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IADIS International Conference on Applied Computing 2005
AN EXTENDED OTIS-CUBE INTERCONNECTION
NETWORK
Jehad Al-Sadi
Computer Science Department
Arab Open University
P. O. Box 926832
AMMAN 11110 - Jordan
ABSTRACT
This paper proposes a new extended OTIS-Cube Interconnection network of the well known class of network, OTIS-Cube.
This paper presents attractive topological properties for the new network which carry a way the weakness of some topological
properties of the OTIS-Cube found in the degree and the diameter. The new network has a regular degree, a small diameter,
and a semantic structure. An efficient routing algorithm is proposed. In this new algorithm, an optimal path is selected based
on the two locations of the source node and the destination node. The paper presents also a comparative analysis proving the
superiority of the proposed interconnection network over the OTIS-Cube. The comparison is obtained in terms of topological
properties measures and routing distances.
KEYWORDS
Interconnection Networks, OTIS-Cube, Topological Properties, Routing Algorithm, Performance Evaluation.
1. INTRODUCTION
Recently, there has been an increasing interest in a class of interconnection networks called Optical Transpose
Interconnection Systems “OTIS-networks” [1, 11, 12]. Marsden et al were the first to propose the OTIS-networks
[7]. Extensive modelling results for the OTIS have been reported in [5]. The achievable terabit throughput at a
reasonable cost makes the OTIS a strong competitor to the electronic alternatives [6, 7]. These encouraging
findings prompt the need for further testing of the suitability of the OTIS for real-world parallel applications.
A number of computer architectures have been proposed in which the OTIS was used to connect different
processors [7]. Krishnamoorthy et al [6] have shown that the power consumption is minimised and the bandwidth
rate is maximised when the OTIS computer is partitioned into N groups of N processors each. [3, 6]. Furthermore,
the advantage of using the OTIS as optoelectronic architecture lies in its ability to manoeuvre the fact that free
space optical communication is superior in terms of speed and power consumption when the connection distance
is more than few millimetres [6]. In the OTIS, shorter (intra-chip) communication is realised by electronic
interconnects while longer (inter-chip) communication is realised by free space interconnects.
OTIS technology processors are partitioned into groups, where each group is realised on a separate chip with
electronic inter-processor connects. Processors on separate chips are interconnected through free space
interconnects. The philosophy behind this separation is to utilise the benefits of both the optical and electronic
technologies. Processors within a group are connected by a certain interconnecting topology, while transposing
group and processor indexes achieve inter-group links. Using cube as a factor network will yield the OTIS-Cube
in denoting this network.
The binary n-cube has been one of the most popular network topologies for multicomputers due to its
attractive topological properties, e.g. regular structure, low diameter, and ability to exploit communication
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locality. Several experimental and commercial systems have been built using the factor cube network including
the NCUBE-2 [2], Intel iPSC [8], Cosmic Cube [10], and SGI Origin 2000 multiprocessor [4].
OTIS-Cube is basically constructed by "multiplying" a cube topology by itself. The set of vertices is equal to
the Cartesian product on the set of vertices in the factor cube network. The set of edges E in the OTIS-Cube
consists of two subsets, one is from the factor cube, called cube-type edges, and the other subset contains the
transpose edges. The OTIS approach suggests implementing cube-type edges by electronic links since they
involve intra-chip short links and implementing transpose edges by free space optics. Throughout this paper the
terms “electronic move” and the “OTIS move” (or “optical move”) will be used to refer to data transmission
based on electronic and optical technologies, respectively.
Although the OTIS-Cube network has many attractive topological properties but it suffers from having
limited optical links between the different groups. When source and destination nodes are in two different groups,
the fact that only one optical link connects two distinguished groups directly create a congestion problem to most
of the shortest paths that have to pass through this particular optical link. Furthermore, alternative paths are too
long compared to the short path because they have to be routed via a third group which required passing via two
optical links in addition to the electronic moves in each group to reach the destination.
Motivating from the above observation, this paper proposes a new interconnection network called an
Extended OTIS-Cube based on the “OTIS-Cube” network [1, 3]. Our motivation of this research is to overcome
the weakness of the topological properties of the OTIS-Cube such as the degree and the diameter by introducing
attractive topological properties of the new network. The Extended OTIS-Cube network is semantic, regular, and
has a small diameter.
2. NOTATIONS AND DEFINITIONS
The n-dimensional undirected graph binary n-cube Qn is one of the well known networks which have been used
in real life systems [2, 4, 8, 10].
Definition 1: The undirected graph n-cube with 2 n vertices, representing nodes, which are labelled by the 2 n
binary strings of length n. Two nodes are joined by an edge if, and only if, their labels differ in exactly one bit
position. The label of node A is written anan-1…a1, where ai ∈ {0, 1} is the i-th bit (or bit at i-th dimension) [9].
The OTIS-Cube is obtained by "multiplying" a cube topology by itself. The vertex set is equal to the
Cartesian product on the vertex set in the factor cube network. The edge set consists of edges from the factor
network and new edges called the transpose edges. The formal definition of the OTIS-Cube is given below.
Definition 2:
Let cube = (V0, E0) be an undirected graph representing a cube network. The OTIS-Cube = (V,
E) network is represented by an undirected graph obtained from cube as follows V = {〈x, y〉 | x, y ∈ V0} and E =
{(〈x, y〉, 〈x, z〉) | if (y, z)∈E0} ∪ {(〈x, y〉, 〈y, x〉) | x, y ∈ V0} [3].
In the OTIS-Cube the address of a node u = 〈x, y〉 from V is composed of two components. Fig. 1 shows a 16
processor OTIS-Cube, the notation 〈g, p〉 is used to refer to the group and processor addresses, respectively. Two
nodes 〈g1, p1〉 and 〈g2, p2〉 are connected if, and only if, g1 = g2 and (p1, p2)∈E0 (such that E0 is the set of
edges in cube network) or g1 = p2 and p1 = g2, in this case the two nodes are connected by transpose edge.
Definition 3: The Topological properties of the OTIS-Cube are defined as follows [3]:
1- Size: If the cube factor network of size N, then the size of the OTIS-Cube is N2.
2- Degree: Let 〈g, p〉 be any node in OTIS-Cube. Then the degree (or deg) of the OTIS-Cube is as follows:
if g = p
⎧deg G0 ( p)
deg OTIS −CUBE ( g , p) = ⎨
deg
(
p
)
1
if g ≠ p
+
⎩ G0
3- Number of Links: Let N0 be the number of links and M be the number of nodes in the cube network, then the
number of links in the OTIS-Cube = (M 2 − M ) / 2 + N0 ⋅ M .
4- Length: Let 〈g1, p1〉 and 〈g2, p2〉 be two different nodes in the OTIS-Cube. To transmit data originated in the
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source node 〈g1, p1〉 to the destination node 〈g2, p2〉 we follow one of the three possible paths shown above i, ii,
and ii. The length of the shortest path between the nodes 〈g1, p1〉 and 〈g2, p2〉 is:
if g1 = g 2
⎧ d ( p1 , p2 )
Length = ⎨
if g1 ≠ g 2
⎩min(d ( p1 , g 2 ) + d ( p2 , g1 ) + 1, d ( p1 , p2 ) + d ( g1 , g 2 ) + 2)
where d(p1, p2) is the length of the shortest path between any two processors 〈g1, p1〉 and 〈g1, p2〉.
5- Diameter: Let n is the diameter of the cube network; the diameter of the OTIS-Cube is 2n+1.
Fig.1 shows a 16 processor OTIS-Cube connection where the bold arrows represent an optical links between two
processors of two different 2-cube groups.
group 0
group 1
〈0,0〉
〈0,1〉
〈1,0〉 〈1,1〉
〈0,10
〉
〈0,1
〈1,10〉 〈1,11〉
〈10,1〉
〈11,0〉
〈10,0〉
〈10,10〉 〈10,11〉
group 2
〈11,1〉
〈11,10 〈11,11〉
〉
group 3
Figure 1. 16-processor OTIS-Network
3. THE EXTENDED OTIS-CUBE TOPOLOGICAL PROPERTIES
This section introduces the new interconnection network topological properties based on the well known OTISCube networks. First we present the rules of constructing the new network.
The Extended OTIS-Cube is obtained by "multiplying" a cube topology by itself. The vertex set is equal to
the Cartesian product on the vertex set in the factor cube network. The edge set consists of edges from the factor
network and new edges called the transpose edges. The formal definition of the Extended OTIS-Cube is given
below.
Definition 4:
Let n-cube = (V0, E0) be an undirected graph representing a cube network where n is the cube
degree. The Extended OTIS-Cube = (V, E) network is represented by an undirected graph obtained from n-cube
as follows V = {〈x, y〉 | x, y ∈ V0} and E = {(〈x, y〉, 〈x, z〉) | if (y, z)∈E0} ∪ {(〈x, y〉, 〈y, x〉) | x, y ∈ V0} ∪ {(〈x, x〉,
〈y, y〉) | x, y ∈ V0 ∩ x is an opposite of y}
Definition 5: Let x, y be group addresses of two nodes in an Extended OTIS-Cube labelled as series of bits
〈xn…x2x1〉, 〈yn…y2y1〉 consequently where each bit is either 0 or 1. x is called an opposite of y if and only if they
are differ only in the first bit.
In the Extended OTIS-Cube the address of a node u = 〈x, y〉 from V is composed of two components. Fig. 2
shows a 16 processor Extended OTIS-Cube, the notation 〈g, p〉 is used to refer to the group and processor
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addresses, respectively. Two nodes 〈g1, p1〉 and 〈g2, p2〉 are connected if one of the following cases occurs:
1- If g1 = g2 and (p1, p2)∈E0 (such that E0 is the set of edges in n-cube network), in this case the two nodes
are connected by electronic edge.
2- If g1 = p2 and p1 = g2, in this case the two nodes are connected by transpose edge.
3- If g1 = p1 and g2 = p2 and g1 is an opposite of g2, in this case the two nodes are connected by transpose
edge too.
Definition 6: The Topological properties of the Extended OTIS-Cube are defined as follows:
1- Size: If the cube factor network of size N, then the size of the Extended OTIS-Cube is N2.
2- Degree: Let deg G0 be the degree of the n-cube network and let 〈g, p〉 be any node in an Extended OTIS-Cube.
group 0
group 1
〈0,0〉
〈0,1〉
〈1,0〉
〈0,10〉
〈0,1
〈1,10〉 〈1,11〉
〉
〈10,0 〈10,1〉
〈10,10〉
〈10,11〉
〈11,0〉
〈1,1〉
〈11,1〉
〈11,10〉 〈11,11〉
group 2
group 3
Figure 2. 16-processor Extended OTIS-Network
Then the degree of the Extended OTIS-Cube is as follows:
degOTIS−CUBE ( g , p) = degG0 ( p) + 1 = n + 1
3- Number of Links: The number of links in Extended OTIS-Cube = ( n + 1)2 2n / 2 where n is the dimension of
the cube factor network.
The distance in the Extended OTIS-Cube is defined as the shortest path between any two processors, 〈g1, p1〉
and 〈g2, p2〉, and involves one of the following forms:
i- When g1 = g2 then the path involves only electronic moves from source node to destination node.
ii- When g1 ≠ g2 and if the number of optical moves is an even number of moves, then the paths can be
E 〈g , p 〉 O 〈p , g 〉 E 〈p , g 〉 O
compressed into a shorter path of the form: 〈g1, p1〉 ⎯⎯→
⎯→ 2 1 ⎯⎯→ 2 2 ⎯⎯→
1
2 ⎯
〈g2, p2〉 where the symbols O and E stand for optical and electronic moves respectively.
iii- When g1 ≠ g2, and the path involves an odd number of OTIS moves. In this case the paths can be
E 〈g1, g2〉 O 〈g2, g1〉 E 〈g2, p2〉.
compressed into a shorter path of the form: 〈g1, p1〉 ⎯⎯→
⎯⎯→
⎯⎯→
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IADIS International Conference on Applied Computing 2005
iv- When g1 is opposite of g2, and if the number of optical moves is an even number of moves, then the
v-
E 〈g , p 〉 O 〈p , g 〉 E
paths can be compressed into a shorter path of the form: 〈g1, p1〉 ⎯⎯→
⎯→ 2 1 ⎯⎯→
1
2 ⎯
O
〈g
,
p
〉
where
the
symbols
O
and
E
stand
for
optical
and
electronic moves
〈p2, g2〉 ⎯⎯
→ 2 2
respectively.
When p1 is opposite of p2, and the path involves an odd number of OTIS moves. In this case the
E 〈g1, g2〉 O 〈g2, g1〉 E
paths can be compressed into a shorter path of the form: 〈g1, p1〉 ⎯⎯→
⎯⎯→
⎯⎯→
〈g2, p2〉.
4- Length: Let 〈g1, p1〉 and 〈g2, p2〉 be two different nodes in the Extended OTIS-Cube. To transmit data
originated in the source node 〈g1, p1〉 to the destination node 〈g2, p2〉 we follow one of the five possible paths
shown above i, ii, …, v. The length of the shortest path between the nodes 〈g1, p1〉 and 〈g2, p2〉 is:
if g1 = g 2
⎧d ( p1 , p2 )
⎪
Length = ⎨min(d ( p1 , g1 ) + d ( g1 Opposite , p2 ) + 1, d ( p1 , g1 ) + d ( p1 Opposite , g 2 ) + 2) if g1 = g 2 Opposite or p1 = p2 Opposite
⎪min(d ( p , g ) + d ( p , g ) + 1, d ( p , p ) + d ( g , g ) + 2)
if g1 ≠ g 2
1
2
2
1
1
2
1
2
⎩
where d(p1, p2) is the length of the shortest path between any two processors 〈g1, p1〉 and 〈g1, p2〉.
5- Diameter: Let n be the diameter of the n-cube network, then the diameter of the Extended OTIS-Cube is 2n.
To show the attractive properties of the new network, we present performance statistical results of the
proposed Extended OTIS-Cube network. To this end, an extensive comparison study has been carried out
between the OTIS-Cube and the Extended OTIS-Cube networks.
Table 1. A comparison between OTIS-Cube and Extended OTIS-Cube networks.
Cube (n)
Degree
Nodes
2
16
3
64
4
256
5
1024
6
4096
7
16384
8
65563
OTIS-Cube
Diameter
Avg Dist
5
2.93
7
3.93
9
4.96
11
5.97
13
6.97
15
7.99
17
9.00
Extended OTIS-Cube
Diameter
Avg Dist
4
2.53
6
3.74
8
4.86
10
5.92
12
6.96
14
7.98
16
8.99
Table 1 shows diameter, average distances, and number of nodes for different OTIS-Cube and Extended OTISCube network sizes. The presented results are based on the cube factor network degree ranged from to 2 up to 8.
Fig. 3 shows the routing algorithm that each node <gc,pc>.in the network applies to route a message towards
its destination node <gd,pd>. The algorithm checks first whether the source and the destination nodes are in the
same group or not. If both nodes are in the same group then the cube routing rules are applied by selecting a
preferred neighbour to guarantee an optimal routing toward the destination. Otherwise, the algorithm selects a
move that leads to make an optical move to reach the destination's group, then, to reach the destination node.
4. CONCLUSION
This paper has proposed a new interconnect network based on the concept of OTIS-Cube networks. We have
introduced attractive topological properties for the new network which overcome the weakness of some
topological properties of the OTIS-Cube found in the degree and the diameter. The new network has a regular
degree, a small diameter, and a semantic structure. An efficient routing algorithm has proposed. In this new
algorithm, an optimal path is selected based on the two locations of the source node and the destination node. The
paper presented also a comparative analysis proving the superiority of the proposed interconnection network over
the OTIS-Cube. The comparison is obtained in terms of topological properties measures and routing distances.
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Algorithm Unsafety_Vectors (M: message; <gc,pc>,<gd,pd>: node)
/* called by current node <gc,pc> to route the message M to
its destination node <gd,pd> */
if gc= gd and pc= pd then exit; /* destination reached */
if gc= gd then route(<gc,pc>,<gd,pd>) /* curr & dest. at the same group */
if dist(pc,pc)+dist(gc gd)= diameter then move m to < gc Opposite, pc Opposite>
if gc=gd Opposite or pc=pd Opposite then
{ if (dist(pc,gc)+dist(pc Opposite, gd)+2)<(dist(pc,gc)+dist(gc Opposite, pd)+1) then
{ if pc= pd Opposite and pc=gc then move m to < pc Opposite, gc Opposite>
else if pc= pd Opposite and pc≠gc then move m to < pc, gc>
else route(<gc,pc>,< gc, pd>) }
else
{ if pc= gd then move m to < pc, gd>
else if pc= gc and pc Opposite =pd then move m to < pc Opposite, gc Opposite>
else route(<gc,pc>,<gc,gd>) }
}
if (dist(pc,pd)+dist(gc,gd)+2)<(dist(pc,gd)+dist(gc,pd)+1) then
{ if pc= pd then move m to < pc, gc>
else route(<gc,pc>,< gc, pd>) }
else
{ if pc= gd then move m to < pc, gd> else route(<gc,pc>,<gc,gd>) } }
Function route(<gc,pc>,<gd,pd>:node)
{ send M to (<gc,
pc(i )
> /* select a preferred neighbour toward the destination <gd,pd>:where gc=gd */ }
Figure 3. The routing algorithm
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