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
1
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
2
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
3
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
4
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
5
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
6
Outline
• Motivation for our work
 Recap Existing 2D routing algorithms in mesh networks
• U2TURN routing algorithm
• Simulation results
• Extensions and future work
7
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
8
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)
9
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
10
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
11
Outline
• Motivation for our work
• Recap Existing 2D routing algorithms in mesh networks
 U2TURN routing algorithm
• Simulation results
• Extensions and future work
12
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+….. )
13
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
14
Outline
• Motivation for our work
• Recap Existing 2D routing algorithms in mesh networks
• U2TURN routing algorithm
 Simulation results
• Extensions and future work
15
Worst-Case Throughput
16
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
17
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
18
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.
19
Thank You
Questions?
20
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
21
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
22
Extend to Asymmetric Mesh
23