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![Page 1: 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,](https://reader036.fdocuments.us/reader036/viewer/2022081515/5697bfdc1a28abf838cb1071/html5/thumbnails/1.jpg)
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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
![Page 3: 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,](https://reader036.fdocuments.us/reader036/viewer/2022081515/5697bfdc1a28abf838cb1071/html5/thumbnails/3.jpg)
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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
![Page 5: 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,](https://reader036.fdocuments.us/reader036/viewer/2022081515/5697bfdc1a28abf838cb1071/html5/thumbnails/5.jpg)
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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
![Page 13: 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,](https://reader036.fdocuments.us/reader036/viewer/2022081515/5697bfdc1a28abf838cb1071/html5/thumbnails/13.jpg)
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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
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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
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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
![Page 22: 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,](https://reader036.fdocuments.us/reader036/viewer/2022081515/5697bfdc1a28abf838cb1071/html5/thumbnails/22.jpg)
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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