1 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

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

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 ϒ wc (R) –The worst-case channel load ϒ wc (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 throughput ϴ wc (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, ϒ wc (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, ϴ wc (R), of minimal-length routing in odd radix one-dimensional mesh is ((K/2)/(k/4)) -1 = ½ for even; ((K-1)/2)/((k 2 -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, ϴ wc (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 Existing Routing Algorithms 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

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

10 Orthogonal 1-TURN (O1TURN) Source Destination Use both minimal XY and YX routing to the destination ( ½ XY + ½ YX) 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/k 2 )

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

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(X 1 YX 1 +X 2 YX 2 +X 3 YX 3 +….. ) + 1/2k (Y 1 XY 1 +Y 2 XY 2 +Y 3 XY 3 +….. )

14 Analytical Results For 2-dimensional mesh, the worst-case channel load, ϒ wc (R) of minimal- length routing is (k-1)/2 in Y-dimension, (k 2 -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, ϒ wc (R) for XYX-routing is (k-1)/2 for k= odd and (k 2 -1)/2k for YXY-routing Therefore the worst-case throughput, ϴ wc (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(k 2 -1)/2k)/((k 2 -1)/4k)) -1 = ((2k 2 -k-1/4k)/((k 2 -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

Throughput compared in ODD mesh 17 3X3 meshVALDORO1TURNU2TURN Worst-case Average-case Transpose Random DOR-WC Complement Nearest-Neighbor X5VALDORO1TURNU2TURN

Throughput compared in EVEN mesh 18 4X4 meshVALDORO1TURNU2TURN Worst-case Average-case Transpose Random DOR-WC Complement0.5 Nearest-Neighbor X6VALDORO1TURNU2TURN

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 Existing Routing Algorithms 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

22 ROMM Only choose intermediate node from restriction area either YX or XY routing to restricted intermediate node Source Destination Then either XY or YX routing to destination node

23 Extend to Asymmetric Mesh