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Weighted Random Oblivious Routing on Torus Networks Rohit Sunkam Ramanujam Bill Lin Electrical and Computer Engineering University of California, San Diego

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Networks-On-Chip Chip-multiprocessors (CMPs) increasingly popular Torus, Mesh, Flattened Butterfly – candidate architectures for on-chip networks Intel Larrabee Tilera Tile64

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Networks-On-Chip Chip-multiprocessors (CMPs) increasingly popular Torus, Mesh, Flattened Butterfly – candidate architectures for on-chip networks Folded Torus 2D Torus

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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 ✔

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Outline Motivation Related Work Optimal routing for rings Optimal routing for 2D torus

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Optimal Oblivious Routing Cast as a Multi-commodity flow problem – Maximize worst-case throughput – Minimize hop-count Solve using Linear Programming Impractical for large networks – Number of paths too large (exponential) – Hard to make it deadlock-free – LP not scalable

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Optimal Oblivious Routing IdealOptimal 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 ✔ X

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Optimal 2TURN Optimum oblivious routing with only 2TURN paths. 1,2 2,2 3,2 1,12,13,1 0,2 1,32,33,3 0,0 0,1 0,3 1,02,03,0

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1,22,23,2 1,12,13,1 0,2 1,32,33,3 0,0 0,1 0,3 1,02,03,0 Optimal 2TURN Optimum oblivious routing with only 2TURN paths. 1,22,23,2 1,12,13,1 0,2 1,32,33,3 0,0 0,1 0,3 1,02,03,0

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Optimal 2TURN IdealOptimal Oblivious Optimal 2TURN Optimum worst-case throughput ✔✔✔ Low latency ✔✔✔ Good average-case throughput ✔✔✔ Easy to guarantee deadlock freedom ✔ X ✔ Low implementation complexity ✔ XX Closed-form algorithmic description ✔ XX

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Valiant Load Balancing (VAL) 2 phases of X-Y routing 1,22,23,2 1,12,13,1 0,2 1,32,33,3 0,0 0,1 0,3 1,02,03,0

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Improved Valiant Routing (IVAL) Phase1: X-Y, Phase2: Y-X 1,22,23,2 1,12,13,1 0,2 1,32,33,3 0,0 0,1 0,3 1,02,03,0

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Improved Valiant Routing (IVAL) Phase1: X-Y, Phase2: Y-X 1,22,23,2 1,12,13,1 0,2 1,32,33,3 0,0 0,1 0,3 1,02,03,0

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VAL and IVAL IdealOptimal Oblivious Optimal 2TURN VALIVAL Optimum worst-case throughput ✔✔✔✔✔ Low latency ✔✔✔ XX Good Average-case throughput ✔✔✔ X ✔ Deadlock freedom ✔ X ✔✔✔ Low implementation complexity ✔ XX ✔✔ Closed-form description ✔ XX ✔✔

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Latency Comparison 13.5%

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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 2TURN – Developed Weighted random 2TURN routing (W2TURN) – Analytically showed latency of W2TURN strictly better than I2TURN

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Outline Motivation Related Work Optimal routing for rings Optimal routing for 2D torus

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Routing on Rings Randomized Load Balancing (RLB) – Optimal worst-case throughput for rings Same routing strategy for both odd and even radix networks

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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 k

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Rings – The Difference Between Odd and Even RLB: Route minimally with probability (k-∆)/k Why 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)/2

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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-∆)/k

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WRD Algorithm Odd radix: – Route minimally with probability (k-∆)/k – Route non-minimally with probability ∆/k Even radix: – Route minimally with probability (k-∆-1)/(k-2) when k > 2 and ∆ > 0 – Route non-minimally with probability (∆-1)/(k-2) when k > 2 and ∆ > 0

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Latency Evaluation 25%

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WRD=Optimal

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WRD - Ideal for 1D Rings IdealWRD Optimum worst-case throughput ✔✔ Low latency ✔✔ Good average-case throughput ✔✔ Easy to guarantee deadlock freedom ✔✔ Low implementation complexity ✔✔ Closed-form algorithmic description ✔✔

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Outline Motivation Related Work Optimal routing for rings Optimal routing for 2D torus

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I2TURN Describe 2TURN paths in terms of 1D segments. 2TURN paths: X-Y-X or Y-X-Y 1,22,23,2 1,12,13,1 0,2 1,32,33,3 0,0 0,1 0,3 1,0 2,03,0 X-Y-X routing XSelect intermediate X position x* at uniform random Route minimally to x* YRoute using RLB on the Y ring at X=x*

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I2TURN Describe 2TURN paths in terms of 1D segments. 2TURN paths: X-Y-X or Y-X-Y 1,22,23,2 1,12,13,1 0,2 1,32,33,3 0,0 0,1 0,3 1,02,03,0 X-Y-X routing XSelect intermediate X position x* at uniform random Route minimally to x* YRoute using RLB on the Y ring at X=x* 1/4

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I2TURN Describe 2TURN paths in terms of 1D segments. 2TURN paths: X-Y-X or Y-X-Y 1,22,23,2 1,12,13,1 0,2 1,32,33,3 0,0 0,1 0,3 1,02,03,0 X-Y-X routing XSelect intermediate X position x* at uniform random Route minimally to x* YRoute using RLB on the Y ring at X=x* XRoute minimally to the destination 3/4 1/4

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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-ring Can easily derive analytical expression for average packet latency Can be proved to be equivalent to IVAL. Hence, it is worst-case throughput optimal Can define YXY routing by swapping dimensions

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W2TURN – Even Radix Reduces latency over I2TURN Use WRD instead of RLB Interpolate X-Y-X and Y-X-Y 2TURN routing with minimal X-Y and Y-X routing – XYX : k/2(k+1) – YXY : k/2(k+1) – XY: 1/2(k+1) – YX: 1/2(k+1)

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X-Y-X W2TURN 1,22,23,2 1,12,13,1 0,2 1,32,33,3 0,0 0,1 0,3 1,0 2,03,0 X-Y-X routing XSelect intermediate X position x* at uniform random Route minimally to x* YRoute using WRD on the Y ring at X=x*

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1,3 X-Y-X W2TURN 1,22,23,2 1,12,13,1 0,2 2,33,3 0,0 0,1 0,3 1,0 2,03,0 X-Y-X routing XSelect intermediate X position x* at uniform random Route minimally to x* YRoute using WRD on the Y ring at X=x* 1

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1,3 X-Y-X W2TURN 1,22,23,2 1,12,13,1 0,2 2,33,3 0,0 0,1 0,3 1,0 2,03,0 X-Y-X routing XSelect intermediate X position x* at uniform random Route minimally to x* YRoute using WRD on the Y ring at X=x* XRoute 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.

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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 probability Most of the intuition gained by observing optimal 2TURN paths

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Latency Evaluation 13.5%

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W2TURN ≈ Optimal-2TURN W2TURN = Optimal-2TURN for odd radix W2TURN within 0.72% of Optimal-2TURN for even radix

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Back to our Wishlist … IdealOptimal Oblivious Optimal 2TURN VALIVALW2TURN Optimum worst-case throughput ✔✔✔✔✔✔ Low latency ✔✔✔ XX ✔ Good average-case throughput ✔✔✔ X ✔✔ Easy to guarantee deadlock freedom ✔ X ✔✔✔✔ Low implementation complexity ✔ XX ✔✔✔ Closed-form algorithmic description ✔ XX ✔✔✔

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Summary of Contributions WRD: Optimal routing algorithm for rings – Worst-case throughput optimal – Minimum hop count W2TURN-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 radix WRD and W2TURN are best performing closed-form algorithms for 1D and 2D torus!!

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Thank You !!

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Average case throughput

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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.

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