rn static net

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Static Interconnection Networks

Dr Rajender Nath



Static and Dynamic

Interconnection Networks

Classification of interconnection networks: (a) a static

network; and (b) a dynamic network.



Interconnection Networks

• Switches map a fixed number of inputs to outputs.

• The total number of ports on a switch is the degree of the

switch.

• The cost of a switch grows as the square of the degreeof the switch, the peripheral hardware linearly as the

degree, and the packaging costs linearly as the number

of pins.



Network Topologies

• A variety of network topologies have been proposed and

implemented.

• These topologies tradeoff performance for cost.

• Commercial machines often implement hybrids ofmultiple topologies for reasons of packaging, cost, and

available components.



5

• Each node has a direct communication link toevery other node in the network.

• Ideal in the sense that a node can send amessage to another node in a single step.

• Static counterpart of crossbar switchingnetworks

• Nonblocking

Completely-connection Network



Network Topologies: Completely Connected

and Star Connected Networks

• N=8

• d=N-1

• D=1• # of Links=N(N-1)/2 =>cost

• Symmetric

• B=(N/2)2



Star-Connected Network

One Node acts as the central Node. Everyother Node has a communication link

connecting it to this central Node.Similar to bus-based network.The central Node is the bottleneck.Two-level tree



Star Connected Networks

N=9

d=N-1, d=1

D=2# of Links=N-1 =>cost

Asymmetric

B=N/2



Linear Array

N=5

d=2, d=1

D=N-1# of Links=N-1 =>cost

Asymmetric

B=1



Ring

N=8

d=2

D=⎣N/2

⎦ # of Links=N =>cost

Symmetric

B=2



Chordal Ring

N=8

d=3

D=3# of Links=12 =>cost

Symmetric

B=4-6



Chordal Ring

N=8

d=4

D=2# of Links=16 =>cost

Symmetric

B=6



Barrel Shifter

Number of nodes N = 2n

Start with a ring

Add extra edges from each

node to those nodeshaving power of 2

distance

i & j are connected if |j-I| =

2r

, r = 0, 1, 2, …, n-1



Barrel Shifter

N=8

d=2n-1

D=n/2Symmetric



Tree Network



Binary Tree

N =

d = 1,2,3

Height of the tree = h =⎡log2N⎤

D=2(h-1)B=1

Asymmetric



Fat Tree



Network Topologies:

Two- and Three Dimensional Meshes

Two and three dimensional meshes: (a) 2-D mesh with nowraparound; (b) 2-D mesh with wraparound link (2-D torus); and

(c) a 3-D mesh with no wraparound.



2D Mesh Network

N=nk = 9

k-dimensional

mesh=2

d=2, 3, 4(2k)

#of Links=kN-kn

D=k(n-1) Asymmetric

B=n



2D Torus Network

N=nk = 9

d=4(2k)

#of Links=2ND=2⎣n/2⎦ Symmetric

B=2n



2D Illiac Mesh Network

Equivalent to 4 degree

Chordal RingN=

d=4

#of Links=2ND=n-1

Asymmetric

B=2n



Hypercube

If a system has N = 2n nodes

Each node is allowed n links (i.e. d = n)

It is a hypercube of dimension n



Dr. Rajender Nath

Hypercubes

n = 0d = 0

n = 1d = 1

n = 2d = 2

n = 3d = 3

0

1

0100

1110

000

001

100 110

111

011

101

010



Hypercube

N=2n

d=n

D=n#of links = nN/2

B=N/2

Symmetric



Hypercube of dimension 4

1110

1111

1010

1011

0110 0111

0010

0011

1101

1010

1000

1001

0100

0101

0010

0000

0001

S

d = 4



Cube Connected Cycles (CCC)

k-cube 2k nodes

k-CCC from k-cube, replace each vertex of the k cube witha ring of k nodes

K-CCC k* 2k nodes

Degree, diameter 3, 2k

Try it for 3-cube



3-CCC



K-Cube Connected Cycles (CCC)

N=kx2k

d=3=k

D=2k-1+

k/2

# of links = k*N/2

B=N/(2k)

Symmetric



K-ary n-Cube

n = cube dimension

K = # nodes along each dimensionN = kn

Wraparound

Hupercube binary n-cube

Tours k-ary 2-cube



K-ary n-Cube

• Generalization of hypercubes (k-nodes in a string)

• Total # of nodes = N = k^n.

• k > 2 reduces # of channels atbisection, thus allowing forwider channels but more hops.



K-ary n-Cube

n = cube dimension

K = # nodes along each dimensionN = kn

Wraparound

Hupercube binary n-cube

Tours k-ary 2-cube



Dr. Rajender Nath



Thank You

rn static net

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