Processing Along the Way: Forwarding vs. Coding

Processing Along the Way:Forwarding vs. Coding Christina Fragouli Joint work with Emina Soljanin and Daniela Tuninetti

Do credit cards work in paradise? A field with many interesting questions… • Problem Formulations and Ongoing Work

If the min-cut to each receiver is h 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 h=2

An Example Source 2 Source 1 k 2 1 3 RN R2 R3 R1

An Example Source 2 Source 1 Network Coding: assign a coding vector to each edge so that each receiver has a full rank set of equations k 2 1 3 Coding vector: vector of coefficients RN R2 R3 R1

An Example Source 2 Source 1 For h=2, it is sufficient to consider q+1 coding vectors over Fq: k 2 1 3 Any two such vectors form a basis of the 2-dimensional space RN R2 R3 R1

An Example Source 2 Source 1 For h=2, it is sufficient to consider q+1 coding vectors over Fq: k 2 1 3 RN R2 R3 R1

Source 2 Source 1 R3 k k 2 2 1 1 3 3 R2 R1 Connection with Coloring RN R2 R3 R1

Source 2 Source 1 R3 k k 2 2 1 1 3 3 R2 R1 Fragouli, Soljanin 2004 Connection with Coloring RN R2 R3 R1

Source 2 Source 1 R2 k 2 1 3 R1 If min-cut >2 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

R2 R1 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 N<q m-1 then the family is q-colorable. 4 k 2 1 q > N 1/(m-1) 3

R2 R1 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 N<q m-1 then the family is q-colorable. 4 k 2 1 3

Source 2 Source 1 k 2 1 3 RN R2 R3 R1 2. What if the alphabet size is not large enough? • N receivers • Alphabet of size q • Min-cut to each receiver m

R2 R1 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 4 k 2 1 3

Source 2 Source 1 R2 k 2 1 3 R1 RN R2 R3 R1 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

R2 R1 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 m=5 and every set intersects 9 other sets, • three colors – a binary alphabet is sufficient. k 2 1 3

What if links are not error free?

1-p 0 0 p p 1 1 1-p Network of Discrete Memoryless Channels Source Receiver Binary Symmetric Channel (BSC) Edges Capacity

1-p 0 0 p p 1 1 1-p Network of Discrete Memoryless Channels Source Receiver Min Cut = 2 (1-H(p)) Binary Symmetric Channel (BSC) Edges Capacity

1-p 0 0 p p 1 1 1-p Network of Discrete Memoryless Channels Source Receiver Binary Symmetric Channel (BSC) Edges Terminals that have processing capabilities in terms of complexity and delay Vertices

1-p 0 0 p p 1 1 1-p 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.

1-p 0 0 p N p 1111010001001111000 1 1 1-p Network of Discrete Memoryless Channels N Source Receiver N N Binary Symmetric Channel (BSC) Edges Terminals that have processing capabilities Vertices Complexity - Delay

Two Cases: allow intermediate nodes finite Partial Processing Perfect Processing Perfect and Partial Processing N Receiver Source N N

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

X X X X 1 2 1 2 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

Partial Processing N Source Receiver N N We can no longer think of links as error free.

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

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.

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.

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.

Each edge: 1-p 0 0 p • Nodes B, C and D can process N bits • Nodes A and E have infinite complexity processing C A B E D p 1 1 1-p Simple Example Source Receiver

E B C D E A C B A D N infinite X1 Source Source Receiver Receiver X2 Min Cut = 2 (1-H(p)) X1, X2 iid

E B C E D A B A D C N=0: Forwarding X1 Source Source Receiver Receiver X2

A C B A E D D C B E N=0: Forwarding X1 Source Source Receiver 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

Each edge: 1-p 0 0 p • Nodes B, C and D can process one bit • Nodes A and E have infinite complexity processing C A B E D p 1 1 1-p N=1 Source Receiver

Each edge: 1-p 0 0 p • Nodes B, C and D can process one bit • Nodes A and E have infinite complexity processing E C D A B p 1 1 1-p N=1 X1 Source Receiver

Each edge: 1-p 0 0 p • Nodes B, C and D can process one bit • Nodes A and E have infinite complexity processing C B A D E p 1 1 1-p N=1 X1 Source Receiver

Each edge: 1-p 0 0 p • Nodes B, C and D can process one bit • Nodes A and E have infinite complexity processing E C D A B p 1 1 1-p N=1 X1 Source Receiver X2

X1 X2 A B C D E Optimal Processing at node D? Source Receiver Three choices to send through edge DE: f1) X1 f2) X1+X2 f3) X1 and X2

C A B D E All edges: BSC(p) X1 X1 X1 X2 X2 X2 Network coding offers benefits for unicast connections

C A B D E All edges: BSC(p) X1 X1 X1 X2 X2 X2 The optimal processing depends on the channel parameters

A B C D E Edges BD and CD: BSC(0) All other edges: BSC(p) X1 X1 X1 X2 X2 X2 Network and channel coding cannot be separated

A B C D E Edges AB, AC, BD and CD: BSC(0) Edges BE, DE and CE: BSC(p) X1 X1 X1 X2 X2 X2

Y1 X1 Y3 X2 Y2 A B C D E Linear Processing Choose matrix A to maximize

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

Processing Along the Way: Forwarding vs. Coding