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Oblivious Routing Design for Mesh Networks to Achieve a New Worst-Case Throughput Bound

Oblivious Routing Design for Mesh Networks to Achieve a New Worst-Case Throughput Bound. Guang Sun 1,2 , Chia-Wei Chang 1 , Bill Lin 1 , Lieguang Zeng 2 , 1 University of California, San Diego, USA 2 Tsinghua University, China. Motivation: Networks-on-Chip.

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Oblivious Routing Design for Mesh Networks to Achieve a New Worst-Case Throughput Bound

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  1. Oblivious Routing Design for Mesh Networks to Achieve a New Worst-Case Throughput Bound Guang Sun1,2, Chia-Wei Chang1, Bill Lin1, Lieguang Zeng2, 1University of California, San Diego, USA 2Tsinghua University, China

  2. Motivation: Networks-on-Chip • Chip-multiprocessors (CMPs) increasingly popular • 2D-mesh networks often used as on-chip fabric 12.64mm I/O Area single tile 1.5mm 2.0mm 21.72mm Tilera Tile64 Intel 80-core I/O Area

  3. Routing Algorithm Objectives • Maximize throughput (much important) • How much load the network can handle • Minimize hop count (within acceptable range) • Minimize routing delay between source and destination

  4. Challenges • 1/2 network capacity is often believed to be the limit of worst-case throughput for mesh networks • For 2D-case, a near-optimal throughput routing algorithm with minimal hop count called O1TURN is known [Seo’05] • Only known optimal throughput routing algorithm is Valiant (VAL) load-balancing, but VAL performs poorly on hop count (latency), twice that of minimal routing • However, 1/2 network capacity is not the limit of worst-case throughput for odd radix mesh networks

  5. Definitions • Maximal channel load ϒ(R, Λ) • for a given routing algorithm R and traffic matrix Λ, the maximal channel load ϒ(R, Λ) is the expected traffic loads crossing the heaviest loaded channel under R , Λ • Worst-case channel load ϒwc(R) • The worst-case channel load ϒwc(R) is the maximal channel load that can be caused by any admissible traffic • The worst-case channel load is the inverse of worst-case throughput • Worst case throughputϴwc(R) • we use the normalized worst-case throughput, which is normalized to the network capacity, as worst-case performance metric: • Network capacity C=1/ϒ* • Network capacity is defined by the maximal sustainable channel load ϒ* when a network is loaded with uniformly distributed traffic • where ϒ* is the inverse of the network capacity

  6. Observations • For one-dimensional mesh, the worst-case channel load, ϒwc(R) of minimal-length routing is (k-1)/2 when the radix k is odd and k/2 when k is even • Therefore the worst-case throughput, ϴwc(R), of minimal-length routing in odd radix one-dimensional mesh is ((K/2)/(k/4))-1= ½ for even; ((K-1)/2)/((k2-1)/4k))-1= (2k/k+1) -1 =(K+1)/2K for odd which is > ½(!= ½) • Next we are interested in • finding what is the limit/bound of worst-case throughput, ϴwc(R), in odd radix two-dimensional mesh networks • Develop a near-optimal throughput routing algorithm with acceptable hop count called U2TURN to achieve this worst-case throughput bound for odd radix meshes

  7. Outline • Motivation for our work • Recap Existing 2D routing algorithms in mesh networks • U2TURN routing algorithm • Simulation results • Extensions and future work

  8. The 2D case Dimension-Ordered Routing (DOR), 1977 Route minimal XY Orthogonal 1-TURN (O1TURN), 2005 Route minimal XY and YX with equal probability Valiant load-balancing (VAL), 1981 Route source → randomly chosen intermediate node → destination Route minimal XY in both phases Existing Routing Algorithms

  9. Dimension-Ordered Routing (DOR) Destination Source Issue: With Minimal routing but poor throughput in the worst-case throughput either minimal XYorYX routing to the destination (here it uses XY route with probability 1.0)

  10. Orthogonal 1-TURN (O1TURN) Destination Source Issue: With Minimal routing and thought to be worst-case throughput optimal for even radices and near worst-case throughput optimal for odd radices (1/k2) Use both minimal XYandYX routing to the destination (½ XY + ½ YX)

  11. Valiant load-balancing (VAL) Destination Randomly chosen intermediate node Source Issue: thought to be worst-case throughput optimal with 1/2 network capacity but latency 2X of DOR Minimal XY routing to any intermediate node, then minimal XY routing to destination node

  12. Outline • Motivation for our work • Recap Existing 2D routing algorithms in mesh networks • U2TURN routing algorithm • Simulation results • Extensions and future work

  13. U2TURN • In the beginning, U2TURN also considers 50% go XY direction and 50% go YX direction • Then U2TURN takes the left one-dimensional freedom to load-balance the link/channel-load : 20% (1/K) for each one-dimension choice • Therefore the total routing decision is • ½ XYX + ½ YXY = 1/2k(X1YX1+X2YX2+X3YX3+….. )+ 1/2k (Y1XY1+Y2XY2+Y3XY3+….. )

  14. Analytical Results • For 2-dimensional mesh, the worst-case channel load, ϒwc(R) of minimal-length routing is (k-1)/2 in Y-dimension, (k2-1)/2k in X-dimension when the radix k is odd and k/2 in X, Y when k is even • Therefore the worst-case channel load, ϒwc(R) for XYX-routing is (k-1)/2 for k= odd and (k2-1)/2k for YXY-routing • Therefore the worst-case throughput, ϴwc(R), of minimal-length routing in odd radix one-dimensional mesh is ((k/2)/(k/4))-1= ½ for even; ((0.5(k-1)/2+ 0.5(k2-1)/2k)/((k2-1)/4k))-1= ((2k2-k-1/4k)/((k2-1)/4k)) -1 =(k+1)/(2k+1) > ½ better then any existed routing algorithms

  15. Outline • Motivation for our work • Recap Existing 2D routing algorithms in mesh networks • U2TURN routing algorithm • Simulation results • Extensions and future work

  16. Worst-Case Throughput

  17. Throughput compared in ODD mesh

  18. Throughput compared in EVEN mesh

  19. Main Contributions • We derived a new worst-case throughput bound, which is higher than 1/2 network capacity, for odd radix two-dimensional mesh networks • Developed a newly discovered oblivious routing algorithm called “U2TURN” routing for 2D odd radix meshes to achieve the new discovered bound with analytical results • U2TURN provably guarantees optimal worst-case throughput in 2D odd radix mesh networks • However U2TURN is a non-minimal routing, which has 1.5X average hop count when compared with O1TURN and DOR.

  20. Thank You Questions?

  21. The 2D case Dimension-Ordered Routing (DOR) Route minimal XY Orthogonal 1-TURN (O1TURN) Route minimal XY and YX with equal probability Valiant load-balancing (VAL) Route source → randomly chosen intermediate node → destination Route minimal XY in both phases ROMM Same as VAL, but intermediate node restricted to minimal direction Existing Routing Algorithms

  22. ROMM Destination Only choose intermediate node from restriction area Source either YXorXY routing to restricted intermediate node Then either XYorYX routing to destination node

  23. Extend to Asymmetric Mesh

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