Network Coding for Routability Improvement in VLSI

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Network Coding for Routability Improvement in VLSI

• Nikhil Jayakumar
• Kanupriya Gulati
• Sunil P. Khatri
• Alexander Sprintson
• Department of Electrical Computer Engineering,
• Texas AM University, College Station, TX

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Outline

• Introduction to Network Coding
• Algorithm
• Using Network Coding in VLSI
• Examples
• Experimental Results
• Conclusions

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Communication Networks
s1

• Directed graph G(V,E).
• Source nodes S.
• Terminal nodes T.
• Requirement transfer information from S to T.

s2
t3
t2
t1
G(V,E)
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The Multicast Setting

• Multicast
• One source transmits data to all terminals.
• There may be several multicast groups.

s
t3
t2
t1
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What is Network Coding?

• Nodes of a network in a regular multicast setting
  
• transmit, forward, duplicate or receive messages
  (or packets).
• Network coding uses nodes of the network
• generate new packets.
• Perform algebraic operations on incoming packets.

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Standard Approach

• Steiner tree packing.
• Assume s1, s2 have messages m1 and m2
  respectively.
• Transmit each message on Steiner tree between S
  and T.
• m1 green.
• m2 blue.
• m1 and m2 transmitted to t1, t2 and t3.
• Nodes forward duplicate messages.

s1
t3
s2
t1
t2
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Network Coding

• Nodes do more than forwarding duplication.
• Information on outgoing link is
• Encoding of information received on incoming
  links.
• Ensuring that all messages can be decoded at the
  terminal nodes.

m1
F1(m1,m2,m3)
m2
F2(m1,m2,m3)
m3
encoding
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Network Coding helps!

• Network coding can help reduce wiring.

t3
s1
s2
t3
s1
s2
m2
m1
m2
m1
m2
m2
m1
m2
m1
m1
m1?m2
m2
m1
t1
t1
m1?m2
t2
m1?m2
t2
Steiner Tree Packing
Network Coding
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Advantage of Network Coding

• Improves throughput of multicast networks.
• Coding advantage (gain in throughput) as high as
• where, n number of nodes k number of
  terminals
• Charikar et al. Chekuri et al.
• For a network in which the routing topology is
  not pre-determined
• Coding advantage is at most 2 .
• Since the Steiner tree packing already exploits
  the additional flexibility.

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Preliminaries Cut

• A cut (V1,V2) is a partition of the nodes of the
  graph G into two subsets V1 and V2 V \ V1
• Size of the cut Number of links between nodes in
  V1 and V2.

V1
G
s
c
V2
t
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Preliminaries Cut (contd)

• A cut (V1,V2) separates s and t if s?V1 and t?V2.
  
• MinCutG(s,t) the minimum size of a cut that
  separates s and t in G.
• Mengers Theorem MinCutG(s,t) is equal to
  maximum number of disjoint paths between s and t.

V1
G
s
c
V2
t
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Algorithm

• Algorithm Jaggi et al.
• Input An acyclic coding network N(G,s,T,h)
• Output A feasible network code for N over a
  field of size k T.

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Algorithm

• Stage 1
• Find for each terminal t1 ?T, a set Pi of h
  disjoint paths between s and ti.

s
t
5
t
1
t
4
t
2
t
3
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Algorithm

• Stage 2
• Visit each node v of the graph G and determine
  the encoding function of all outgoing edges of v.
• Visit edges in topological order towards
  terminals
• Choose encoding function of each edge e such that
  all terminals downstream of e receive h linearly
  independent symbols.

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An Example
s
S
001
100
010
C
B
A
010
010
001
100
D
E
001
100
110
011
???
???
F
G
011
110
110
011
H
101
101
101
I
t1
t2
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An Example
s
S
001
100
I
II
010
C
B
A
010
010
III
001
100
D
E
001
100
110
011
F
G
011
110
110
011
I
H
II
101
III
101
101
I
t1
t2
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Algorithm

• Complexity of algorithm
• O(EkhVk2h2(kh))
• Jaggi et al.

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Using Network Coding in VLSI

• Signals on a VLSI chip can be viewed as instances
  of multicast communications
• Any signal with fanout gt 1 is an example of a
  multicast communication within an IC.
• Network coding can be useful in scenarios where
  wires are long (long buses)
• For short wires, overhead due to coding logic may
  overshadow wire-length reductions.
• Buffers are inserted in long wires anyway
• Augmenting them with coding logic incurs minimal
  overhead.
• Network coding actually reduces area power.
• Reduction of wires and buffers for long
  interconnects.

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A Generic Example

• Instruction (I) and data (D) buses routed to two
  terminals t1 and t2.
• Overlapping segment induces network nodes A, B

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Experiments

• Assumed 0.1um BPTM process
• VDD1.2V
• Assumed dimensions of chip 1cm X 1cm
• Considered two buses
• Instruction bus driven by instruction decode
  unit.
• Data bus driven by data cache.
• Used metal 3, metal 4 layers for route
• 0.5um wide wires.
• Bit-wise XOR used as coding logic.

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Test Case 1
No Coding
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Test Case 2
No Coding
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Future Work

• 3D routing instance.

S
b
S
a
a
b
b
a
a
t1
b
t1
b
a?b
a
a
b
t3
t3
a
a
a?b
b
t2
t2
b
a?b
Without Network Coding
With Network Coding
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Future Work

• 3D routing instance.
• Search for patterns in routing problems that can
  exploit network coding.
• Embedding this search in routers or buffer
  insertion routines.

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Conclusions

• We have extended the Network Coding paradigm to
  VLSI routing.
• Network coding for VLSI routing is useful in many
  commonly occurring situations.
• Several multicast nets with sections routed in
  parallel.
• In particular, long wires/buses running in
  parallel help reduce the overheads.
• Reduces wiring area (6 to 10), power (7 to
  9), active area (2 to 3).
• Small delay penalty (2 to 3)

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Thank you