Processing Along the Way: Forwarding vs' Coding

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Processing Along the WayForwarding vs. Coding

• Christina Fragouli
• Joint work with Emina Soljanin and Daniela
  Tuninetti

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A field with many interesting questions

• Problem Formulations and Ongoing Work

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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?

Sufficient for h2
4
An Example
Source 1
Source 2
k
1
3
2
RN
R2
R3
R1
5
An Example
Source 1
Source 2
Network Coding assign a coding vector to each
edge so that each receiver has a full rank set of
equations
k
1
3
2
Coding vector vector of coefficients
RN
R2
R3
R1
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An Example
Source 1
Source 2
For h2, it is sufficient to consider q1
coding vectors over Fq
k
1
3
2
Any two such vectors form a basis of the
2-dimensional space
RN
R2
R3
R1
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An Example
Source 1
Source 2
For h2, it is sufficient to consider q1
coding vectors over Fq
k
1
3
2
RN
R2
R3
R1
8
An Example
Source 1
Source 2
For h2, it is sufficient to consider q1
coding vectors over Fq
k
1
3
2
RN
R2
R3
R1
9
An Example
Source 1
Source 2
For h2, it is sufficient to consider q1
coding vectors over Fq
k
1
3
2
RN
R2
R3
R1
10
An Example
Source 1
Source 2
For h2, it is sufficient to consider q1
coding vectors over Fq
k
1
3
2
RN
R2
R3
R1
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Connection with Coloring
RN
R2
R3
R1
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Connection with Coloring
RN
R2
R3
R1
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If min-cut gt2
4
k
2
1
3
RN
R2
R3
R1
Each receiver observes a set of vertices
Find a coloring such that every receiver
observes at least two distinct colors
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Coloring families of sets
A coloring is legal if no set is monochromatic.
Erdos (1963) Consider a family of N sets of
size m. If Nltq m-1 then the family is
q-colorable.
4
k
2
1
q gt N 1/(m-1)
3
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Coloring families of sets
A coloring is legal if no set is monochromatic.
Erdos (1963) Consider a family of N sets of
size m. If Nltq m-1 then the family is
q-colorable.
4
k
2
1
3
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2. What if the alphabet size is not large enough?

• N receivers
• Alphabet of size q
• Min-cut to each receiver m

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2. What if the alphabet size is not large enough?
If we have q colors, how many sets are going to
be monochromatic?
There exists a coloring that colors at most
Nq1-m sets monochromatically
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k
2
1
3
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And if we know something about the structure?
Erdos-Lovasz 1975 If every set intersects at
most qm-3 other members, then the family is
q-colorable.
4
k
2
1
3
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And if we know something about the structure?
Erdos-Lovasz 1975 If every set intersects at
most qm-3 other members, then the family is
q-colorable.
4

• If m5 and every set intersects 9 other sets,
• three colors a binary alphabet is sufficient.

k
2
1
3
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What if links are not error free?
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Network of Discrete Memoryless Channels
Source
Receiver
Binary Symmetric Channel (BSC)
Edges
Capacity
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Network of Discrete Memoryless Channels
Source
Receiver
Min Cut 2 (1-H(p))
Binary Symmetric Channel (BSC)
Edges
Capacity
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Network of Discrete Memoryless Channels
Source
Receiver
Binary Symmetric Channel (BSC)
Edges
Terminals that have processing capabilities
in terms of complexity and delay
Vertices
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Network of Discrete Memoryless Channels
Source
Receiver
Binary Symmetric Channel (BSC)
Edges
Capacity
We are interested in evaluating possible benefits
of intermediate node processing from an
information-theoretic point of view.
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Network of Discrete Memoryless Channels
N
Source
Receiver
N
N
Binary Symmetric Channel (BSC)
Edges
Terminals that have processing capabilities
Vertices
Complexity - Delay
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Perfect and Partial Processing
N
Receiver
Source
N
N
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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
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Network Coding
Receiver 1

Source
Receiver 2
Employing additional coding over the error free
links allows to better share the available
resources when multicasting
Network Coding Coding across independent
information streams
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Partial Processing
N
Source
Receiver
N
N
We can no longer think of links as error free.
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Partial Processing

• We will show that
• Network and Channel Coding cannot be separated
  without loss of optimality.

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Partial Processing

• We will show that
• Network and Channel Coding cannot be separated
  without loss of optimality.
• 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.

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Partial Processing

• We will show that
• Network and Channel Coding cannot be separated
  without loss of optimality.
• 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.
• For a unicast connection over the same network,
  the optimal processing depends on the channel
  parameters.

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Partial Processing

• We will show that
• Network and Channel Coding cannot be separated
  without loss of optimality.
• 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.
• For a unicast connection over the same network,
  the optimal processing depends on the channel
  parameters.
• There exists a connection between the optimal
  routing over a specific graph and the structure
  of error correcting codes.

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Simple Example
Source
Receiver
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N infinite
X1
Source
Receiver
Source
Receiver
X2
Min Cut 2 (1-H(p)) X1, X2 iid
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N0 Forwarding
X1
Source
Receiver
Source
Receiver
X2
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N0 Forwarding
X1
Source
Receiver
Source
Receiver
X2
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N0 Forwarding
X1
Source
Receiver
Source
Receiver
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
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N1
Source
Receiver
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N1
X1
Source
Receiver
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N1
X1
Source
Receiver
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N1
X1
Source
Receiver
X2
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Optimal Processing at node D?
Source
Receiver
Three choices to send through edge DE f1) X1
f2) X1X2 f3) X1 and X2
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All edges BSC(p)
X1
X1
X1
X2
X2
X2
Network coding offers benefits for unicast
connections
45
All edges BSC(p)
X1
X1
X1
X2
X2
X2
The optimal processing depends on the channel
parameters
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Edges BD and CD BSC(0) All other edges
BSC(p)
X1
X1
X1
X2
X2
X2
Network and channel coding cannot be separated
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Edges AB, AC, BD and CD BSC(0) Edges BE, DE
and CE BSC(p)
X1
X1
X1
X2
X2
X2
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Edges AB, AC, BD and CD BSC(0) Edges BE, DE
and CE BSC(p)
X1
X1
X1
X2
X2
X2
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Linear Processing
Choose matrix A to maximize
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Connection to C oding
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
Choose matrix A to maximize
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Conclusions