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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, USA2Tsinghua University, China

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Motivation: Networks-on-Chip• Chip-multiprocessors (CMPs) increasingly popular• 2D-mesh networks often used as on-chip fabric

I/O Area

I/O Area

single tile

1.5mm

2.0mm

21.7

2mm

12.64mm

65nm, 1 poly, 8 metal (Cu)Technology

100 Million (full-chip) 1.2 Million (tile)

Transistors

275mm2 (full-chip) 3mm2 (tile)

Die Area

8390C4 bumps #

65nm, 1 poly, 8 metal (Cu)Technology

100 Million (full-chip) 1.2 Million (tile)

Transistors

275mm2 (full-chip) 3mm2 (tile)

Die Area

8390C4 bumps #

Tilera Tile64Intel 80-core

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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

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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

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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 throughput ϴwc(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

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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

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Outline

• Motivation for our work

Recap Existing 2D routing algorithms in mesh networks

• U2TURN routing algorithm

• Simulation results

• Extensions and future work

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Existing Routing AlgorithmsThe 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

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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

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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/k2)

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Valiant load-balancing (VAL)

Randomly chosenintermediate 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

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Outline

• Motivation for our work

• Recap Existing 2D routing algorithms in mesh networks

U2TURN routing algorithm

• Simulation results

• Extensions and future work

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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+….. )

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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

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Outline

• Motivation for our work

• Recap Existing 2D routing algorithms in mesh networks

• U2TURN routing algorithm

Simulation results

• Extensions and future work

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Worst-Case Throughput

Throughput compared in ODD mesh

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3X3 mesh VAL

DOR O1TURN U2TURN

Worst-case 0.5 0.33 0.44 0.57

Average-case 0.5 0.405 0.477 0.604

Transpose 0.5 0.33 0.67 0.8

Random 0.5 1 1 0.72

DOR-WC 0.5 0.33 0.67 0.8

Complement 0.5 0.67 0.67 0.57

Nearest-Neighbor 0.5 1.33 1.33 0.75

5X5 VAL DOR O1TURN U2TURN0.5 0.3 0.48 0.55

0.5 0.44 0.53 0.632

0.5 0.3 0.6 0.75

0.5 1 1 0.685

0.5 0.3 0.6 0.75

0.5 0.6 0.6 0.55

0.5 2.4 2.4 1.17

Throughput compared in EVEN mesh

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4X4 mesh VAL

DOR O1TURN U2TURN

Worst-case 0.5 0.33 0.5 0.5

Average-case 0.5 0.48 0.54 0.64

Transpose 0.5 0.33 0.67 0.8

Random 0.5 1 1 0.7

DOR-WC 0.5 0.33 0.67 0.8

Complement 0.5 0.5 0.5 0.5

Nearest-Neighbor 0.5 2 2 1.1

6X6 VAL DOR O1TURN U2TURN0.5 0.3 0.5 0.5

0.5 0.47 0.556 0.65

0.5 0.3 0.6 0.75

0.5 1 1 0.682

0.5 0.3 0.6 0.75

0.5 0.5 0.5 0.5

0.5 3 3 1.27

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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.

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Thank You

Questions?

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Existing Routing AlgorithmsThe 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

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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

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Extend to Asymmetric Mesh