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

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

3. 1. Alphabet size and min-cut tradeoff Directed graph with unit capacity edges, coding over F q. 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 Sufficient for h=2

4. An Example Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k

5. An Example Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k 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

6. An Example Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k For h=2, it is sufficient to consider q+1 coding vectors over F q : Any two such vectors form a basis of the 2-dimensional space

7. An Example Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k For h=2, it is sufficient to consider q+1 coding vectors over F q :

8. An Example Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k For h=2, it is sufficient to consider q+1 coding vectors over F q :

9. An Example Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k For h=2, it is sufficient to consider q+1 coding vectors over F q :

10. An Example Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k For h=2, it is sufficient to consider q+1 coding vectors over F q :

11. R3R3 R1R1 R2R2 Connection with Coloring Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k 13 2 k

12. R3R3 R1R1 R2R2 Connection with Coloring Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k 13 2 k Fragouli, Soljanin 2004

13. R1R1 If min-cut >2 Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k 1 3 2 k 4 R2R2 Each receiver observes a set of vertices Find a coloring such that every receiver observes at least two distinct colors

14. R1R1 Coloring families of sets 1 3 2 k 4 R2R2 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)

15. R1R1 Coloring families of sets 1 3 2 k 4 R2R2 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.

16. 2. What if the alphabet size is not large enough? Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k  N receivers  Alphabet of size q  Min-cut to each receiver m

17. R1R1 1 3 2 k 4 R2R2 There exists a coloring that colors at most Nq 1-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?

18. R1R1 1 3 2 k 4 R2R2 Erdos-Lovasz 1975: If every set intersects at most q m-3 other members, then the family is q-colorable. And if we know something about the structure? Source 1 Source 2 R1R1 RNRN R3R3 R2R2 13 2 k

19. R1R1 1 3 2 k 4 R2R2 Erdos-Lovasz 1975: If every set intersects at most q m-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.

20. What if links are not error free?

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

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

23. Network of Discrete Memoryless Channels 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

24. Network of Discrete Memoryless Channels 1-p p p 0 0 1 1 Binary Symmetric Channel (BSC) Edges Source Receiver Capacity We are interested in evaluating possible benefits of intermediate node processing from an information-theoretic point of view.

25. Network of Discrete Memoryless Channels 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

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

27. 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

28. Network Coding Receiver 1 Employing additional coding over the error free links allows to better share the available resources when multicasting Receiver 2 Source X 1 X 2 X 1 X 2 + Network Coding: Coding across independent information streams

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

30. Partial Processing We will show that: 1.Network and Channel Coding cannot be separated without loss of optimality.

31. 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.

32. 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.

33. 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.

34. Simple Example Source Receiver A B C D E 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

35. N infinite Source Receiver A B C D E Source Receiver A B C D E X1X1 X2X2 Min Cut = 2 (1-H(p)) X 1, X 2 iid

36. N=0: Forwarding Source Receiver A B C D E Source Receiver A B C D E X1X1 X2X2

37. N=0: Forwarding Source Receiver A B C D E Source Receiver A B C D E X1X1 X2X2

38. N=0: Forwarding Source Receiver A B C D E Source Receiver A B C D E X1X1 X2X2 Path diversity: receive multiple noisy observations of the same information stream and optimally combine them to increase the end-to-end rate X 1, X 2 iid

39. N=1 Source Receiver A B C D E 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

40. N=1 Source Receiver A B C D E 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 X1X1

41. N=1 Source Receiver A B C D E X1X1 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

42. N=1 Source Receiver A B C D E X1X1 X2X2 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

43. Optimal Processing at node D? Source Receiver A B C D E X1X1 X2X2 Three choices to send through edge DE: f1) X1 f2) X1+X2 f3) X1 and X2

44. All edges: BSC(p) A B C D E RateDE R1R1 X1X1 R2R2 X 1 +X 2 R3R3 X 1 & X 2 X1X1 X2X2 X2X2 X1X1 X1X1 X2X2 Network coding offers benefits for unicast connections

45. All edges: BSC(p) A B C D E RateDE R1R1 X1X1 R2R2 X 1 +X 2 R3R3 X 1 & X 2 X1X1 X2X2 X2X2 X1X1 X1X1 X2X2 The optimal processing depends on the channel parameters

46. Edges BD and CD: BSC(0) All other edges: BSC(p) A B C D E RateDE R1R1 X1X1 R2R2 X 1 +X 2 R3R3 X 1 & X 2 X1X1 X2X2 X2X2 X1X1 X1X1 X2X2 Network and channel coding cannot be separated

47. Edges AB, AC, BD and CD: BSC(0) Edges BE, DE and CE: BSC(p) A B C D E RateDE R1R1 X1X1 R2R2 X 1 +X 2 R3R3 X 1 & X 2 X1X1 X2X2 X2X2 X1X1 X1X1 X2X2

48. Edges AB, AC, BD and CD: BSC(0) Edges BE, DE and CE: BSC(p) A B C D E RateDE R1R1 X1X1 R2R2 X 1 +X 2 R3R3 X 1 & X 2 X1X1 X2X2 X2X2 X1X1 X1X1 X2X2

49. Linear Processing A B C D E X1X1 X2X2 Y2Y2 Y1Y1 Y3Y3 Choose matrix A to maximize

50. Connection to Coding Choose matrix A to maximize “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

51. Conclusions