosability by assuming an additional condition that any fault-set can't contain all the neighbors of any node in a multiprocessor system. This paper considers the conditional diagnosability of an (n,k)-arrangement graph An,k, a flexible interconnection network model for multiprocessor systems, under the classical PMC diagnostic model, and determines that the conditional diagnosability of An,k(k≥2,n≥k+2) is (4k−4)(n−k)−3, which is about four times its traditional diagnosability.

Performance modelling of adaptive routing communication networks in multi-cluster systems under bit-reversal traffic

Several analytical models of interconnection networks of multi-cluster systems under uniform traffic pattern have been proposed in the literature. However, there has been hardly any work reported yet that deals with other important non-uniform traffic patterns found in many parallel applications. In this paper we propose a new analytical model of adaptive routing in fat-tree-based interconnection networks in the presence of traffic pattern generated by bit-reversal permutation, which is an important communication operation in parallel applications such as FFT computations. The model is validated through comprehensive simulation, through which the proposed model exhibited a good degree of accuracy for various system organisations and under different working conditions.

A new characterization and a recognition algorithm of Lucas cubes

Graph Theory Fibonacci and Lucas cubes are induced subgraphs of hypercubes obtained by excluding certain binary strings from the vertex set. They appear as models for interconnection networks, as well as in chemistry. We derive a characterization of Lucas cubes that is based on a peripheral expansion of a unique convex subgraph of an appropriate Fibonacci cube. This serves as the foundation for a recognition algorithm of Lucas cubes that runs in linear time.

L-Networks: A Topological Model for Regular 2D Interconnection Networks

A complete family of Cayley graphs of degree four, denoted as L-networks, is considered in this paper. L-networks are 2D mesh-based topologies with wrap-around connections. L-networks constitute a graph-based model which englobe many previously proposed 2D interconnection networks. Some of them have been extensively used in the industry as the underlying topology for parallel and distributed computers of different scales. Tori, twisted and doubly twisted tori, toroidal diagonal meshes, chordal rings, and circulant graphs are, among others, members of the L-network family. Therefore, many results obtained in previous studies on these networks can be deduced from the general framework presented in this work. In addition, the network model presented in this work allows for new results on the domain of low-degree interconnection networks. Particularly, closed expressions for the graph distance properties have been derived and an optimal routing algorithm of constant complexity is provided. Since symmetry has a big impact on network performance, we have also identified which L-networks are symmetric by studying their group of automorphisms. Finally, a very simple model that predicts the performance of L-networks is also presented. Such model has been contrasted with empirical evaluation.

The (Δ, D) and (Δ, N) Problems for New Amsterdam and Manhattan Digraphs

We give a quasi-complete solution of the (Δ, N) problem for two well-known families of digraphs used as good models for large interconnection networks. In our study we also relate both families, the New Amsterdam and Manhattan digraphs, with the double-step graphs (or circulant graphs with degree two).

An accurate performance model for network-on-chip and multicomputer interconnection networks

In this paper, we present a mathematical background for a new approach for performances modeling of interconnection networks, based on analyzing the packet blocking and waiting time spent in each channel passing through all possible paths in the channel dependency graph. We have proposed a new, simple and very accurate analytical model for deterministic routing in wormhole networks, which is general in terms of the network topology and traffic distribution. An accurate calculation of the variance of the service time has been developed, which overcomes the rough approximation used, as a rule, in the existing models. The model supports two-dimensional mesh topologies, widely used in network-on-chip architectures, and multidimensional topologies, popular in multicomputer architectures. It is applicable even for irregular topologies and arbitrary application-specific traffic. Results obtained through simulation show that the model achieves a high degree of accuracy.

Evaluation of Controllable Load Management for Frequency Regulation in Cluster Interconnected Power Networks

This paper presents the effectiveness evaluation approach of demand side management (dema nd control) for frequency regulation in regional power networks (so called Cluster-oriented networks). The analytical model of the frequency fluctuation for a three-cluster interconnected network model was developed with MATLAB/SIMULINK. The control method is similar to the conventional TBC (Tie Line Bias Control) widely used in power grids. In order to maintain the quality of frequency, Battery Energy Storage Systems (BESS) and controllable loads such as heat pumps installed in the customer side will play an effective role in clusters. By the results of conducted simulations, it shows that the proposed new frequency regulation scheme by utilizing controllable loads such as battery charges and heat pumps retains effectiveness to suppress frequency deviation when a large amount of natural renewable energy is installed in the regional cluster-oriented networks.