International Journal of Computer Engineering and Applications, Volume VII, Issue II, Part II,
August 14
COMPARISON OF OCTAGON-CELL NETWORK WITH OTHER
INTERCONNECTED NETWORK TOPOLOGIES AND ITS
APPLICATIONS
Sanjukta Mohanty 1, Dr. Prafulla Kr. Behera 2
1
Department of Mathematics, North Orissa University, Srirama chandra vihar, Takatpur, Baripada,
Mayurbhanj, India.
2
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,
Multiprocessor network.
Interconnection
topology,
Routing,
Network
services,
[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[8,9,10]. 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[2,7].
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
201
S. Mohanty and P. K. Behera
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.
[2] 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 [2].
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 2 represents eight octagon-cells
surrounding the octagon-cell at level 1. Level 3 represents 16 octagon-cells surrounding the 8
octagon-cells at level 2 and so on [3].
Figure: 1. Octagon-Cell level: 1
Figure: 2. Addressing nodes in Octagon-Cell with level:1
(X,Y represents line no-X with node no-Y)
202
S. Mohanty and P. K. Behera
International Journal of Computer Engineering and Applications, Volume VII, Issue II, Part II,
August 14
Each level i has Ni nodes, representing processing elements and interconnected in a ring
structure. In an octagon-cell network, the number of nodes at level i is: Ni = 8(4i-3)
Now at level 1, N1 = 8, since there is a single octagon-cell with 8 vertices. Level 2
introduces 8 octagon-cells. Therefore at level 2 the number of nodes
N2 = 8(4*2-3) = 8*5 = 40, N3 = 8(4*3–3 ) = 8*9 = 72
In octagon-cell the level (i+1) has 32 nodes in addition to corresponding nodes to those
at level i. Therefore
Ni = 8+(i-1)*32 = 8+32*i–32 = 32*i–24 = 8(4*i–3)
The total number of nodes in a octagon-cell network is,
d
N = ∑ 8(4i–3) = 32∑i -∑24 = 32∑i–24∑1 = 32d (d+1)/2–24d = 16d2– 8= 8d(2d-1)
i=1
Or we can write N = 8i(2i-1),
Now N = 16d2–8d or 16d2 = N+8d or d2 = N+8d/16 or d = [1 + Sqrt(1+N)] / 4
Therefore the total no of nodes at level 1 is N = 8(2*1-1) = 8
At level 2, N = 8(2*4-2) = 48
At level 3, N = 8(2*9-3) = 120 and so on.
1,1
1,2
1,3
1,4
1,5
1, 6
2,1
3,1
2,4
2
2
2
3,4
4,1
4,6
5,1
6,1
5,4
2
1
2
6,4
7,1
8,1
7,6
2
2
8,4
2
9,1
9,4
10,1
10,2
10,3
10,4
10,5
10,6
Figure: 3. Addressing nodes in Octagon-Cell with level:2
Here we have shown the addressing nodes of only 1st and 10th line but not all the lines,
for Example 2,2 and 2,3 haven‟t been shown, because the figure will be more complex.
203
S. Mohanty and P. K. Behera
Comparison Of Octagon-Cell Network With Other Interconnected Network Topologies And Its
Applications
[2.1] DIAMETER
The diameter D of a network is defined as the maximum shortest path between
any two nodes [2,4,5,6]. 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 [2].
The diameter of octagon-cell is 4(2i-1)
At level 1 the diameter is 4(2i-1) = 4(2.1-1) = 4
At level 2 the diameter is 4(2i-1) = 4(2.2-1) = 12 and so on.
Representing „d‟ as the depth, we have the diameter D = 4(2d-1) =
2[1+Sqrt(1+N)] - 1/4
The graph diameter verses number of nodes is drawn below.
Table I
Number of Nodes
Diameter
8
48
120
224
360
4
12
20
28
36
Graph of octagon-cell with number of nodes verses diameter
40
35
30
25
Diameter
20
15
10
5
0
Diameter
0
100
200
300
400
Number of nodes
Figure: 4. Graph of octagon-cell with number of nodes verses diameter
[2.2] NODE DEGREE
The node degree on an interconnection network is defined as the maximum
number of edges that a node can have in the network [2].
204
S. Mohanty and P. K. Behera
International Journal of Computer Engineering and Applications, Volume VII, Issue II, Part II,
August 14
The node degree of octagon-cell with depth 1 is 2. If d > 1, then node degree
remains constant, that is 3. The network topology which secures constant node degree
is highly desirable. Constant node degree facilitates modularity in building blocks for
scalable systems [2,4,5,6]. 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
8
48
120
224
2
3
3
3
Graph of octagon-cell with number of nodes verses node degree
3.5
3
2.5
Node degree 2
1.5
1
0.5
0
Node Degree
0
1
2
3
4
5
6
Number of nodes
Figure: 5. Graph of octagon-cell with number of nodes verses node degree
[2.3] NUMBER OF LINKS
The number of links at each level i is Ni = 8, 52, 100, 148….for levels 1, 2, 3,
4…..respectively.
The total number of links in octagon-cell are given by 8, 60, 160, 308….for levels 1, 2,
3, 4…..respectively.
The total number of links are given by the following formulas,
N1 = 8(2i2 – 1), N2 = 8(2i2 – 1) + 4, N3 = 8(2i2 – 1) + 24, N4 = 8(2i2 – 1) + 44 ….for
levels 1, 2, 3, 4…. respectively.
The graph of octagon-cell with number of nodes verses number of links is shown
below:
205
S. Mohanty and P. K. Behera
Comparison Of Octagon-Cell Network With Other Interconnected Network Topologies And Its
Applications
Table III
Number of Nodes
Number of links
8
8
48
120
60
160
224
308
Graph of octagon-cell with number of nodes verses number of
350
300
250
Number of
links
200
Number of links
150
100
50
0
0
50
100
150
200
250
Number of nodes
Figure: 6. Graph of octagon-cell with number of nodes verses number of links
[2.4] 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 [2,4,5,6]. The bisection width of octagon-cell
network is 2d = [1+Sqrt(1+N)]/2.
[3] COMPARISON OF PARAMETERS
NETWORK WITH SEVERAL TOPOLOGIES
OF
OCTAGON-CELL
In octagon-cell network topology the node degree with depth 1 is 2. If depth d>1, then
the node degree remains constant, that is 3. 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 log2N for depth d > 1. Binary tree and
Cube-connected cycles have the node degree 3 for depth d > 1. Linear array, Ring have node
degree 2 for depth d > 1. 2D-Torus has the node degree 4 for depth d > 1. 3D 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
206
S. Mohanty and P. K. Behera
International Journal of Computer Engineering and Applications, Volume VII, Issue II, Part II,
August 14
message density in the links of the network. The diameter of octagon-cell network is 4(2i-1),
where i is the level number. Hex-cell has the diameter 4√(N/6)-1[2]. Hypercube has the
diameter log2N. Binary tree has the diameter 2(log2N -1). Ring has the diameter N/2. 3D
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[2]. The bisection width of octagon-cell network is 2d. Hex-cell has the bisection
width 2√(N/6)[2]. Hypercube and 2D-Torus has the bisection width N/2. Ring has the bisection
width 2. Binary tree and Linear array have the bisection width 1.
Table III
Topology
Name
Diameter
Number of Links
Bisection
Bandwidth
Remarks
Octagon-Cell
Maximum
Node
Degree
3
2[1+√(1+N)]
- 1/4
*1+√(1+N)+/
2.
N is the
number of
nodes.
Hex-Cell
3
4√(N/6) - 1
[1+√(1+N)]2-8+L
(L=0 for level 1,
L=[1+√(1+N)]/2 for
level 2, L=
2[1+√(1+N)] for
level 3,
L=11/4[1+√(1+N)]
for level 4.
3N/2 -3√(N/6)
2√(N/6)
Hypercube
log2N
log2N
log2N (N/2)
(N/2)
Binary Tree
3
2(log2N-1)
N-1
1
Linear Array
Ring
2D-Torus
2
2
4
N-1
N/2
2(r/2)
N-1
N
2N
1
2
N/2
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
CubeConnected
Cycles
3
2k-1+[k/2]
3N/2
N/(2k)
3D
Hexagonal
6
1.16√N
3N-8.66√N
2.32√N
N = k x 2k
nodes with a
cycle length
k≥3
N is number of
nodes
207
S. Mohanty and P. K. Behera
Comparison Of Octagon-Cell Network With Other Interconnected Network Topologies And Its
Applications
[4] COMPARISON OF DIFFERENT TOPOLOGIES WITH DIAMETER
VERSUS NO OF NODES
Graph with diameter against network size
40
Diameter
30
Octagon-cell
Linear array
Ring
Hex-cell
Binart tree
20
10
0
0
50
100
150
200
250
Number of nodes
Figure: 7. Graph of different topologies with number of nodes verses diameter
[5] COMPARISON OF DIFFERENT TOPOLOGIES WITH NODE
DEGREE
VERSUS NO OF NODES
Graph with node degree against number of nodes
3
2
Octagon-cell
1
Binary tree
Ring
0
0
200
400
600
800
Number of nodes
Figure: 8. Graph of different topologies with number of nodes verses node degree
Node degree against the network size
4
Node degree
Node degree
4
3
2
Octagon-Cell
1
Linear array
Hex-cell
0
0
200
400
600
800
Number of nodes
Figure: 9. Graph of different topologies with number of nodes verses node degree
208
S. Mohanty and P. K. Behera
International Journal of Computer Engineering and Applications, Volume VII, Issue II, Part II,
August 14
[6] COMPARISON OF DIFFERENT TOPOLOGIES WITH NO OF
LINKS VERSUS NO OF NODES
Number of links against number of nodes
1200
Number of links
1000
800
600
Hex-cell
Octagon-cell
400
Binary tree
200
0
0
200
400
600
800
1000
1200
1400
No of nodes
Figure: 10. Graph of different topologies with number of nodes verses number of links
Number of links against number of nodes
800
Number of links
700
600
500
400
Octagon-cell
300
Linear array
200
Ring
100
0
0
200
400
600
800
No of nodes
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 [12]. In direct network, MPSoCS node processors are connected directly with each
other. System –on-Chip applications demand higher data-processing capability that can perform
209
S. Mohanty and P. K. Behera
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 [13]. 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 2D-torus. The degree and total number of
links of the octagon-cell is less than hypercube, 2D-torus and 3D 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 3 links per node.
210
S. Mohanty and P. K. Behera
International Journal of Computer Engineering and Applications, Volume VII, Issue II, Part II,
August 14
REFERENCES
[1]
Boxer, L. and Miller. R., 1998. “Dynamic Computational Geometry on Meshes and
Hypercubes”, In Proceedings of the 1998 International Conference on Parallel Processing,,
pp.323-330.
[2]
Ahmad Sharieh, Mohammad Qatawneh, Wesam Almobaideen, Azzam Sleit, “Hex-Cell:
Modeling, Topological Properties and
Routing Algorithm”, In European Journal of
Scientific Research, ISSN No. 1450-216X Vol.22 No.2(2008), pp.457-468.
[3]
Sanjukta Mohanty and Prafulla Ku. Behera, “Optimal Routing Algorithm in a OctagonCell Network”,In International Journal of Advanced Research in Computer Science, ISSN
No. 0976-5697, Vol 2, No. 5, Sept-Oct 2011, pp.625-637.
[4]
Della Vecchia. G. and C. Sanges. 1995. “A Recursively Scalable Network for VLSI
Implementation”, In Future Generation Computer Systems. Pp.235-243
[5]
Kai
Hwang,
1993.
Advanced
Computer
Architecture:
Parallelism,
Scalability,
Pragrammability:, McGraw-Hill Book Co. International Edition.
[6]
Parhami., 1999. “Introduction to Parallel Processing: Algorithms and Architectures”,
Plenum.
[7]
Ghose, K. and K.R. Desai, 1995. “Hierarchical Cubic Network”, In IEEE Transaction
Parallel and Distributed Systems, Vol. 6, No. 4, pp. 427-435.
[8]
J. Bosh, D. Maltz, D. Johnson, Y. Hu, and J. Jetcheva, “ A Performance Comparison of
Multi-Hop Wireless Ad Hoc Network Routing Protocols”, In Proceedings of 4th Annual
AAM/IEEE International Conference On Mobile Computing and Networking , 1998.
[9]
R. Schoonderwoerd, O. Holland, J. Bruten, and L. Rothkrantz, “Ant-based Load Balancing
in telecommunications Networks, Adaptive Behavior”, Vol. 5, pp. 169-207, 1997.
[10]
C. K. Toh, Maximum Battery Life Routing to Support Ubiquitous Mobile Computing in
Wireless AdHoc Networks, In IEEE Communications Magazine, pp. 138-147, 2001
[11]
Benini, L. and De Micheli, G.
“Networks on chips: a new SoC paradigm”, IEEE
Computer, Vol. 35, No. 1, January, pp.70–78, 2002
[12]
Duato, J., Yalamanchili, S. and Ni, L. “Interconnection Networks, an Engineering
Approach”, In IEEE Computer Society Press, 1997.
[13]
Terry Tao Ye, Giovanni De Micheli, “On-chip implementation of multiprocessor networks
and switch fabrics”, In Int. J. Embedded Systems, Vol. 3, No. 4, 2008 209.
211
S. Mohanty and P. K. Behera