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Gaussian
Interconnections for
On-Chip Networks
Ramón Beivide and Enrique Vallejo
University of Cantabria, Spain
[email protected]
Gaussian Interconnections for On-chip Networks
Index
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Introduction: Why a Topology?
Dense Gaussian Networks and other topologies
Different layouts
Routing:
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
ideas: Adaptive routing, deadlock avoidance, fault tolerance
Unicast routing
 Broadcast Routing
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Perfect placement of resources
Expansibility:
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
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Increasing number of nodes in a Gaussian network
Hierarchical Gaussian networks
Some ideas about cache coherence
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Gaussian Interconnections for On-chip Networks
Introduction
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Future trends: many PE on a chip
Possible interconections: bus, MIN, direct
network
Bus-based interconnections do not scale – they
do not provide a sufficient bandwith when there
are many PEs. MIN hard to implement in a chip.
Direct networks with a given topology: The way
to connect different routers in the chip
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Gaussian Interconnections for On-chip Networks
Mesh Network
b
Number of Nodes N:
N=bxb=
-2+2i
-1+2i
2i
1+2i
2+2i
-2+i
-1+i
i
1+i
2+i
-2
-1
0
1
2
-2-i
-1-i
-i
1-i
2-i
-2-2i
-1-2i
-2i
1-2i
2-2i
b2
b N
Diameter k:
k = (b-1) + (b-1) = 2b-2
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Gaussian Interconnections for On-chip Networks
Diamond Network
Number of Nodes N:
3i
N = (b+1)2 + b2
b
2N 1 1
2
Diameter k= b + b = 2b
2 b+1
-3
-1+2i
2i
1+2i
-2+i
-1+i
i
1+i
2+i
-2
-1
0
1
2
-2-i
-1-i
-i
1-i
2-i
-1-2i
-2i
1-2i
3
-3i
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Gaussian Interconnections for On-chip Networks
Torus Network
b
Number of Nodes N:
N = b x b = b2
-2+2i
-1+2i
2i
1+2i
2+2i
-2+i
-1+i
i
1+i
2+i
-2
-1
0
1
2
-2-i
-1-i
-i
1-i
2-i
-2-2i
-1-2i
-2i
1-2i
2-2i
b N
Diameter k = b -1
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Gaussian Interconnections for On-chip Networks
Dense Gaussian Network
3i
Number of Nodes N:
N=
b
(b+1)2
+
2+i
2i
1+2i
2+i
2+i
i
1+i
2+i
-2
-1
0
1
2
2+i
2+i
-i
2+i
2+i
2+i
-2i
2+i
b2
2N 1 1
2
Diameter k = b
2b+1
-3
• Same # links as torus, with
peripheral links.
• Lower mean distance and
Diameter.
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Gaussian Interconnections for On-chip Networks
Topological properties comparative
Topology
Nodes
Diameter
Aprox.
Diam.
2-D Mesh
N W 2
2W  2
2 N
2-D Torus
N W 2
W 
2 
2
N
k
N
2
Dense
N  2k 2  2k  1
Gaussian
Average
Distance
2 N
3
2W
3
W  W  1  W
2  
 2   2  N  1

Aprox.
Aver. Dist

 2 k 2 1 
k 1 

 3N  1 
N
2
2N
3
Lower average distance and diameter
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Gaussian Interconnections for On-chip Networks
Area comparative
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Different Layouts
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3
3i
-2-i
-1-i
-i
1-i
2-i
-1+2i
2i
1+2i
1-2i
-2i
1-2i
-2+i
-1+i
i
1+i
2+i
-3i
-3
-2
-1
0
12
10
15
14
9
13
16
8
7
5
6
18
17
19
20
4
1
3
2
24
21
23
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Different layouts for the same network:
•Mesh-like layout
•Without peripheral links, bounded link length
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Gaussian Interconnections for On-chip Networks
Routing ideas
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Adaptive routing: in-fligh packets can choose their
(minimal) path from info in the Routing Record (jumps in
each direction), depending on congestion or other
parameters.
Deadlock avoidance: Bubble routing proposed as a
cost-effective deadlock avoidance mechanism (used in
IBM’s Blue Gene). Only 2 virtual channels needed per
link.
Fault-tolerant routing: Inmunet proposed as a fast,
efficient mechanism to detect link failures and restore
network performance.
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Gaussian Interconnections for On-chip Networks
Unicast Routing
3i
-3
-1+2i
2i
1+2i
-2+i
-1+i
i
1+i
2+i
-2
-1
0
1
2
-2-i
-1-i
-i
1-i
2-i
-1-2i
-2i
-3i
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Route from a to b:
Routing record generated
From the difference:
dest-source (x, y)
3
Example:
i – 2:
(-1-i)
1+2i (x=1, y=2)
Example
The= movement
makes
1 jump
to the right,
2 jumps up
the arrow
fall outside
the original
network  Peripheral links used
Movement from source
node to
the surrounding
origin (node 0)
Translations
from
generates
routingare
record.
replicas
of the network
considered, to obtain an optimal RR
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Gaussian Interconnections for On-chip Networks
Broadcast Routing
3i
NW
-3
SW
NE
-1+2i
2i
1+2i
-2+i
-1+i
i
1+i
2+i
-2
-1
P
1
2
-2-i
-1-i
-i
1-i
2-i
-1-2i
-2i
1-2i
3
• Triangle-based broadcast
• Minimum number of steps
• The same for any node
(due to peripheral links)
• Balanced use of resources
• Simple routing based on
labels (see abstract)
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Perfect placement of resources
Resource distribution over
the network.
All nodes have resources
within a given distance
(example: distance 1)
Resource example:
I/O ports
Processing elements
Memory banks
...
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Gaussian Interconnections for On-chip Networks
Expansibility: Increasing # nodes
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Increasing Gaussian network: Network can be expanded
with the number of nodes necessary to increase
diameter in 1 unit: 4k +4.
Alternatively, hierarchical Gaussian networks have been
proposed, joining several Gaussian networks with
another gaussian pattern. Useful for CMPs sceneries, for
example (different latency, link length, etc. in each
hierarchical level):
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Lower level: interconnection between different cores inside a
chip. Fast and reliable, with low latency
Higher level: interconnection between different chips. Slower
and with higher latency.
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Gaussian Interconnections for On-chip Networks
Expansibility: Increasing # nodes
Lower level (on-chip) with a
dense Gaussian pattern.
Higher level, with the same
pattern.
Central routers will have 8 links:
4 internal links
4 external links
Route from one node to another:
1) Route to the central router of
the same gaussian
2) Route in the higher level to
the desired gaussian.
3) Route from the central router
of the dest. Gaussian, to the
destiny node.
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Gaussian Interconnections for On-chip Networks
Cache coherence in Gaussian
networks
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Recent proposals based in broadcast, such as TokenB
(M. Hill) can beneficiate from a Gaussian
interconnection:
Broadcast block requests (latency optimized with
Gaussian interconection).
Unicast response with grants (Tokens) to use memory
blocks, between different nodes and main memory.
There is no need to maintain a directory for coherence.
Broadcast network can work as a bus with a snoopy-like
protocol.
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Additional comments (not
presented)
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Dense Gaussian Networks are isomorphic to
Dense Midimew Networks. However, these two
topologies are not isomorphic in the general
case (not dense). In this work, related to Dense
Gaussian networks, properties studied for both
Gaussian and Midimew topologies are
presented.
References in the next slide will be thus referred
to both Midimew and Gaussian networks
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Commented References (I)
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Midimew networks were first introduced in:
R. Beivide, E. Herrada, J.L. Balcázar, Agustín Arruabarrena, “Optimal
Distance Networks of Low Degree for Parallel Computers”. IEEE
Transactions on Computers, Vol. 40, No 10, Oct 1991, pp. 1109-1124.
This paper introduces properties, analysis and some rectangular (mesh-like)
layouts. Unicast routing is also proposed, but based on labeling nodes with
integer labels (instead of Gaussian numbers).
Bounded link-length layouts were introduced in:
E. Vallejo, R. Beivide y C. Martínez, “Practicable Layouts for Optimal
Circulant Graphs”. Proceedings of the “13th Euromicro Conference on
Parallel, Distributed and Network-based Processing”, Lugano, Switzerland,
Feb. 2005.
A previous work on Midimew folding, which obtained a worse result
(maximum link length 4) is the following one:
Francis C. M. Lau, Guihai Chen, “Optimal Layouts of Midimew Networks”.
IEEE Transactions on Parallel and Distributed Systems, Vol 7, No 9, pp
954-961
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Commented References (II)
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Gaussian Networks will be introduced in:
C. Martínez, R. Beivide, J. Gutierrez and E. Gabidulin. "On the Perfect tDominating Set Problem in Circulant Graphs and Codes over Gaussian
Integers". Accepted for presentation at ISIT’05, September, Australia.
This paper also deals with perfect resource placement.
Broadcasting in Dense Gaussian Networks will be introduced in:
R. Beivide, C. Martínez, E. Vallejo, J. Gutierrez, C. Izu, “Gaussian
Interconnection Networks”. Accepted for presentation at the Spanish
Parallelism Conferences, Sept. 05, Granada, Spain.
This paper also introduces unicast routing in terms of the Gaussian
numbers (instead of integer labels)
Hierarchical Gaussian Networks will be introduced in:
Miquel Moretó, Carmen Martínez, Enrique Vallejo, Ramón Beivide, Mateo
Valero, “Hierarchical Topologies for Large-scale Two-level Networks”,
Accepted for presentation at the Spanish Parallelism Conferences, Sept.
05, Granada, Spain.
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Commented References (III)
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Bubble routing is described in
V. Puente, C. Izu, R. Beivide, J.A. Gregorio, F. Vallejo and J.M.
Prellezo, “The Adaptative Bubble Router”, Journal of Parallel and
Distributed Computing. Vol 61 - nº 9, September 2001
Inmunnet was introduced in
V. Puente, J.A. Gregorio, F. Vallejo and R. Beivide. "Immunet: A
Cheap and Robust Fault-Tolerant Packet Routing Mechanism". 31th
Annual International Symposium on Computer Architecture (ISCA31), pp. 198-209, 2004.
Token Coherence was presented in:
M. M. K. Martin, M. D. Hill, and D. A. Wood. "Token Coherence:
Decoupling Performance and Correctness". 30th Annual
International Symposium on Computer Architecture (ISCA-30), pp.
182-193, 2003.
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