GRAPH MODELS OF CLOS NETWORKS

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P O Z N A N U N I V E R S I T Y O F T E C H N O L O G Y A C A D E M I C J O U R N A L S

No 54 Electrical Engineering 2007

__________________________________________ * Gdansk University of Technology.

Andrzej JASTRZĘBSKI*

GRAPH MODELS OF CLOS NETWORKS

In the article, graph models for multicast Clos networks are proposed. In literature there haven’t shown any graph models for multicast 3-stage Clos networks. There are four common physical models of multicast 3-stage Clos networks: all switches have got fan-out capability or one stage hasn’t got fan-out capability. We focused on model-0 and model-1. Any Clos network can be transformed into graph and coloring of the graph can be transformed into network controlling.

Keywords: multicast Clos networks, graph coloring

1. INTRODUCTION

The first breakthrough on how to build minimum-cost multistage connecting networks starting form small crossbars has been known since 1953 and is due to Charles Clos. Nowadays, these three-stage networks are known as Clos networks. They have been widely used for data communications and parallel computing systems under the assumption that the service supported is always unicast. That is each connection is established between an idle input and only one idle output of the network.

Besides unicast calls, there are also multicast calls. It means that switching node must be able to set-up connection from an idle input to more than one idle outputs. Interest in this kind of networks has been rising due to growing needs for supporting multicast communication services.

2. MULTICAST CLOS NETWORKS

In a 3-stage Clos network the first (input) stage consist of

r

₁ switches with

n

₁

inputs and

m

outputs, the second (middle) stage consists of

m

switches with

r

1

inputs and

r

₂ outputs, and the third stage consist of

r

₂ switches with

m

inputs and

n

₂ outputs. In each stage there are same sized switches and links exists only

2007

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142 Andrzej Jastrzębski

between adjacent stages. Fig. 1. illustrates such a 3-stage Clos network. We denote such a Clos network by

C

(

n

₁

,

r

₁

,

m

,

n

₂

,

r

₂

)

.

Fig. 1. An example of 3-stage Clos network

In multicast traffic, an input can request to connect to up to a certain number of outputs. Suppose ( Oi, ) is the current multicast call where i is an input and O is a set of idle outputs.

There are four main hardware model described. Let model-0 denote hardware model in which all stages have fan-out capability, and model-i, for i=1,2,3, denote hardware model in which stage i has no fan-out capability and other stages have fan-out capability.

4. MODEL-0

As it was said in model-0 all switches have fan-out capability. We name input switches 1 , , , 2 1 a ar a K _{, middle switches} m

b

1, 2,K, , output switches 2 , , , ₂ 1

c

r

c

K _{(see Fig. 1.). One could consider a bipartite graph}

)

,

(

V

1

V

2

E

G

=

∪

and a collection

F

where set

{

}

1 , , 1 1

a

r

V

= K _{is the input}

partition and set

{

}

1

, ,

1 2

c

r

V

= K _{is the output partition.}

A multicast call

(

{

}

)

k j j

c

a

, , , 1

1 K is transformed into

k

edges

{

a

i,

c

j1

}

, ,

{

a

i, ,

c

jk

}

K

K _{and set of these edges is an element of the collection}

F

_.

One can notice that the collection

F

is dividing the set of edges into equivalence

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Graph models of Clos networks 143

classes. In other words calls are transformed into edges and an equivalence relation between them.

After transformation one has bipartite graph. Every vertex in output partition has the same degree as the number of adjacent equivalence classes. Edges of the graph can be colored by the following algorithm: if edges

e

,

f

are adjacent and they aren’t in the same equivalence class then they receive different colors.

We assume that each color represents a middle switch – an edge

{ }

a

_i,

c

_j in color

λ

means the call will be routed through the middle switch

b

_λ. It is easy to show that represented above coloring is equivalent to controlling of 3-stage Clos network in model-0 [4].

Example of transformation Clos network in model-0 is shown at Figure 2.

Fig. 2. An example of a model-0 Clos network

4. MODEL-1

As it was said, in model-1 switches belong to input stage have no fan-out capability. Switches is named as in model-0. One could consider hypergraph

)

,

(

V

₁

V

₂

E

H

=

∪

where set

{

}

1 , , 1 1

a

r

V

= K _{is input partition and set}

{

1, , 1

}

2

c

r

V

= K _{is output partition. Call}

(

{

}

)

k j j

c

a

, , , 1 1 K is transformed into hyperedge

{

}

k j j i

c

a

, , , 1 K _.

Aftet transformation hyperedges are colored natural so two adjacent hyperedges have distinct colors. Colors represent middle switches.

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144 Andrzej Jastrzębski

Fig. 3. Example model-1

5. REMARKS

Model-2 can be transformed like Clos networks with unicast calls. It means that call from switch

a

_i to

c

_j changes into an edge

{ }

a

_i,

c

_j . Edge coloring of the graph can be used to control the Clos network [1].

Modelling of model-3 is similar to model-0 with the exception that collection

F

_{is substituted with function}

_f

_:

_V

_×

_E

_→_Ν₍_Ν_{is set of natural numbers)}

which also divide edges into local equivalence classes. Local means that every vertex have own division for equivalence classes. The coloring of graph is also similar to model-0 coloring.

REFERENCES

[1] Hwang F.K.: Rearrangeability of Multi-Connection Three-Stage Clos Networks, Networks, 2: 301-306, 1972

[2] Pattavina A., Tesei G.L.: Modelling the Blocking Behavior of Multicast Clos Networks, IEEE INFOCOM 2003

[3] Fu H.-L., Hwang F.K.: On 3-stage Clos Networks with Different Nonblocking requirements on Two Types of Calls, Journal of Combinatorial Optimization, 9, 263-266, 2005

[4] Jastrzębski A.: Sterowanie rozsiewczymi polami Closa, master thesis, 2006