Weighted Random Oblivious Routing on Torus Networks - PowerPoint Presentation

Slide1

Weighted Random Oblivious Routing on Torus Networks

Rohit Sunkam Ramanujam

Bill Lin

Electrical and Computer Engineering

University of California, San DiegoSlide2

Networks-On-Chip

Chip-multiprocessors (CMPs) increasingly popular

Torus, Mesh, Flattened Butterfly – candidate architectures for on-chip networks

Intel Larrabee

Tilera Tile64Slide3

Networks-On-Chip

Chip-multiprocessors (CMPs) increasingly popular

Torus

, Mesh, Flattened Butterfly – candidate architectures for on-chip networks

Folded Torus

2D TorusSlide4

Routing Algorithm Wishlist

Ideal

Optimum worst-case

throughput

✔

Low latency

✔

Good average-case throughput

✔

Easy to guarantee deadlock freedom

✔

Low implementation complexity

✔

Closed-form algorithmic description

✔Slide5

Outline

Motivation

Related Work

Optimal routing for rings

Optimal routing for 2D torusSlide6

Optimal Oblivious Routing

Cast as a

Multi-commodity flow

problemMaximize worst-case throughput

Minimize hop-countSolve using Linear ProgrammingImpractical for large networksNumber of paths too large (exponential)Hard to make it deadlock-free

LP not scalableSlide7

Optimal Oblivious Routing

Ideal

Optimal Oblivious

Optimum worst-case

throughput

✔

✔

Low latency

✔

✔

Good average-case throughput

✔

✔

Easy to guarantee deadlock freedom

✔

X

Low implementation

complexity

✔

X

Closed-form algorithmic description

✔

XSlide8

Optimal 2TURN

Optimum oblivious routing with only 2TURN paths.

1,2

2,2

3,2

1,1

2,1

3,1

0,2

1,3

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0Slide9

1,2

2,2

3,2

1,1

2,1

3,1

0,2

1,3

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0

Optimal 2TURN

Optimum oblivious routing with only 2TURN paths.

1,2

2,2

3,2

1,1

2,1

3,1

0,2

1,3

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0Slide10

Optimal 2TURN

Ideal

Optimal Oblivious

Optimal

2TURN

Optimum worst-case

throughput✔

✔

✔

Low latency

✔

✔

✔

Good average-case throughput

✔

✔

✔

Easy to guarantee deadlock freedom

✔

X

✔

Low implementation complexity

✔

X

X

Closed-form algorithmic description

✔

X

XSlide11

Valiant Load Balancing

(VAL)

2 phases of

X-Y routing

1,2

2,2

3,2

1,1

2,1

3,1

0,2

1,3

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0Slide12

Improved Valiant Routing (IVAL)

Phase1:

X-Y

, Phase2:

Y-X

1,2

2,2

3,2

1,1

2,1

3,1

0,2

1,3

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0Slide13

Improved Valiant Routing (IVAL)

Phase1:

X-Y

, Phase2:

Y-X

1,2

2,2

3,2

1,1

2,1

3,1

0,2

1,3

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0Slide14

VAL and IVAL

Ideal

Optimal Oblivious

Optimal

2TURN

VAL

IVAL

Optimum worst-case

throughput

✔

✔

✔

✔

✔

Low latency

✔

✔

✔

X

X

Good Average-case throughput

✔

✔

✔

X

✔

Deadlock freedom

✔

X

✔

✔

✔

Low implementation

complexity

✔

X

X

✔

✔

Closed-form description

✔

X

X

✔

✔Slide15

Latency Comparison

13.5%Slide16

Evolution of W2TURN

Step 1

. Started with the simple case of 1D rings

Developed Weighted Random Direction (WRD)

Step 2

. Described 2TURN paths in IVAL in terms of routing on 1D segments (I2TURN)I2TURN has analytical expression for hop count.Step 3

. Combined the intuition gained from WRD, I2TURN and optimal 2TURNDeveloped Weighted random 2TURN routing (W2TURN)

Analytically showed latency of W2TURN strictly better than I2TURNSlide17

Outline

Motivation

Related Work

Optimal routing for rings

Optimal routing for 2D torusSlide18

Routing on Rings

Randomized Load Balancing (RLB) – Optimal worst-case throughput for rings

Same routing strategy for both odd and even radix networksSlide19

Some Facts …

Worst-case throughput determined by maximum channel load under most adversarial traffic

For a torus network with radix

k

,Maximum channel for worst-case throughput optimality = k/4 Even k

= k/4 – 1/4k Odd kSlide20

Rings – The Difference Between

Odd

and Even

RLB: Route minimally with probability (k-∆)/kWhy can’t we route minimally more often?

Total Channel load = (k-1)/2 * (k+1)/2k =

k/4 - 1/4k

= Maximum load for worst-case throughput optimality

Tornado traffic

∆ = (k-1)/2Slide21

Rings – The Difference Between Odd and

Even

RLB: Route

minimally

with probability (k-∆)/k.Can we route minimally more often?

Total Channel load = (k/2 – 1) * (k+2)/2k = k/4 – 1/k < Maximum load for worst-case throughput optimality

Tornado traffic

∆ = k/2-1

Route minimally with a probability of (k-∆-1)/(k-2) > (

k-∆)/kSlide22

WRD Algorithm

Odd radix:

Route minimally with probability (k-∆)/k

Route non-minimally with probability ∆/kEven radix:

Route minimally with probability (k-∆-1)/(k-2) when k > 2 and ∆ > 0Route non-minimally with probability (∆-1)/(k-2) when k > 2 and ∆ > 0Slide23

Latency Evaluation

25%Slide24

WRD=OptimalSlide25

WRD - Ideal for 1D Rings

Ideal

WRD

Optimum worst-case

throughput

✔

✔

Low latency

✔

✔

Good average-case throughput

✔

✔

Easy to guarantee deadlock freedom

✔

✔

Low implementation complexity

✔

✔

Closed-form algorithmic description

✔

✔Slide26

Outline

Motivation

Related Work

Optimal routing for rings

Optimal routing for 2D torusSlide27

I2TURN

Describe 2TURN paths in terms of 1D segments.

2TURN paths:

X-Y-X

or

Y-X-Y

1,2

2,2

3,2

1,1

2,1

3,1

0,2

1,3

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0

X-Y-X

routing

Select intermediate X position

x

* at uniform random

Route minimally to

x

*

Route using RLB on the Y ring at X=

x

*Slide28

I2TURN

Describe 2TURN paths in terms of 1D segments.

2TURN paths:

X-Y-X

or

Y-X-Y

1,2

2,2

3,2

1,1

2,1

3,1

0,2

1,3

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0

X-Y-X

routing

Select intermediate X position

x

* at uniform random

Route minimally to

x

*

Route using RLB on the Y ring at X=

x

*

1/4Slide29

I2TURN

Describe 2TURN paths in terms of 1D segments.

2TURN paths:

X-Y-X

or

Y-X-Y

1,2

2,2

3,2

1,1

2,1

3,1

0,2

1,3

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0

X-Y-X

routing

Select intermediate X position

x

* at uniform random

Route minimally to

x

*

Route using RLB on the Y ring at X=

x

*

Route minimally to the destination

3/4

1/4Slide30

I2TURN – Main Idea

For XYX routing,

load balance across the Y-rings

to make traffic along every Y-ring admissible

Use worst-case throughput optimal routing (RLB) on the Y-ringCan easily derive analytical expression for average packet latency

Can be proved to be equivalent to IVAL. Hence, it is worst-case throughput optimalCan define YXY routing by swapping dimensionsSlide31

W2TURN – Even Radix

Reduces latency

over I2TURN

Use WRD

instead of RLBInterpolate X-Y-X and Y-X-Y 2TURN routing with minimal X-Y and Y-X routingXYX : k/2(k+1)

YXY : k/2(k+1)XY: 1/2(k+1)YX: 1/2(k+1)Slide32

X-Y-X W2TURN

1,2

2,2

3,2

1,1

2,1

3,1

0,2

1,3

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0

X-Y-X

routing

Select intermediate X position

x

* at uniform random

Route minimally to

x

*

Route using WRD on the Y ring at X=

x

*Slide33

1,3

X-Y-X W2TURN

1,2

2,2

3,2

1,1

2,1

3,1

0,2

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0

X-Y-X

routing

Select intermediate X position

x

* at uniform random

Route minimally to

x

*

Route using WRD on the Y ring at X=

x

*

1Slide34

1,3

X-Y-X W2TURN

1,2

2,2

3,2

1,1

2,1

3,1

0,2

2,3

3,3

0,0

0,1

0,3

1,0

2,0

3,0

X-Y-X

routing

Select intermediate X position

x

* at uniform random

Route minimally to

x

*

Route using WRD on the Y ring at X=

x

*

Route minimally to the destination

1

When number of hops in both directions are equal, avoid using links used by minimal X-Y or Y-X routing.Slide35

W2TURN – Odd Radix

W2TURN = Optimal 2TURN

for odd radix

More elaborate description but easy to implement

Uses X-Y-X and Y-X-Y 2TURN routing with equal probabilityMost of the intuition gained by observing optimal 2TURN pathsSlide36

Latency Evaluation

13.5%Slide37

W2TURN ≈ Optimal-2TURN

W2TURN = Optimal-2TURN for odd radix

W2TURN within 0.72% of Optimal-2TURN for even radix Slide38

Back to our Wishlist …

Ideal

Optimal Oblivious

Optimal

2TURN

VAL

IVAL

W2TURN

Optimum worst-case

throughput

✔

✔

✔

✔

✔

✔

Low latency

✔

✔

✔

X

X

✔

Good average-case throughput

✔

✔

✔

X

✔

✔

Easy to guarantee deadlock freedom

✔

X

✔

✔

✔

✔

Low implementation

complexity

✔

X

X

✔

✔

✔

Closed-form algorithmic description

✔

X

X

✔

✔

✔Slide39

Summary of Contributions

WRD:

Optimal routing algorithm for rings

Worst-case throughput optimal

Minimum hop countW2TURN-Odd: Optimal 2TURN routing with a closed form description for 2D torus with odd radix

W2TURN-Even: Latency within 0.072% of optimal 2TURN routing for 2D torus with even radixWRD and W2TURN are

best performing closed-form algorithms for 1D and 2D torus!! Slide40

Thank You !!Slide41

Average case throughputSlide42

Proof of worst-case throughput optimality

Optimal worst-case channel load = 2*(Channel load for uniform traffic)

To prove a routing is worst-case throughput optimal, sufficient to prove that maximum channel load:

= k/4 when k is even.

= k/4 – 1/4k when k is odd.