Transcript slides

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
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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
21.72mm
2.0mm
Technology
65nm, 1 poly, 8 metal (Cu)
Transistors
100 Million (full-chip)
1.2 Million (tile)
Die Area
275mm2 (full-chip)
3mm2 (tile)
C4 bumps # 8390
Tilera Tile64
I/O Area
Intel 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 worstcase 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 minimallength 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 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
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Dimension-Ordered Routing (DOR)
Destination
Source
Issue:minimal
With Minimal
routing
throughput
either
XY or YX
routingbut
to poor
the destination
in the worst-case throughput
(here it uses XY route with probability 1.0)
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Orthogonal 1-TURN (O1TURN)
Destination
Source
Issue:
With Minimal
routing
and
thought
to be
Use both
minimal
XY and YX
routing
to the
destination
worst-case throughput optimal for even radices and
(½ XY + ½
YX) for odd radices (1/k2)
near worst-case throughput
optimal
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Valiant load-balancing (VAL)
Destination
Randomly chosen
intermediate node
Source
Issue: Minimal
thought XY
to be
worst-case
optimal
routing
to anythroughput
intermediate
node,with
1/2
network
capacity
but latency
2X of DOR
then
minimal
XY routing
to destination
node
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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 minimallength 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
3X3 mesh
VAL DOR
O1TURN
U2TURN
Worst-case
0.5
0.33
0.44
Average-case
0.5
Transpose
5X5
VAL DOR
O1TURN U2TURN
0.57
0.5
0.3
0.48
0.55
0.405 0.477
0.604
0.5
0.44
0.53
0.632
0.5
0.33
0.67
0.8
0.5
0.3
0.6
0.75
Random
0.5
1
1
0.72
0.5
1
1
0.685
DOR-WC
0.5
0.33
0.67
0.8
0.5
0.3
0.6
0.75
Complement
0.5
0.67
0.67
0.57
0.5
0.6
0.6
0.55
Nearest-Neighbor
0.5
1.33
1.33
0.75
0.5
2.4
2.4
1.17
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Throughput compared in EVEN mesh
4X4 mesh
VAL DOR
O1TURN
U2TURN
Worst-case
0.5
0.33
0.5
Average-case
0.5
0.48
Transpose
0.5
Random
6X6
VAL DOR
O1TURN U2TURN
0.5
0.5
0.3
0.5
0.5
0.54
0.64
0.5
0.47
0.556
0.65
0.33
0.67
0.8
0.5
0.3
0.6
0.75
0.5
1
1
0.7
0.5
1
1
0.682
DOR-WC
0.5
0.33
0.67
0.8
0.5
0.3
0.6
0.75
Complement
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
Nearest-Neighbor
0.5
2
2
1.1
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 twodimensional 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 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
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ROMM
Destination
Only choose
intermediate node
from restriction area
Source
either YX or XY routing to restricted intermediate node
Then either XY or YX routing to destination node
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Extend to Asymmetric Mesh
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