On the Topological Properties of the Arrangement-Star Network

Journal of Systems Architecture 48 (2003) 325–336 www.elsevier.com/locate/sysarc On the topological properties of the arrangement–star network A.M. Awwad a c a,* , A. Al-Ayyoub b, M. Ould-Khaoua c Department of Computer Science, Zarka Private University, Zarka 13110, Jordan b Faculty of Computer Studies, Arab Open University, Amman 11953, Jordan Department of Computer Science, University of Glasgow, Glasgow G12, 8RZ, UK Abstract This paper proposes a new interconnection network, referred to as the arrangement–star network, which is constructed from the product of the star and arrangement networks. Studying this new network is motivated by the good qualities it exhibits over its constituent networks, the star and arrangement networks. The star network has been a research focus for quite a long time until recently when the algorithm development on the star network turned out to be cumbersome. The arrangement network as a generalized class for the star network offers no solution in that direction. The arrangement–star network, on the other hand, makes it possible to efficiently embed grids, pipelines, as well as other computationally important topologies in a very natural manner. Furthermore, the fact that the product of the star and arrangement networks comes with little increase in the network diameter and a better result on communication cost, motivates further investigation for this new alternative, the arrangement–star network. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Star network; Arrangement network; Product network; Hierarchical structure; Vertex symmetry; Parallel algorithms 1. Introduction During the last decade a wide variety of interconnection networks have been investigated [3,4,9,10]. The star network [3,6] is one example of networks that have been thoroughly investigated. Among the investigated issues for the star network are the basic topological properties [3], parallel path characterization [12], and embedding [21,22]. Akers and coworkers [2,3] have shown that the star network has several advantages over the bi- * Corresponding author. E-mail addresses: ahmad_awwad@zpu.edu.jo (A.M. Awwad), ayyoub@acm.org (A. Al-Ayyoub), mohamed@dcs. gla.ac.uk (M. Ould-Khaoua). nary n-cube including a smaller diameter, smaller average diameter and lower degree for a fixed network size. The star network has also been shown to be edge and vertex symmetric [3] and is maximally fault tolerant [12]. Furthermore, a limited number of parallel algorithms for solving some well-known problems on the star network have been reported in the literature, including computing fast Fourier transforms [12], matrix decomposition [5], broadcasting [20], and sorting [23]. The star network, however, has some drawbacks [4]. One major problem of the star network is related to its scalability. The size of the star network increases according to a factorial function, and thus grows very rapidly. Despite its attractive topological properties, the star network 1383-7621/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S1383-7621(03)00020-1 326 A.M. Awwad et al. / Journal of Systems Architecture 48 (2003) 325–336 has not been used in practical systems yet. One reason for this may be related to the difficulty in developing efficient parallel algorithms for common parallel applications, e.g. matrix computation, on this network. Mapping of data and tasks on the star network is not as obvious as it is the case for the hypercube and mesh [15,25]. In an attempt to address the scalability problem in the star network, Day and Tripathi [10] have proposed the arrangement network as a generalization of the star network. The arrangement network slightly improves the scalability problem of the star network while preserving its desirable properties. Since its introduction, there has been little work devoted to the development of new algorithms for this topology. In fact, the arrangement network inherits the major difficulties in developing efficient algorithms that can fully take advantage of its attractive topological properties. The network product has recently been investigated in [13,27] as a network-theoretical framework for generating and analysing interconnection network topologies. Day and Al-Ayyoub [13] have used this framework to investigate properties of existing networks such as scalability, vertex symmetry, routing, broadcasting, embedding, recursive structure, and the existence of maximum-size families of node-disjoint paths along with some results about node-connectivity and an upper bound for the fault-diameter. Several other researchers have investigated the network product on existing networks. For instance, Das and Banerjee [9] has studied the network product of the binary n-cube and Peterson networks. Al-Ayyoub and Day [4] have shown that the hyperstar (a product of star networks) outperforms many other product networks in various respects. Other examples of product networks that have been studied in the literature include the hyper-deBruijn [19], star-cube [14], and mesh-connected tree [17]. This paper considers the network product of the arrangement and star networks in an attempt to enhance the topological characteristics of these two networks with the elegant capabilities of product networks [13]. As we shall see below, our study reveals that the arrangement–star has superior topological properties over both the star and arrangement networks. Furthermore, a major contribution of this paper is the development of efficient frameworks for developing parallel algorithms on the proposed network. This study will show that these algorithmic frameworks enable the new network to provide efficient support for an important class of parallel applications that are based on grid and pipeline views. The rest of the paper is organized as follows. Section 2 provides the necessary notation and definitions, and then formally presents the arrangement–star. Section 3 discusses some general topological properties of the arrangement–star network. Section 4 conducts a comparison on some of the basic properties of the star, arrangement and arrangement–star networks. Section 5 discusses and proves the hierarchical structure of the arrangement–star network. Section 6 develops algorithmic frameworks to support important classes of parallel applications that are based on grid and pipeline views. Section 7 shows that the algorithmic frameworks enable the arrangement– star network to outperform both star and cube networks in terms of communication cost required to support grid and pipeline-based applications. Finally, Section 8 concludes this study. 2. Notation and definitions The n-star network, denoted by Sn , has n! nodes each labelled with a unique permutation on hni ¼ f1; . . . ; ng. Any two nodes are connected if, and only if, their corresponding permutations differ in exactly the first and any other position. Fig. 1 shows the 4-star network with 4 groups each containing 6 vertices (i.e. four copies of 3-star networks). The diameter and the node degree of Sn are b32ðn  1Þc and n  1, respectively [3]. The ðm; kÞ-arrangement network, denoted by Am;k where 1 6 k 6 m, has m!=ðm  kÞ! nodes. Each node is labelled with a unique arrangement of k symbols chosen from hmi. The network has a diameter b32kc and the node is degree kðm  kÞ [10]. Two nodes are connected if, and only if, they differ in exactly one of their k symbols. Fig. 2 shows the topology of A4;2 . The network product is an elegant mathematical representation for studying interconnection A.M. Awwad et al. / Journal of Systems Architecture 48 (2003) 325–336 327 the set of edges in G1 , and E2 is the set of edges in G2 , the network-product of G1 and G2 is an undirected network G1 G2 ¼ ðV ; EÞ, where V ¼ fhx; yijx 2 V1 and y 2 V2 g and E ¼ fðhx1 ; yi; hy1 ; yiÞj ðx1 ; y1 Þ 2 E1 g [ fðhx; x2 i; hx; y2 iÞjðx2 ; y2 Þ 2 E2 g. A node X ¼ hx1 ; x2 i in G ¼ G1 G2 has an address consisting of two parts, one coming from G1 and the other coming from G2 . We will denote the earlier part by lpðX Þ ¼ x1 and the later part by rpðX Þ ¼ x2 . Definition 2. The arrangement–star network is the cross product of the n-star network and arrangement network given by ASn;m;k ¼ Am;k Sn such that n > 1 and 1 6 k 6 m. Fig. 3 shows an example of multiplying A3;2 by Fig. 1. The 4-star network, S4 . S2 . 42 3. General topological properties 12 32 14 34 24 13 31 43 23 21 41 Fig. 2. The arrangement network, A4;2 . networks. It has been used as a tool for generating new attractive interconnection networks [4,9,17,19,24]. Below, a formal definition of the product network is given along with that of the arrangement star network. Definition 1. Given any two undirected networks G1 ¼ ðV1 ; E1 Þ and G2 ¼ ðV2 ; E2 Þ, where V1 is the set of vertices in G1 , V2 is the set of vertices in G2 , E1 is This section discusses some of the basic topological properties of the arrangement–star network including size, degree, diameter, average diameter, optimal routing, and optimal broadcasting. Table 1 summarizes the topological properties of the star and arrangement networks along with those of the arrangement–star for comparison purposes. These topological properties have been derived using the theoretical framework for analyzing product networks, proposed by Day and Al-Ayyoub [13] and Youssef [27]. For instance, if G1 and G2 are two undirected networks of respective sizes s1 and s2 and of respective diameters d1 and d2 then the size s and the diameter of the product G1 G2 is equal to s1 s2 and d1 þ d2 respectively. It therefore follows that the size, node degree and diameter of the arrangement–star network are m!n!=ðm  kÞ!, n þ kðm  kÞ  1 and b32ðn  1Þc þ b32kc respectively. Similarly, it can be easily shown that the average diameter of the arrangement–star network is n þ 2=nHn  4 þ Hk þ kðk  2Þ=m where n þ 2=n þ Hn  4 and Hk þ kðk  2Þ=m are the average diameters of the n -star network and ðm; kÞ-arrangement network, respectively. In the table P Hn ¼ ni¼1 1=i is the Harmonic number. The terms 328 A.M. Awwad et al. / Journal of Systems Architecture 48 (2003) 325–336 ab13 ab12 ab23 ab32 ab21 ab31 ba21 ba31 ba32 ba23 ba13 ba12 Fig. 3. The arrangement–star network, AS2;3;2 . Table 1 Basic topological properties for the three networks Property Star Arrangement Arrangement–star Size [13,27] Node degree [13,27] Diameter [13,15,27] Average diameter [3,8,27] n! n  1  3 ðn  1Þ 2 N þ 2=n þ Hn  4 m!=ðm  kÞ! kðm 3  kÞ k 2 Hk þ kðk  2Þ=m Optimal routing [13] R1 R2 Optimal broadcasting [13] B1 B2 m!n!=ðm  kÞ! n3þ kðm  kÞ   1 ðn  1Þ þ 32 k 2 N þ 2=n þ Hn  4 þ Hk þ kðk  2Þ=m  R1 ðx1 ; y1 Þ if x1 6¼ y1 R1 ðhx1 ; x1 ihy1 ; y1 iÞ ¼ R2 ðx2 ; y2 Þ if x1 6¼ y1 Apply B1 then B2 or B2 then B1 R1 and R2 stand for the optimal routing algorithms for the star and arrangement networks, respectively. The terms B1 and B2 stand for the optimal broadcasting algorithms for the star and arrangement networks, respectively. 4. Comparison of the topological properties This section conducts a comparative study between the three networks: star, arrangement, and arrangement–star. This study shows the superiority of the arrangement–star over the star and arrangement networks. We base our comparison on the most widely used criteria such as degree, diameter, scalability, number of links and broadcasting cost [4,12,20]. Furthermore, we will use a new criterion referred to as the degree of accuracy which gives indication on the network fit to the desired size. In what follows, we compare the three static parameters of size, degree and diameter for the star network, Sn , arrangement network, Am;k , and arrangement–star network, ASn;m;k . We plot in Fig. 4 the network size against the matching probability of the desired size for all the networks of sizes in the range of [210 , 226 ], where the network sizes on the x-axis are presented in log scale. In this criterion the percentage of integer numbers that correspond to an actual network size from 210 up to a certain desired size are plotted for the three network families. The results reveal that the arrangement–star network provides relatively better fit to the desired network size. Fig. 5 shows the node degree for the three network families. The figure shows that the arrangement–star network has a lower degree than the arrangement network and the same degree as the star network. Fig. 6 shows the diameter of the three networks. The arrangement–star improves A.M. Awwad et al. / Journal of Systems Architecture 48 (2003) 325–336 329 7% Matching Probability 6% Star 5% Arrangement 4% Arrangememt-star 3% 2% 1% 0% 10 11 12 13 14 15 16 17 18 Net work size (logarithmic) Fig. 4. The matching probability for the three networks. 15 13 Degree 11 9 7 Star 5 Arrangement 3 Arrangement-star 1 10 12 14 16 18 20 22 24 26 Network size (Logarithmic) Fig. 5. Node degree for the three networks. the diameter of the star network, however the arrangement network has a better diameter than the other two both networks. This gain in network diameter in the arrangement network comes at a higher cost in node degree. A new measure called the degree of accuracy, which gives an indication how far the closest network size stands from a desired size. In this measure, the closer the value to 100% the better the fit to the desired network size. In Fig. 7, the arrangement–star network exhibits exact fit to the desired network size. The other two networks provide fluctuating network sizes; sometimes much larger and sometimes much lower than the desired network size. For instance, almost 100% of the actual arrangement–star sizes are within of the desired network size. While 88% of the actual sizes of the arrangement network are within the desired network sizes, and for the star network, 55% from the actual values are within the desired network sizes. In the figure, we allowed 0.1 error in the size fit. Percentages higher that 100% mean networks with sizes larger than the desired size are offered. The number of links that are required by a given network is an important factor that affects its implementation cost. This measure captures both the real wiring cost and the number of pins required at each node. Fig. 8 plots the number of links against network size for the three networks. 330 A.M. Awwad et al. / Journal of Systems Architecture 48 (2003) 325–336 18 Star 16 Arrangement 14 Arrangement-star Diameter 12 10 8 6 4 2 0 10 12 14 16 18 20 22 24 26 Network size ( Logarithmic ) Degree of accuracy [%] Fig. 6. Diameter for the three networks. 200 Star 180 160 Arrangement Arrangement-star 140 120 100 80 60 40 20 0 10 12 14 16 18 20 22 24 26 Network size ( Logarithmic ) Fig. 7. Degree of accuracy for the three networks. 1.E+10 Number of links 1.E+09 1.E+08 1.E+07 1.E+06 Star 1.E+05 Arrangement 1.E+04 Arrangement-star 1.E+03 1.E+02 10 12 14 16 18 20 22 Network size (Logarithmic) Fig. 8. Number of links for the three networks. 24 26 A.M. Awwad et al. / Journal of Systems Architecture 48 (2003) 325–336 331 Table 2 The broadcasting cost for the three networks Network One-to-all broadcasting cost ffi2 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Ma 3 k 1 b þ bkðmkÞ 2 ffi2 qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Ma 3 ðn  1Þ  1 b þ bðn1Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2    Ma 3 þ ðn  1Þ þ 32 k  1 b b½ðn1ÞþkðmkÞ 2 Arrangement Am;k Star Sn Arrangement–star ASn;m;k 5. Hierarchical structure of the arrangement–star network Once again the arrangement–star outperforms its counterparts in terms of number links required to implement the network. One of the most widely used criteria to evaluate interconnection networks is the cost of broadcasting; an important communication operation required by many parallel applications [20]. Table 2 shows lower bounds on the communication cost for one-to-all broadcasting in the three networks. These expressions have been derived using the results in [20]. The parameters M, a, and b denote the message length, unit transmission cost, and the message latency, respectively. For the sake of the present discussion M, a and b have been set to 1024 byte, 1 and 1000 ls, respectively, as suggested in similar previous studies [5,20]. Fig. 9 depicts broadcasting cost in the three networks based on expressions in Table 2. The figure shows that the arrangement–star outperforms the star network. Thew arrangement network offers better performance than the other two networks in this measure. In this section, the hierarchical structure of arrangement–star networks is discussed. The hierarchical structure of a network relates to the ability to build large networks from smaller networks of the same nature. The hierarchical networks have attractive symmetry properties that are important in the design of routing algorithms and in constructing pipeline and grid views that will be discussed in the next section. Proposition 1. The ASn;m;k can be decomposed into n! m! disjoint copies of ASnq;mp;kp where ðnqÞ! ðmpÞ! 1 6 q < n and 1 6 p 6 m. Proof. It has been shown in [10] that Sn can be n! disjoint copies of Snq while decomposed into ðnqÞ! m! Am;k can be decomposed into ðmpÞ! [3,12] disjoint copies of Amp;kp . Let X be a node in ASn;m;k and let Broadcasting cost Star 20000 18000 16000 Arrangement Arrangement-star 14000 12000 10000 8000 6000 4000 2000 0 10 12 14 16 18 20 22 Network size (Logarithmic) Fig. 9. Broadcasting cost for the three networks. 24 26 332 A.M. Awwad et al. / Journal of Systems Architecture 48 (2003) 325–336 q and p be two integers such that 1 6 q < n and 1 6 p < m. Pick any q symbols out of hni and fix them in the last q positions of the left part, lpðX Þ. Similarly, pick any p symbols from hmi and fix them in the last p positions of the right part, rpðX Þ. Now, varying the remaining symbols in lpðX Þ and the remaining symbols in rpðX Þ will produce a new copy of the ASnq;mp;kp sub-network. For each new ðn  qÞ-permutations there are n q different ways of choosing the q symbols from hni and there are q! ways of fixing these symbols in lpðX Þ. Also, for each new ðk  pÞ-arrangement there are m p different ways of choosing p symbols from hmi and there are p! ways of fixing these symbols in rpðX Þ. Hence, the number of disjoint copies that made is n! m! n! m! q! p!ðmpÞ! p! ¼ ðnqÞ! .  equal to q!ðnqÞ! ðmpÞ! Corollary 1. The ASn;m;k can be decomposed into disjoint copies of ASq;mkþp;p , where 1 6 q < n and 1 6 p < m. n! m! q! ðmkþpÞ! Proof. For the star network this decomposition can be achieved by fixing n  q position out of n symbols and changing the remaining q positions producing n!  Sq subnetworks. On the other hand q! m! we can decompose Am;k network into ðmkþpÞ!  Amkþp;p subnetworks. This decomposition can be achieved by fixing k  p positions out of m positions and varying the remaining p positions which will yields the claimed results.  Corollary 2. ASn;m;k can be decomposed into m! k n  1 n! non-disjoint copies of q ðnqÞ! ðmpÞ! p ASnq;mp;kp: m! k n  1 n! q ðnqÞ! ðmpÞ! p ASnq;mp;kp:  non-disjoint copies of 6. Grids and pipelines in the arrangement–star network In this section we present two frameworks for algorithm development on the arrangement–star network. These frameworks are based on computationally important topologies; the pipeline and the grid networks. The choice of these topologies stems from the fact that pipelines and grids have been extensively employed to develop vast bodies of parallel algorithms for many real life applications such as Fourier transform [18], matrix decomposition [5] and ascend/descend-type of divide-and-conquer algorithms [7,26]. The pipeline and grid frameworks for developing algorithms, which will be presented in this section, are important for at least three reasons. First, they fill the gap in algorithm development for the star and the arrangement networks. Such studies have been badly overlooked in the literature. Second, the presented frameworks are general in the sense that they can be used to reproduce any existing pipeline and grid based application to the new network, hence saving and reusing the enormous research results of the past decades. Third, the presented frameworks provide convenient structural views for developing new algorithms of various kinds. 6.1. Rectangular grid view Proof. From the above proposition and the corollary it is concluded that ASn;m;k can be decomm! k 1 n! non-disjoint posed into n  q ðnqÞ! ðmpÞ! p copies of ASnq;mp;kp: The decomposing process here is similar to that in proposition, except there is an additional free parameter which is the set of positions in lpðX Þ and rpðX Þ. We can change all the positions in 1 lpðX Þ except the first one. So, we will have n  q different ways of setting positions. Furthermore, we can change all the positions in lpðX Þ, hence we will have pk different ways of set positions. Hence, the ASn;m;k can be decomposed into The previous section discusses various ways of decomposing the arrangement–star network. In fact, as a product network, the arrangement–star network has rich decomposition capabilities. In particular ASn;m;k embeds n!  m!=ðm  kÞ! grids where columns are Sn -connected and rows are Am;k connected. These two decompositions are orthogonal partitions that each node in the grid belongs to exactly one copy of Sn and to exactly one copy of Am;k . That is to say, the grid has m!=ðm  kÞ disjoint columns each of which has n! nodes interconnected by Sn . Similarly, the grid has n! A.M. Awwad et al. / Journal of Systems Architecture 48 (2003) 325–336 disjoint rows each of which has m!=ðm  kÞ nodes interconnected by Am;k . The above partition is a particular case of the general decompositions discussed in the previous section where q and p are set to n  1 and k respectively. The important issue of these two orthogonal partitions is that the vertices of each sub-network in the first partitioning are contained in the second sub-network, one vertex per subnetwork. This decomposition will facilitate the broadcasting of data from one node to another in different rows and columns in parallel and without interference. This is an important feature for many applications in linear algebra, sorting and selection. For instance, it is very common to broadcast matrix elements across different rows/columns at the same time. With the above setup, this can be done very efficiently by running sufficient instances of the broadcast algorithms of the constitute networks. 6.2. Pipeline view The orthogonal nature is a pleasant property for the arrangement–star network which is inherited from the elegant structure of the product networks. For any network H that can be embedded in the arrangement (respectively the star) network and a path P can be characterized in the star (respectively the arrangement) network, a pipeline of jP j stages each of which is isomorphic to H can be obtained from the arrangement star. In the obtained pipeline, nodes in successive stages are order-preserving in the sense there exists a ranking function for stage nodes that gives same ranks to peer nodes in successive stages. The order-preserving condition is important to allow parallel data shift between peer nodes. As an example, the arrangement–star network has a pipeline of n! stages, each of which has m!=ðm  kÞ! nodes. Nodes in each stage communicate using Am;k communication primitives. Since Am;k is Hamiltonian [10,11,16], there exits a function that ranks nodes in the arrangement network, and hence nodes in successive stages can be coupled according to this function so the pipeline is order-preserving. 333 Of course, there are many pipelines in the arrangement–star as we can characterize H and P in the arrangement and the star networks. 7. Performance evaluation for the grids and pipelines frameworks In this section, we conduct a comparison study between four networks: the star, the hypercube, the mesh and the arrangement–star. The comparison is based on how these four networks perform when put into service. Again, the focus is the communication overhead induced by each of these four networks under two realistic communication scenarios. The first scenario involves a grid structure where nodes in each gets engaged in a oneto-all communicate across rows and columns. A realistic communication pattern would require a set of parallel row (column) broadcasts followed by a set of parallel column (row) broadcasts. The total communication cost is then equal to the sum of row and column communication costs. For this comparison to be possible, we should first derive the optimal grid structure for the other networks. Fortunately, there are few known results for the star network [1,5,7]. As for the hypercube, an optimal grid structure can be simply obtained by dividing the n-cube into two sub-networks, resulting in a 2n=2  2n=2 grid where each row/column is n=2-cube configured. The mesh is a grid by its nature. In this section we excluded the arrangement network from the comparison since no grid embeddings are known nor the construction of grid and pipeline views seems to be possible. In estimating the communication cost in the above described scenario, the model given in Table 2 is used. Again, we set the parameters M, a and b to 1024 byte, 1 and 1000 ls, respectively. Fig. 10 plots the obtained communication cost against network size for the star, the cube, the mesh and the arrangement–star network. The figure shows that the arrangement–star network outperforms its counterparts in this scenario. The second scenario involves a pipeline structure where nodes in each stage get engaged in a one-to-all communication followed by a shift to peer nodes in the successive stage. This cost is 334 A.M. Awwad et al. / Journal of Systems Architecture 48 (2003) 325–336 Broadcasting cost 10000000 1000000 Star Cube Mesh Arrangement-star 100000 10000 1000 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Network size (Logarithmic) Fig. 10. Broadcasting cost for the four networks based on grid view. Broadcasting cost 10000000 1000000 Star Cube Mesh Arrangement-star 100000 10000 1000 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Network size ( Logarithmic ) Fig. 11. Broadcasting cost for the four networks based on pipeline view. equal to the lower bound on the cost of one-to-all communication across the stage plus the cost of shifting the data to the next stage. The cost of shifting the data to the next stage is better estimated by the model dðb þ MaÞ, where the parameters M, b and a are as defined above. Fig. 11 shows the estimated communication cost for the four networks. Once again the arrangement–star network outperforms its counterparts in this scenario as well. It is well-know that fixed degree networks are scalable and cost-effective in terms of system up- grade, yet there are several shortcomings of fixed degree networks such as weak fault-tolerance, high-dilation embeddings, semi-linear diameters and average internode distance (although less significant with wormhole routing). The hypercube for example is a variable degree network that have received lots of attention and popularity for years and even have been implement and commercially used. The arrangement–star network is variable degree network much like the hypercube and the well-known star network, yet outperforms both in various aspects. A.M. Awwad et al. / Journal of Systems Architecture 48 (2003) 325–336 8. Conclusions Over the past two decades many interconnection networks have been proposed in the literature, including the star, hyperstar, hypercube, and arrangement networks. Most existing research on these networks has focused on analysing their topological properties. Consequently, there has been relatively little work devoted to designing efficient parallel algorithms for important parallel applications. In an attempt to fill this gap, this paper aims to propose efficient frameworks for algorithm development beside deriving and discussing the topological properties of the arrangements-star network and show the superiority of this network in terms of topological properties over its factors; the star and the arrangement networks. These frameworks are based on grid and pipeline views as popular structures that support a vast body of applications that are encountered in many areas of science and engineering, including matrix computation, divide-and-conquer type of algorithms, sorting, and Fourier transforms. The proposed frameworks are applied to the proposed arrangement–star along with the star, cube and mesh networks. Results from a performance study conducted in this paper reveal that the proposed arrangement–star supports efficiently applications based on the grid or pipeline structural outlooks. The comparative study between the mesh, star, arrangement and arrangement–star has revealed that the proposed network possesses superior topological properties over its counterparts in terms of degree, diameter and more flexibility in choosing the desired network size, and suitability for real applications. References [1] A. Menn, A.K. Somani, An efficient sorting algorithm for the star network interconnection network, Proceedings of the International Conference Parallel Processing, 1990, pp. 1–8. [2] S. Akers, B. Krishnamurthy, A group-networks theoretical model for symmetric interconnection networks, IEEE Trans. Comput. 38 (4) (1998) 555–566. [3] S. Akers, D. Harel, B. 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Circuits, Syst. Comput. 1 (1) (1991) 43–74. [23] S. Rajasekaran, D. Wei, Selection, routing, and sorting on the star network, J. Parallel Distr. Comput. 41 (1997) 225– 233. [24] A. Rosenberg, Product-shuffle networks: towards reconciling shuffles and butterflies, Discr. Appl. Math. 37/38 (1992) 465–488. [25] D. Saika, R.K. Sen, Two ranking schemes for efficient computation on the star interconnection network, IEEE Trans. Parallel Distr. Syst. 7 (1996) 321–327. [26] D. Saika, R.K. Sen, Order preserving communication on a star network, Parallel Comput. 21 (1995) 1292– 1300. [27] A.Youssef, Design and analysis of product networks, Proceedings of the 5th Symposium Frontiers of Massively Parallel Computation (Frontiers Õ95), 1995, pp. 521– 528. Abdel-Elah Al-Ayyoub is an Associate Professor of Computer Science at the Arab Open University. He received his B.Sc. degree in Computer Science in 1986 from Yarmouk University, Jordan. He then joined the Middle East Technical University, Turkey, where he obtained his MS and Ph.D. degrees in Computer Engineering in 1987 and 1992, respectively. Before joining the Arab Open University, Dr. Al-Ayyoub severed in the University of Bahrain, Sultan Qaboos University––Oman, The University of Akron––Ohio, and Jordan University of Science and Technology. His experience in teaching extends to 14 years. He has received more than US$ 230,000 in research grants, won two major prizes in Computer Science (the State Prize and Abdul-Hameed Shoman Prize), and published over 50 papers in well-known journals and conference proceedings. His areas of interest include interconnection networks, parallelizing compilers, the design of parallel algorithms, mobile computing, and artificial intelligence. Dr. AlAyyoub is an IEEE Senior Member. Ahmad Awwad finished both of his B.Sc. degree in computer science and his M.Sc. degree in Mathematical science from Tennessee State University, USA, in 1987 and 1989 respectively. He gained his Ph.D. in computer science from the University of Glasgow, United Kingdom, 2002. He held a teaching position as senior lecturer at the Sohar College, Oman. He is currently the head of Computer Science Department at Zarka Private University. Also he is a member of the general secretariat for Arab Conference on Information Technology (ACIT). His research interests are in field of parallel processing, interconnection networks, product networks, vertex product networks, opto-electronic networks, and algorithm design for HSPC. Dr. Awwad also is a member of the IEEE-CS. Mohamed Ould-Khaoua received his B.Sc. degree from the University of Algiers, Algeria, in 1986, and the M.App.Sci. and Ph.D. degrees in Computer Science from the University of Glasgow, UK, in 1990 and 1994 respectively. He is currently a lecturer in the Department of Computing Science at the University of Glasgow, UK. His research focuses on applying theoretical results from stochastic processes and queuing theory to the quantitative study of hardware and software architectures. He was the cochair of the first and second international workshops on performance modeling, evaluation, and optimization of parallel and distributed systems (PMEO-PDSÕ2002 and PMEOPDSÕ2003). His current research interests are performance modelling/evaluation of parallel and distributed systems, parallel algorithms, and computer networks. He is a member of the IEEE-CS and BCS.