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Processing Along the Way:Forwarding vs. Coding
Christina FragouliJoint work with Emina Soljanin and Daniela Tuninetti
A field with many interesting questions…
• Problem Formulations and Ongoing Work
Do credit cardswork in paradise?
1. Alphabet size and min-cut tradeoff
• Directed graph with unit capacity edges, coding over Fq.
• What alphabet size q is sufficient for all possible configurations
with h sources and N receivers?
If the min-cut to each receiver is h
NqN
2
1
4
72
Sufficient for h=2
An Example
Source 1 Source 2
R1RNR3R2
1 32 k
2
101x
x
Coding vector: vector of coefficients
Network Coding: assign a coding vector to each edge so that each receiver has a full rank set of equations
An Example
Source 1 Source 2
R1RNR3R2
1 32 k
For h=2, it is sufficient to consider q+1 coding vectors over Fq:
0 1 1 0 1 a 1 a2 1 aq 1
2
101x
x
Any two such vectors form a basis of the 2-dimensional space
An Example
Source 1 Source 2
R1RNR3R2
1 32 k
For h=2, it is sufficient to consider q+1 coding vectors over Fq:
0 1 1 0 1 a 1 a2 1 aq 1
2
101x
x
An Example
Source 1 Source 2
R1RNR3R2
1 32 k
For h=2, it is sufficient to consider q+1 coding vectors over Fq:
0 1 1 0 1 a 1 a2 1 aq 1
2
101x
x
An Example
Source 1 Source 2
R1RNR3R2
1 32 k
For h=2, it is sufficient to consider q+1 coding vectors over Fq:
0 1 1 0 1 a 1 a2 1 aq 1
2
101x
x
An Example
Source 1 Source 2
R1RNR3R2
1 32 k
For h=2, it is sufficient to consider q+1 coding vectors over Fq:
0 1 1 0 1 a 1 a2 1 aq 1
2
101x
x
R3
R1
R2
Connection with Coloring
Source 1 Source 2
R1RNR3R2
1 32 k
1 32 k
12 1110110 qaaa
Fragouli, Soljanin 2004
R1
If min-cut >2
Source 1 Source 2
R1RNR3R2
1 32 k
1
3
2k
4R2
12 1110110 qaaa
Each receiver observes a set of vertices
Find a coloring such that every receiver observes at least two distinct colors
R1
Coloring families of sets
1
3
2k
4R2
12 1110110 qaaa
Erdos (1963): Consider a family of N sets of size m.If N<q m-1 then the family is q-colorable.
A coloring is legal if no set is monochromatic.
q > N 1/(m-1)
R1
Coloring families of sets
1
3
2k
4R2
12 1110110 qaaa
Erdos (1963): Consider a family of N sets of size m.If N<q m-1 then the family is q-colorable.
A coloring is legal if no set is monochromatic.
2. What if the alphabet size is not large enough?
Source 1 Source 2
R1RNR3R2
1 32 k
N receiversAlphabet of size qMin-cut to each receiver m
R1
1
3
2k
4R2
12 1110110 qaaa
There exists a coloring that colors at most Nq1-m
sets monochromatically
If we have q colors, how many sets are going to be monochromatic?
2. What if the alphabet size is not large enough?
R1
1
3
2k
4R2
12 1110110 qaaa Erdos-Lovasz 1975:If every set intersects at most qm-3 other members, then the family is q-colorable.
And if we know something about the structure?
Source 1 Source 2
R1RNR3R2
1 32 k
R1
1
3
2k
4R2
12 1110110 qaaa Erdos-Lovasz 1975:If every set intersects at most qm-3 other members, then the family is q-colorable.
And if we know something about the structure?
•If m=5 and every set intersects 9 other sets, three colors – a binary alphabet is sufficient.
Network of Discrete Memoryless Channels
1-p
1-p
p
p
0 0
1 1
Binary Symmetric Channel (BSC)
Edges
Source Receiver
)(1 pHC Capacity
Network of Discrete Memoryless Channels
1-p
1-p
p
p
0 0
1 1
Binary Symmetric Channel (BSC)
Edges
Source Receiver
)(1 pHC Capacity
Min Cut = 2 (1-H(p))
Network of Discrete Memoryless Channels
1-p
1-p
p
p
0 0
1 1
Binary Symmetric Channel (BSC)
Edges
Vertices Terminals that have processing capabilities in terms of complexity and delay
Source Receiver
Network of Discrete Memoryless Channels
1-p
1-p
p
p
0 0
1 1
Binary Symmetric Channel (BSC)
Edges
Source Receiver
)(1 pHC Capacity
We are interested in evaluating possible benefits of intermediate nodeprocessing from an information-theoretic point of view.
Network of Discrete Memoryless Channels
1-p
1-p
p
p
0 0
1 1
Binary Symmetric Channel (BSC)
Edges
Vertices Terminals that have processing capabilities
Source Receiver
N
1111010001001111000
Complexity - Delay
N
N
N
Perfect and Partial Processing
Source ReceiverN
N
N
N
Two Cases:allow intermediate nodes
N finite
Perfect Processing Partial Processing
Perfect Processing
Source Receiver
We can use a capacity achieving channel code to transform each edge of the network to a practically error free link.
For a unicast connection: we can achieve the min-cut capacity
Network Coding
Receiver 1
Employing additional coding over the error free links allows to better share the available resources when multicasting
Receiver 2
Source
X1
X2
X1 X2+
Network Coding: Coding across independent information streams
Partial Processing
We will show that:
1. Network and Channel Coding cannot be separated without loss of optimality.
Partial Processing
We will show that:
1. Network and Channel Coding cannot be separated without loss of optimality.
2. Network coding can offer benefits for a single unicast connection. That is, there exist configurations where coding across information streams that bring independent information can increase the end-to-end achievable rate.
Partial Processing
We will show that:
1. Network and Channel Coding cannot be separated without loss of optimality.
2. Network coding can offer benefits for a single unicast connection. That is, there exist configurations where coding across information streams that bring independent information can increase the end-to-end achievable rate.
3. For a unicast connection over the same network, the optimal processing depends on the channel parameters.
Partial Processing
We will show that:
1. Network and Channel Coding cannot be separated without loss of optimality.
2. Network coding can offer benefits for a single unicast connection. That is, there exist configurations where coding across information streams that bring independent information can increase the end-to-end achievable rate.
3. For a unicast connection over the same network, the optimal processing depends on the channel parameters.
4. There exists a connection between the optimal routing over a specific graph and the structure of error correcting codes.
Simple Example
Source ReceiverA
B
C
D E
1-p
1-p
p
p
0 0
1 1
•Each edge:
•Nodes B, C and D can process N bits•Nodes A and E have infinite complexity processing
N=0: Forwarding
Source ReceiverA
B
C
D ESource ReceiverA
B
C
D E
X1
X2
Path diversity: receive multiple noisy observations of the same information stream and optimally combine
them to increase the end-to-end rate
X1, X2 iid
N=1
Source Receiver
A
B
C
D E
1-p
1-p
p
p
0 0
1 1
•Each edge:
•Nodes B, C and D can process one bit•Nodes A and E have infinite complexity processing
N=1
Source Receiver
A
B
C
D E
1-p
1-p
p
p
0 0
1 1
•Each edge:
•Nodes B, C and D can process one bit•Nodes A and E have infinite complexity processing
X1
N=1
Source Receiver
A
B
C
D E
X1
1-p
1-p
p
p
0 0
1 1
•Each edge:
•Nodes B, C and D can process one bit•Nodes A and E have infinite complexity processing
N=1
Source Receiver
A
B
C
D E
X1
X2
1-p
1-p
p
p
0 0
1 1
•Each edge:
•Nodes B, C and D can process one bit•Nodes A and E have infinite complexity processing
Optimal Processing at node D?
Source Receiver
A
B
C
D E
X1
X2
Three choices to send through edge DE: f1) X1 f2) X1+X2 f3) X1 and X2
All edges: BSC(p)
A
B
C
D E
Rate DE
R1 X1
R2 X1+X2
R3 X1 & X2
X1
X2 X2
X1X1
X2
Network coding offers benefits for unicast connections
All edges: BSC(p)
A
B
C
D E
Rate DE
R1 X1
R2 X1+X2
R3 X1 & X2
X1
X2 X2
X1X1
X2
The optimal processing depends on the channel parameters
Edges BD and CD: BSC(0)
All other edges: BSC(p)
A
B
C
D E
Rate DE
R1 X1
R2 X1+X2
R3 X1 & X2
X1
X2 X2
X1X1
X2
Network and channel coding cannot be separated
Edges AB, AC, BD and CD: BSC(0)
Edges BE, DE and CE: BSC(p)
A
B
C
D E
Rate DE
R1 X1
R2 X1+X2
R3 X1 & X2
X1
X2 X2
X1X1
X2
Edges AB, AC, BD and CD: BSC(0)
Edges BE, DE and CE: BSC(p)
A
B
C
D E
Rate DE
R1 X1
R2 X1+X2
R3 X1 & X2
X1
X2 X2
X1X1
X2
Linear Processing
3
2
1
2
1
3
2
1
**
10
01
N
N
N
X
X
Y
Y
Y
A
A
B
C
D E
X1
X2 Y2
Y1
Y3
Choose matrix A to maximize ),,;,( 32121 YYYXXI
Connection to Coding
3
2
1
2
1
3
2
1
**
10
01
N
N
N
X
X
Y
Y
Y
A
Choose matrix A to maximize ),,;,( 32121 YYYXXI
“Equivalent problem”: maximize the composite capacity of a BSC(p) that is preceded by a linear block encoder
Determined by the weight distribution of the code