Benes Network Research Articles

On the Geodetic and Strong Geodetic Parameters of Certain Hypercubic Networks

Networks derived from the hypercube are noted for their efficiency in simulating any bounded-degree communication network and hence they play a significant role in parallel machines. For a graph [Formula: see text], a geodetic cover is a subset [Formula: see text] of [Formula: see text], satisfying the condition that every vertex of [Formula: see text] other than the vertices in [Formula: see text] are covered using geodesics connecting any two vertices in [Formula: see text]. The geodetic number of [Formula: see text] is the least order of such covers in [Formula: see text]. This seminal distance-based parameter is well employed in the areas of location theory, convexity theory and game theory. The strong variant and its edge version are recent variations of the problem that were introduced in due of their applications in social networks. The most established and strong hypercube derivative networks are studied in this paper with respect to certain geodetic parameters. Notably, among the networks that we have considered, the butterfly and the Benes networks belong to the class of bipartite graphs for which the geodetic number problem and its strong version are NP-complete. We have determined the geodetic parameters that include the geodetic number, the strong geodetic number and their corresponding edge variants of the networks taken into consideration. In the process, we have identified a new class of graphs, as semi-extreme edge geodesic graphs and that the enhanced butterfly networks belong to this class. The study of such geodetic parameters in the structural models of parallel computation could be useful for the design of efficient algorithms for the structures considered.

Making Path Selection Bright: A Routing Algorithm for On-Chip Benes Networks

Making Path Selection Bright: A Routing Algorithm for On-Chip Benes Networks

A comparison between certain graph parameters of biswapped networks and their maximal twin-preserving subgraphs with applications in determining complexity

The study of different objects, such as the properties of algebraic structures, the analysis of chemical graph structures without experiment, and understanding networks through topological indices (TIs), is among the modern directions in graph theory. The analysis of different networks, such as the Benes network, butterfly network, hex-derived network and biswapped network, using TIs is one of the modern research fields. The storage and transfer of data carrying certain information has always remained a modern world problem. The optoelectronic technology used in optical transpose interconnection systems (OTIS) is efficient in outcome over an electrical system due to the minimum power consumption and broad bandwidth. By combining the use of wireless and electrical technologies, OTIS has developed into a valuable network and boosted the efficiency of contemporary optoelectronic computers. A special type of OTIS is the biswapped network over the path graph Pn , denoted by . In this article, a maximal twin-preserving subgraph of a biswapped network over the path graph Pn is constructed. Moreover, the biswapped network and its constructed subgraph are compared through ev-degree- and ve-degree-dependent TIs. In addition to this, to analyse the complexity of and through information functional-based entropies, we define ve-degree- and ev-degree-based information functionals and compute formulae for ve-degree- and ev-degree-based entropies. The analytical formulae, tables and graphical trends are used to describe a comparison between the TIs and entropies of and . It is shown that the values of TIs and entropies only differ by a constant depending on the value of n.

Monitoring edge-geodetic numbers of convex polytopes and four networks

Foucard, Krishna and Lekshmi introduced a new graph-theoretic concept in the area of network monitoring. Let G be a graph with vertex set , there exists a subset , if deleting any edge in G, one can find that there exist at least two vertices in S whose distance is changed, then we call S is a monitoring edge-geodetic set (MEG-set for short). The minimum size of the MEG-set of G is called the monitoring edge-geodetic number of G (meg(G) for short). In this paper, we study the monitoring edge-geodetic number of some well-known networks, including Ladder, butterfly, circulant and Benes networks and convex polytopes.

Arbitrary-size permutation networks using arbitrary-radix switches

Permutation networks accepting an arbitrary number of inputs using arbitrary-radix switches find their applications in a wide range of areas such as datacenter network, network on chip, and network switches. We extend the Benes network to accepting an arbitrary-size input and using arbitrary-radix switches. The resulting network, namely A2-Benes network, has a regular topology as the Benes or Clos network while using fewer switches. For arbitrary n≥1 and w≥2, the number of switches of the n×nA2-Benes network is the total number of 1's and 0's in the radix-w representations of integers 0,…,n−1, not counting the leading 0's; the number of stages is 2⌈logw⁡n⌉−1. The regular topology and fewer switches of the A2-Benes network lead to lower wiring complexity and costs.

Complexity Analysis of Benes Network and Its Derived Classes via Information Functional Based Entropies

The use of information–theoretical methodologies to assess graph-based systems has received a significant amount of attention. Evaluating a graph’s structural information content is a classic issue in fields such as cybernetics, pattern recognition, mathematical chemistry, and computational physics. Therefore, conventional methods for determining a graph’s structural information content rely heavily on determining a specific partitioning of the vertex set to obtain a probability distribution. A network’s entropy based on such a probability distribution is obtained from vertex partitioning. These entropies produce the numeric information about complexity and information processing which, as a consequence, increases the understanding of the network. In this paper, we study the Benes network and its novel-derived classes via different entropy measures, which are based on information functionals. We construct different partitions of vertices of the Benes network and its novel-derived classes to compute information functional dependent entropies. Further, we present the numerical applications of our findings in understanding network complexity. We also classify information functionals which describe the networks more appropriately and may be applied to other networks.

On Some Ev-Degree and Ve-Degree Dependent Indices of Benes Network and營ts Derived Classes

One of the most recent developments in the field of graph theory is the analysis of networks such as Butterfly networks, Benes networks, Interconnection networks, and David-derived networks using graph theoretic parameters. The topological indices (TIs) have been widely used as graph invariants among various graph theoretic tools. Quantitative structure activity relationships (QSAR) and quantitative structure property relationships (QSPR) need the use of TIs. Different structure-based parameters, such as the degree and distance of vertices in graphs, contribute to the determination of the values of TIs. Among other recently introduced novelties, the classes of ev-degree and ve-degree dependent TIs have been extensively explored for various graph families. The current research focuses on the development of formulae for different ev-degree and ve-degree dependent TIs for dimensional Benes network and certain networks derived from it. In the end, a comparison between the values of the TIs for these networks has been presented through graphical tools.

Novel CNN Accelerator Design With Dual Benes Network Architecture

We presented a novel hardware architecture that uses dual Benes networks to accelerate Convolutional Neural Network (CNN) algorithms. This architecture can reduce the need for high-speed buses and maintain a high-speed connection between execution units and memories. Also, this design can work with multiple neural network models by changing the Benes network configuration only due to the reorderability of the non-blocking networks. The proposed architecture can save the time of re-implementing the hardware design. Thus, this architecture can act as a general CNN accelerator. We successfully implemented our CNN accelerator on the Xilinx Virtex UltraScale+ VU19P and evaluated it by running the CNN model LeNet-5.

Twin vertices in fault-tolerant metric sets and fault-tolerant metric dimension of multistage interconnection networks

A set of vertices S⊆V(G) is a resolving set of a graph G if for each x,y∈V(G) there is a vertex u∈S such that d(x,u)≠d(y,u). A resolving set S is a fault-tolerant resolving set if S∖{x} is a resolving set for every x∈S. The fault-tolerant metric dimension (FTMD) β′(G) of G is the minimum cardinality of a fault-tolerant resolving set. It is shown that each twin vertex of G belongs to every fault-tolerant resolving set of G. As a consequence, β′(G)=n(G) iff each vertex of G is a twin vertex, which corrects a wrong characterization of graphs G with β′(G)=n(G) from [Mathematics 7(1) (2019) 78]. This FTMD problem is reinvestigated for Butterfly networks, Benes networks, and silicate networks. This extends partial results from [IEEE Access 8 (2020) 145435–145445], and at the same time, disproves related conjectures from the same paper.Dedicated to Dr. Bharati Rajan on her 70th birthday

Modified Zagreb Connection Indices for Benes Network and Related Classes

The study of networks such as Butterfly networks, Benes networks, interconnection networks, David‐derived networks through graph theoretical parameters is among the modern trends in the area of graph theory. Among these graph theoretical tools, the topological Indices (TIs) have been frequently used as graph invariants. TIs are also the essential tools for quantitative structure activity relationship (QSAR) as well as quantity structure property relationships (QSPR). TIs depend on different parameters, such as degree and distance of vertices in graphs. The current work is devoted to the derivation of 2‐distance based TIs, known as, modified first Zagreb connection index and first Zagreb connection index (ZC1) for r− dimensional Benes network and some classes generated from Benes network. The horizontal cylindrical Benes network (HCB(r)), vertical cylindrical Benes network (VCB(r)), and toroidal Benes network (TB(r)) are the three classes generated by identifying the vertices of the first row with the last row, the first column with the last column of the Benes network. The obtained results are also analyzed through graphical tools.

On the Secure Domination of Benes Networks

This paper focuses on the fascinating and highly active area of research on domination in graphs. In this article, the secure domination problem in graph theory is solved for Benes networks. The results obtained can be used in finding minimum secure dominating sets for an interconnection network whose vertices are the routing servers while it becomes necessary to obtain an optimal solution for designing the network defence strategy.

On some degree-based irregularity indices of certain networks

Irregularity indices are a special type of topological indices, first studied by Paul Erdös In this paper, we compute twelve degree-based irregularity topological indices for some networks like Butterfly networks, BF(r), Benes networks B(r), two important mesh derived networks namely MDN1, MDN2 with respect to the edge partition obtained for each such networks.

On Topological Indices for New Classes of Benes Network

Topological indices (TIs) transform a molecular graph into a number. The TIs are a vital tool for quantitative structure activity relationship (QSAR) and quantity structure property relationship (QSPR). In this paper, we constructed two classes of Benes network: horizontal cylindrical Benes network HCB r and vertical cylindrical Benes network obtained by identification of vertices of first rows with last row and first column with last column of Benes network, respectively. We derive analytical close formulas for general Randić connectivity index, general Zagreb, first and the second Zagreb (and multiplicative Zagreb), general sum connectivity, atom-bond connectivity ( VCB r ), and geometric arithmetic ABC index of the two classes of Benes networks. Also, the fourth version of GA and the fifth version of ABC indices are computed for these classes of networks.

On the isometric path partition problem

The isometric path cover (partition) problem of a graph is to find a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric $k$-path partition problem for $k\geq 3$ are NP-complete on general graphs. Fisher and Fitzpatrick \cite{FiFi01} have shown that the isometric path cover number of $(r\times r)$-dimensional grid is $\lceil 2r/3\rceil$. We show that the isometric path cover (partition) number of $(r\times s)$-dimensional grid is $s$ when $r \geq s(s-1)$. We establish that the isometric path cover (partition) number of $(r\times r)$-dimensional torus is $r$ when $r$ is even and is either $r$ or $r+1$ when $r$ is odd. Then, we demonstrate that the isometric path cover (partition) number of an $r$-dimensional Benes network is $2^r$. In addition, we provide partial solutions for the isometric path cover (partition) problems for cylinder and multi-dimensional grids.

Bi-Directional Benes With Large Port-Counts and Low Waveguide Crossings for Optical Network-on-Chip

We propose a new variant of the Benes network using a merge-replace-fold approach. It is to realize a large port-count optical switch with low waveguide crossings. A bidirectional switch is built instead of a unidirectional switch. Compared to the classical Benes, it requires the same number of switches but far fewer intersections. The novel designs enable a 24-percentage reduction in waveguide intersections. Moreover, it has a worst-case insertion loss of 12.01 dB and a comparable crosstalk level of −13.67 dB between the neighbouring paths. Thus, it offers an energy-efficient way to solve the optical on-chip design problem in a bidirectional manner.

A Flexible and Reconfigurable Optical Add-Drop Multiplexer for Mode Division Multiplexing Systems

Reconfigurable optical add-drop multiplexer (ROADM) is one of the key building blocks for on-chip optical networks, which can download the desired signals from the bus waveguide to the corresponding drop port and upload the local signals from the add port to the bus waveguide to achieve the signals dropping/adding between the local and the bus waveguide. In this letter, we propose a ROADM based on a Benes network for mode-division multiplexing systems. As a proof of the concept, a ROADM with three quasi-transverse electric modes (TE0-TE2) is fabricated and experimentally demonstrated, which can download signals from TEi (i = 0, 1, 2) mode channel in the bus waveguide to an arbitrary local drop port or upload local signals to TEi mode channel via an arbitrary add port. The excess loss of the fabricated device is less than 10.7 dB, and the crosstalk among different space or mode channels is lower than -13.7 dB. Finally, a 12 Gbps data transmission experiment is carried out to verify the performance of the device. The proposed device is reconfigurable and scalable, and thus, it is expected to be used for optical data processing in the mode-division multiplexing systems.

A Wavelength-Selective Multiwavelength Ring-Assisted Mach–Zehnder Interferometer Switch

Wavelength-selective switches have been proposed for datacenter use to enhance the scalability of their networks to help in meeting ever-increasing traffic demands. We have previously demonstrated a 4 × 4 ring-based crossbar silicon photonic switch in which each cross-point contained three ring pairs to partition the free spectral range (FSR) into three equal regions to reduce wavelength tuning range per ring pair—thereby reducing both the tuning power consumption and stress on the rings—while maintaining full routing flexibility. However, the question of scalability remains for such a crossbar switch in which 96 signal pads—one routed to each ring—are required to fully control it. In this article we present a 4-port silicon photonic ring-assisted Mach–Zehnder interferometer (RAMZI) switch with multiple-sized rings per switching elements in a Benes network configuration to reduce the number of electrical pads required to 36 signal pads for the equivalent number of wavelength-selective switch elements per switch cell. The switch is 500 μm × 3 mm in size and is packaged on a custom PCB. Another advantage the RAMZI switch has over the crossbar switch is that the loss through the switch is not path-dependent due to its balanced path configuration. In the crossbar switch, the difference between the shortest and longest paths is 2(N − 1) switching elements. Finally, we present results from the fabricated switch and the outlook for further scaling of the switch architecture.

Assessment of reliability and signature of Benes network using universal generating function

This paper explores the reliability and signature analysis of the multistage interconnection network, especially in a non-blocking network that can be rearranged. In general structure, blocking is a crucial problem faced by multistage interconnection networks. Hence to discuss this issue, here we have taken a most significant part of rearrangeable non-blocking networks, namely Benes network which is the utmost popular rearrangeable non-blocking multistage interconnection networks. In the current study, we propose to assess the reliability having both ‘independent and identically and non-identically distributed components’ of Benes network in the three contexts, viz. terminal, broadcast and network with the help of universal generating function. With the aid of Owen's method, an efficient algorithm was developed to enumerate the signature and mean time to failure of the Benes network having independent, identically distributed lifetime components.

Toward a High-Performance and Low-Loss Clos–Benes-Based Optical Network-on-Chip Architecture

As chip multiprocessors (CMPs) keep growing in capability, on-chip communication efficiency is crucial to the overall performance. However, on-chip networks based on electronic switches suffer from excessive power consumption and limited performance. In order to take advantages of optical interconnect, we propose an optimized design toward a high-performance and low-loss Clos–Benes-based hierarchical optical network-on-chip (NoC) for large-scale CMPs. We propose several key techniques, including a loss-aware adaptive (LAA) routing for intraswitch Benes network, a priority-based round-Robin virtual output queue selection and a $Q$ -learning-based heuristic routing for interswitch Clos network, and a local transfer link (LTL) technique to improve the traffic locality. A case study on a 256-core CMP under uniform traffic shows that the network throughput is increased by 346.7%, 61%, and 12.9%, respectively, than the mesh, fattree, and the traditional generic Clos–Benes optical NoC. On average of a set of real applications, the application end-to-end delay is reduced by 47.6%, 28.2%, and 19.4%, respectively, than the mesh, fattree, and the traditional generic Clos–Benes network. Meanwhile, the average optical power loss is decreased by 11.8% and 49.9%, respectively, as compared to the mesh and fattree. As compared to a baseline Clos–Benes network, the use of LAA routing together with the LTL could reduce the average optical power loss by 28.4%.

Approximation Algorithms for the Achromatic Number of Butterfly and Beneš Networks

Abstract Let G = (V,E) be a graph. Then the achromatic number for the graph is the largest integer m in such a way that there is a partition of V into disjoint independent sets (V1, V2,…,Vm) satisfying the condition that for each pair of distinct sets Vi, Vj, Vi ∪ Vj is a dependent set in G. To determine the achromatic number for Butterfly and Benes networks, in this paper we have presented the approximate algorithms.