CSE 291-a Interconnection Networks

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CSE 291-a Interconnection Networks. Lecture 7: February 7, 2007 Prof. Chung-Kuan Cheng CSE Dept, UC San Diego Winter 2007 Transcribed by Thomas Weng. Topics. Circuit Switching - Definitions: Nonblocking, rearrangeable, strict. Crossbar Clos Networks. 1. x. x. x. 2. x. x.

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CSE 291-aInterconnection Networks

Lecture 7: February 7, 2007

Prof. Chung-Kuan Cheng

CSE Dept, UC San Diego

Winter 2007

Transcribed by Thomas Weng

Topics
• Circuit Switching

- Definitions: Nonblocking, rearrangeable, strict.

• Crossbar
• Clos Networks

1

x

x

x

2

x

x

x

n

x

x

x

Crossbar
• n inputs, n outputs, n2 switches
• Rearrangeable, strict and wide nonblocking
• If n is small, this is usually the best choice.

1

2

n

Physical layout (what to do with many nodes?)

Control (packet switching)

Paper - BlackWidow: High-Radix Clos Networks, S. Scott, D. Abts, J. Kim, W.J. Dally

1

2

n

1

2

3

n

Engineering Issues
Physical Layout (example)

8x8 crossbar

1

. . . . . . . .

8

.

.

.

.

.

.

.

Goes to row 1, row 2, … , row 8

8 wires per row

64 horizontal wires

64 wires in between each signal

8x64 vertical wires in all

64 8x8 switches

Clos Network: Three Stage Clos(m,n,r)

n x m

n

n

1

1

1

n

n

2

2

2

.

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n

n

r

m

r

Clos Network (continued)
• rn inputs, rn outputs
• 2rnm + mr2 switches (this is less than r2n2)

Clos(m,n,r) is rearrangeable iff m >= n

Let m = n

• rn inputs

2rn2 + nr2 switches = (2n + r)rn

(a crossbar is rn2 switches)

Optimal choice of n and r?

Clos Network - Proof

Proof: By induction

Clos(1,1,r) – you have r boxes, each box is 1 x 1

n

1

1

r r

n

2

2

.

.

.

.

n

r

r

This is a crossbar, which we know is rearrangeable.

Clos Network – Proof (cont)

Assume that for the case Clos(n-1, n-1, r), n>=2, the statement is true. For the case Clos(n, n, r), we use the first switch in the middle to reduce the requirement to Clos(n-1, n-1, r).

n x m

n

n

1

1

1

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n

n

r

n

r

Clos Network – Proof (cont)

Permutation

Output side

Each box is a node with degree=n

(i,j) for p(j) = i

1

p1

n

n

2

p2

1

1

.

.

n

n

Bipartite graph

n

pn

2

2

n+1

n

n

.

.

n

n

2n

1

1

(r-1)n+1

2

2

.

Perfect matching

3

3

.

4

4

rxn

5

5

Each box will have n outputs

Each box will have n input edges

Because n inputs, n outputs, we can always find a perfect matching. If we take out a middle box, and now have (n-1) inputs, (n-1) outputs.

Clos Network – Strictly non-blocking

Clos Network is non-blocking in strict sense when m >= 2n-1.

n x (2n–1)

n

n

1

1

1

n

n

2

2

2

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n

n

r

2n-1

r

Each box has 2n-1 output pins

Each box has 2n-1 input pins

Clos Network – Proof

From i to j, we cannot make connection, e.g. from 1 to 2, we cannot make connection.

Only time we can’t make a connection is if all paths are taken.

Input i has taken n-1 signals, output j has taken n-1 signals. Thus, at most 2n-2 paths are taken.

However, we have 2n-1 boxes for 2n-1 distinct paths between i and j. So we will always have at least one path to go through.

Clos Network – Proof (cont)

At most n-1 boxes taken from 1, and n-1 boxes taken from 2, so 2n-2 boxes are taken.

n x (2n–1)

n

n

1

1

1

n-1

n

n

2

2

2

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n-1

n

n

r

2n-1

r

Clos Network – More than three stages

Clos Networks: Adding wires to reduce switches.

Can we do better? Add even more wires to reduce number of switches? Yes! By increasing number of stages.

Change middle stage box into another 3-stage Clos Network, this gives us 5-stage Clos Network. Can repeat this process!

Replace with 3-stage Clos Network

n

n

1

1

1

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n

n

r

m

r

Clos Network – More than three stages (cont)

C(1) = N2 switches (crossbar)

C(3) = 6N3/2 – 3N

C(5) = 16N4/3 – 14N + 3N2/3

C(7) = 36N5/4 – 46N + 20N3/4 – 3N1/2

C(9) = 76N6/5 – 130N + 86N4/5 – 26N3/5 + 3N2/5

(if N is huge, we want more levels)

Benes Network

Start with Butterfly Network. What if we flip this, and repeat this network to the other side? This is Benes Network.

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. . .

2n

0

1

n-1

n

2x2n inputs: (2n+1)2nx4 switches

N inputs: 2N(2log2N-1) switches