COMPARISON OF OCTAGON-CELL NETWORK WITH OTHER INTERCONNECTED NETWORK TOPOLOGIES AND ITS APPLICATIONS

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1 International Journal of Computer Engineering and Applications, Volume VII, Issue II, Part II, COMPARISON OF OCTAGON-CELL NETWORK WITH OTHER INTERCONNECTED NETWORK TOPOLOGIES AND ITS APPLICATIONS Sanjukta Mohanty 1, Dr. Prafulla Kr. Behera 1 Department of Mathematics, North Orissa University, Srirama chandra vihar, Takatpur, Baripada, Mayurbhanj, India. Department of Computer Science and Applications, Utkal University, Vani Vihar, Bhubaneswar, India ABSTRACT: In an Interconnected network topology, the source node first makes a connection with the destination node before sending a packet. The physical or logical arrangement of links in a network is called topology. In this paper an efficient topology octagon-cell network is presented and we have shown the octagon-cell network by means of an undirected Graph G = (V, E), where V is the set of nodes in the graph (hosts in the network) & E is the set of edges in the graph (links in the network). In this paper we have discussed the applications of octagoncell network and compared the node degree, diameter, number of links and bisection width of the octagon-cell network with other interconnected network topologies. Keywords: Octagon-cell, Interconnection topology, Routing, Network services, Multiprocessor network. [1] INTRODUCTION An interconnection network can be viewed as an undirected graph in which the vertices correspond to processors and edges correspond to the bi-directional communication links between processing elements[1]. Routing of data employs routing algorithms. The performance on the routing algorithms depends on the routing algorithms adopted and the physical arrangement of links in a network. Topological properties are the important factors in routing algorithm. Routing algorithms can also be classified as minimal or non-minimal. Minimal routing allows packets to follow only minimal cost paths, while non-minimal routing allows more flexibility in choosing the path by utilizing other heuristics[8,9,1]. Communication data routing is the most fundamental function of interconnection networks. Data routing is the act of moving information across an interconnection network from a source to a destination[,7]. Network topology is the arrangement of the various elements (links, nodes, etc) of a computer network. Essentially, it is the topological structure of a network and may be depicted physically or logically. Physical topology is the placement of the various components of a network, including device location and cable installation, while logical topology illustrates how 1

2 Comparison Of Octagon-Cell Network With Other Interconnected Network Topologies And Its Applications data flows within a network, regardless of its physical design. Distances between nodes, physical interconnections, transmission rates, or signal types may differ between two networks, yet their topologies may be identical. An example is a local area network (LAN): Any given node in the LAN has one or more physical links to other devices in the network; graphically mapping these links results in a geometric shape that can be used to describe the physical topology of the network. Conversely, mapping the data flow between the components determines the logical topology of the network. In an octagon-cell, each device has a dedicated point-to-point connection only with the two devices on either side of it. A signal is passed along the network in one directly, from device to device, until it reaches its destination. Each device in the octagon-cell incorporates a repeater. When a device receives a signal intended for another device, its repeater regenerates the bits and passes them along. Octagon-cell is easy to install and reconfigure. Each device is linked only to its immediate neighbors. [] DESCRIPTION OF OCTAGON-CELL NETWORK An interconnection network can be viewed as an undirected graph, in which vertices correspond to processors and edges correspond to the bidirectional communication links between processing elements []. In this paper we have described octagon-cell network and compared its properties with other interconnected networks. An octagon-cell has eight nodes. It has d levels numbered from 1 to d with depth d. Level 1 represents one octagon-cell. Level represents eight octagon-cells surrounding the octagon-cell at level 1. Level represents 16 octagon-cells surrounding the 8 octagon-cells at level and so on []. Figure: 1. Octagon-Cell level: 1 Figure:. Addressing nodes in Octagon-Cell with level:1 (X,Y represents line no-x with node no-y)

3 International Journal of Computer Engineering and Applications, Volume VII, Issue II, Part II, Each level i has N i nodes, representing processing elements and interconnected in a ring structure. In an octagon-cell network, the number of nodes at level i is: N i = 8(4i-) Now at level 1, N 1 = 8, since there is a single octagon-cell with 8 vertices. Level introduces 8 octagon-cells. Therefore at level the number of nodes N = 8(4*-) = 8*5 = 4, N = 8(4* ) = 8*9 = 7 In octagon-cell the level (i+1) has nodes in addition to corresponding nodes to those at level i. Therefore N i = 8+(i-1)* = 8+*i = *i 4 = 8(4*i ) The total number of nodes in a octagon-cell network is, d N = 8(4i ) = i - 4 = i 4 1 = d (d+1)/ 4d = 16d 8= 8d(d-1) i=1 Or we can write N = 8i(i-1), Now N = 16d 8d or 16d = N+8d or d = N+8d/16 or d = [1 + Sqrt(1+N)] / 4 Therefore the total no of nodes at level 1 is N = 8(*1-1) = 8 At level, N = 8(*4-) = 48 At level, N = 8(*9-) = 1 and so on. 1,1 1, 1, 1,4 1,5 1, 6,1,1 4,1 5,1 6,1 7,1 8,1 9,1 1,4,4 4,6 5,4 6,4 7,6 8,4 9,4 1,1 1, 1, 1,4 1,5 1,6 Figure:. Addressing nodes in Octagon-Cell with level: Here we have shown the addressing nodes of only 1 st and 1 th line but not all the lines, for Example, and, haven t been shown, because the figure will be more complex.

4 Comparison Of Octagon-Cell Network With Other Interconnected Network Topologies And Its Applications [.1] DIAMETER The diameter D of a network is defined as the maximum shortest path between any two nodes [,4,5,6]. The path length is measured by the number of links traversed. The network diameter indicates the maximum number of distinct hops between any two hops between any two nodes. The network diameter should be as small as possible. It will not only reduce the traversing time for messages, but also minimize message density in the links of the network []. The diameter of octagon-cell is 4(i-1) At level 1 the diameter is 4(i-1) = 4(.1-1) = 4 At level the diameter is 4(i-1) = 4(.-1) = 1 and so on. Representing d as the depth, we have the diameter D = 4(d-1) = [1+Sqrt(1+N)] - 1/4 The graph diameter verses number of nodes is drawn below. Number of Nodes Table I Diameter Graph of octagon-cell with number of nodes verses diameter Diameter Diameter [.] NODE DEGREE Figure: 4. Graph of octagon-cell with number of nodes verses diameter The node degree on an interconnection network is defined as the maximum number of edges that a node can have in the network []. 4

5 International Journal of Computer Engineering and Applications, Volume VII, Issue II, Part II, The node degree of octagon-cell with depth 1 is. If d > 1, then node degree remains constant, that is. The network topology which secures constant node degree is highly desirable. Constant node degree facilitates modularity in building blocks for scalable systems [,4,5,6]. Therefore the node degree of octagon-cell is constant when d > 1. The graph of octagon-cell with number of nodes verses node degree is shown below: Table II Number of Nodes Node Degree Graph of octagon-cell with number of nodes verses node degree Node degree Node Degree [.] NUMBER OF LINKS Figure: 5. Graph of octagon-cell with number of nodes verses node degree The number of links at each level i is N i = 8, 5, 1, 148.for levels 1,,, 4..respectively. The total number of links in octagon-cell are given by 8, 6, 16, 8.for levels 1,,, 4..respectively. The total number of links are given by the following formulas, N 1 = 8(i 1), N = 8(i 1) + 4, N = 8(i 1) + 4, N 4 = 8(i 1) + 44.for levels 1,,, 4. respectively. The graph of octagon-cell with number of nodes verses number of links is shown below: 5

6 Comparison Of Octagon-Cell Network With Other Interconnected Network Topologies And Its Applications Number of Nodes Table III Number of links Graph of octagon-cell with number of nodes verses number of 5 5 Number of links Number of links Figure: 6. Graph of octagon-cell with number of nodes verses number of links [.4] BISECTION WIDTH When a given network is cut into two equal halves the minimum number of edges along the cut is called the channel bisection width b [,4,5,6]. The bisection width of octagon-cell network is d = [1+Sqrt(1+N)]/. [] COMPARISON OF PARAMETERS OF OCTAGON-CELL NETWORK WITH SEVERAL TOPOLOGIES In octagon-cell network topology the node degree with depth 1 is. If depth d>1, then the node degree remains constant, that is. Constant node degree facilitates modularity in building blocks for scalable systems. It is a desirable feature in an interconnection that the number of ports does not grow at the same rate as a function of the number of nodes in the network. In other words, a network topology which secures constant node degree is highly desirable. Hypercube has the maximum node degree log N for depth d > 1. Binary tree and Cube-connected cycles have the node degree for depth d > 1. Linear array, Ring have node degree for depth d > 1. D-Torus has the node degree 4 for depth d > 1. D Hexagonal has the node degree 6. Diameter is the important feature of any network topology. The network diameter indicates the maximum number of distinct hops between any two nodes, thus providing the feature of communication merit for the network. Therefore the network diameter should be as small as possible. It will not only reduce the travelling time for messages, but also minimize 6

7 International Journal of Computer Engineering and Applications, Volume VII, Issue II, Part II, message density in the links of the network. The diameter of octagon-cell network is 4(i-1), where i is the level number. Hex-cell has the diameter 4 (N/6)-1[]. Hypercube has the diameter log N. Binary tree has the diameter (log N -1). Ring has the diameter N/. D Hexagonal has the diameter 1.16 N. Bisection width is the important feature of any network, which provides good indicator of the maximum communication bandwidth along the bisection of a network. When a given network is cut into two equal halves, the minimum number of edges (channel) along the cut is called the channel bisection width b. In case of a communication network, each edge corresponds to a channel with w bits wires. Therefore, the wire bisection is B = b*w. This parameter B reflects the wiring density of a network. When B is fixed, the channel width (in bits) w = B/b[]. The bisection width of octagon-cell network is d. Hex-cell has the bisection width (N/6)[]. Hypercube and D-Torus has the bisection width N/. Ring has the bisection width. Binary tree and Linear array have the bisection width 1. Table III Topology Name Octagon-Cell Hex-Cell Hypercube Binary Tree Linear Array Ring D-Torus Maximum Node Degree log N 4 Diameter Number of Links Bisection Bandwidth [1+ (1+N)] - 1/4 4 (N/6) - 1 log N (log N-1) N-1 N/ (r/) [1+ (1+N)] -8+L (L= for level 1, L=[1+ (1+N)]/ for level, L= [1+ (1+N)] for level, L=11/4[1+ (1+N)] for level 4. N/ - (N/6) log N (N/) N-1 N-1 N N *1+ (1+N)+/. (N/6) (N/) 1 1 N/ Remarks N is the number of nodes. N is the number of nodes. N is the number of nodes. N is the number of nodes. N nodes. N nodes. r x r torus where r = N Cube- Connected Cycles D Hexagonal 6 k-1+[k/] 1.16 N N/ N-8.66 N N/(k). N N = k x k nodes with a cycle length k N is number of nodes 7

8 Node degree Node degree Diameter Comparison Of Octagon-Cell Network With Other Interconnected Network Topologies And Its Applications [4] COMPARISON OF DIFFERENT TOPOLOGIES WITH DIAMETER VERSUS NO OF NODES 4 1 Graph with diameter against network size Octagon-cell Linear array Ring Hex-cell Binart tree Figure: 7. Graph of different topologies with number of nodes verses diameter [5] COMPARISON OF DIFFERENT TOPOLOGIES WITH NODE DEGREE VERSUS NO OF NODES 4 Graph with node degree against number of nodes Octagon-cell Binary tree Ring Figure: 8. Graph of different topologies with number of nodes verses node degree 4 Node degree against the network size Octagon-Cell Linear array Hex-cell Figure: 9. Graph of different topologies with number of nodes verses node degree 8

9 Number of links Number of links International Journal of Computer Engineering and Applications, Volume VII, Issue II, Part II, [6] COMPARISON OF DIFFERENT TOPOLOGIES WITH NO OF LINKS VERSUS NO OF NODES 1 Number of links against number of nodes Hex-cell Octagon-cell Binary tree No of nodes Figure: 1. Graph of different topologies with number of nodes verses number of links Number of links against number of nodes No of nodes Octagon-cell Linear array Ring Figure: 11. Graph of different topologies with number of nodes verses number of links [7] APPLICATIONS Many different multiprocessor network topologies are designed for specific applications due to different performance requirements and cost metrics. Multiprocessor systems on chips (MPSoC) networks can be categorized as direct network and indirect networks [1]. In direct network, MPSoCS node processors are connected directly with each other. System on-chip applications demand higher data-processing capability that can perform 9

10 Comparison Of Octagon-Cell Network With Other Interconnected Network Topologies And Its Applications parallel and multi-threading tasks. Multiprocessor systems on chips combine the advantages of computation parallelism of multiprocessors with single chip integration of systems on-chip. Thus MPSoCs are widely employed in today s and tomorrow s network processors, parallel multimedia processors and many application specific array processors [1]. MPSoCs also need to adopt a dedicated on-chip interconnection network that can provide reliable and scalable communication[11]. Octagon network can be used as on-chip communication architecture for network processors due to its scalability property. [7] CONCLUSION The architecture of octagon-cell network has some of the interesting characteristics of hypercube, binary tree, linear array, star, hex-cell and D-torus. The degree and total number of links of the octagon-cell is less than hypercube, D-torus and D Hexagonal. In hypercube, the node degree of each node is a logarithmic function of the total number of nodes, which is the drawback of the topology. In the topological structures such as hypercube, binary tree have some drawbacks. Because the maximum node degree, diameter, number of links are the logarithmic functions of the number of nodes, which are the drawbacks of these topology. Therefore octagon-cell networks have the better features in comparison to these topologies, that can connect hundreds of thousands nodes with links per node. 1

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