1. Network Alignment: Treating Networks as Wireless Interference Channel Chun Meng Univ. of California, Irvine

2. o Motivation: Network ≈ Wireless Interference Channel o Approaches: NA in the middle, Precoding-Based NA o PBNA Feasibility of PBNA o Conclusion 2 Outline

3. Intra-Session NC Achievable rate = min-cut [1,2] LP-formulation [3] Code design: RNC [4], deterministic [5] 3 State of the Art - I [1] R. Ahlswede, et al, “Network information flow” [2] R. Koetter and M. M′edard, “An algebraic approach to network coding” [3] Z. Li, et al, “On Achieving Maximum Multicast Throughput in Undirected Networks” [4] T. Ho, et al, “A random linear network coding approach to multicast” [5] S. Jaggi, et al, “Polynomial Time Algorithms for Multicast Network Code Construction”

4. Inter-Session NC Only approximation of bounds [1] Exponential number of variables Code design: NP-hard [5] LP, evolutionary approach 4 State of the Art - II [1] N. Harvey, et al, “On the Capacity of Information Networks” [2] A. R. Lehman and E. Lehman, “Complexity classification of network information flow problems” [3] D. Traskov, et al, “Network coding for multiple unicasts: An approach based on linear optimization” [4] M. Kim, et al, “An evolutionary approach to inter-session network coding”

5. 5 Restrictive Framework R. Koetter and M. M′edard, “An algebraic approach to network coding” Interference must be canceled out

6. 6 Network vs. Wireless Channel - I Network with multiple unicastsSISO Channel gain: introduced by nature Transfer function: introduced by network Min-cut = 1

7. 7 Networks vs. Wireless Channel - II Network with multiple unicasts MIMO Min-cut > 1 Transfer matrix Channel matrix

8. 8 Interference Alignment Common problem: Too MANY unknowns! Solution: Align interferences to reduce the number of unknowns V. Cadambe and S. Jafar, “Interference Alignment and Degrees of Freedom of the K-User Interference Channel” Benefit: Everyone gets one half of the cake

9. 9 Brief Intro of IA o Originally introduced by Cadambe & Jafar o Approaches: Asymptotic alignment, Ergodic alignment, Lattice alignment, Blind alignment o Applications K-user wireless interference channel, K-user MIMO interference channel, Cellular networks, Multi-hop interference networks, Exact repair in distributed storage Syed A. Jafar, “Interference Alignment — A New Look at Signal Dimensions in a Communication Network”

10. 10 Network Is NOT Wireless Channel

11. o Motivation: Network ≈ Wireless Interference Channel o Approaches: NA in the middle, Precoding-Based NA o PBNA Feasibility of PBNA o Conclusion 11 Outline

12. 12 NA in the Middle t=1 t=2 ≠== NA in the middle: B. Nazer, et al, "Ergodic Interference Alignment"

13. 13 NA in the Middle: Pros & Cons Pros: Achieve ½ in exactly 2 time slots Cons: Finding code is NOT easy

14. 14 Precoding-Based NA - I S1S1 S2S2 S3S3 D 1 D2D2 D3D3 2n+1 uses of network or 2n+1 symbol extension x1x1 x2x2 x3x3 n+1 n n y 1 =V 1 x 1 y 2 =V 2 x 2 y 3 =V 3 x 3 2n+1 V. R. Cadambe and S. A. Jafar, "Interference Alignment and Degrees of Freedom of the K-User Interference Channel“

15. 15 Precoding-Based NA - II M 11 V 1 x 1 M 12 V 2 x 2 M 13 V 3 x 3 M 22 V 2 x 2 M 21 V 1 x 1 M 23 V 3 x 3 M 33 V 3 x 3 M 32 V 2 x 2 M 31 V 1 x 1 Align interferences

16. 16 Precoding-Based NA - III Alignment conditions Rank conditions

17. 17 Precoding-Based NA - Advantages Code design is simple Encoding & decoding are predetermined regardless of topology

18. 18 Get a Better Understanding V 1 can NOT be chosen freely!

19. 19 Reformulated Feasibility Cond. Condensed alignment cond. Reformulated rank cond.

20. 20 Algebraic Formulation - I

21. 21 Algebraic Formulation - II

22. 22 Algebraic Formulation - III is full rank Linearly independent

23. 23 Algebraic Formulation - IV

24. 24 Algebraic Formulation - V

25. 25 Algebraic Formulation - VI p i (x) is not constant

26. 26 Summarization p i (x) is not constant

27. o Motivation: Network ≈ Wireless Interference Channel o Approaches: NA in the middle, Precoding-Based NA o PBNA Feasibility of PBNA o Conclusion 27 Outline

28. 28 Unfriendly Networks - I p i (x) is not constant

29. 29 Unfriendly Networks - II

30. 30 Coupling Relations

31. 31 Coupling Relations are Mostly Bad Bad guys Good guy Arbitrary precoding matrix V 1 is OK

32. 32 Networks vs. Wireless Channel Have structures Coupling relations Feasibility conditions are violated Structureless Can change independently IA is always feasible

33. 33 NOT All Coupling Relations are Realizable Max degree of x ee’ ≤ 2Max degree of x ee’ ≥ 3

34. 34 Topology and Coupling Relations

35. 35 How About Other Precoding Matrices? The ONLY one ?

36. 36 Answer to Q1 Answer:

37. 37 Answer to Q3 Answer: NO !

38. 38 Combining the Answers to Q1 & Q3

39. 39 Key Idea Behind Q-1 Graph-related properties

40. 40 Graph-Related Properties - I How to check p i (x) is not constant? 12 13 1 2 1 3 1 2 13

41. 41 Graph-Related Properties - II Linearization Property Assign values to x Max degree = 1

42. 42 Graph-Related Properties - III Intuition behind Linearization Property 1 1 3 2 e e’

43. 43 Graph-Related Properties - IV Square-Term Property Implication: Assign values to x

44. 44 Graph-Related Properties - V Intuition behind Square-Term Property 12 13 e e’ 13 12 e

45. 45 Finding Realizable Coupling Relations - I Objective: Step I Assign values to x Max degree of f(z) and g(z) = 1

46. 46 Finding Realizable Coupling Relations - II Step II Define No square term in the numerator

47. 47 Finding Realizable Coupling Relations - III Step III [1] J. Han, et al, “Analysis of precoding-based intersession network coding and the corresponding 3-unicast interference alignment scheme” Unrealizable

48. 48 How to Answer Q3 ? How to construct V 1 ?

49. 49 Example: Construct V 1

50. 50 All Precoding Matrices Are Equivalent Any V 1 cannot be used

51. 51 Topology of Coupling Relations - I 1 1 3 2

52. 52 Topology of Coupling Relations - II 1 1 2 3

53. 53 Topology of Coupling Relations - III

54. 54 Trivial Case Perfectly aligned p i (x) is not constant

55. 55 Trivial Case - Example

56. o Motivation: Network ≈ Wireless Interference Channel o Approaches: NA in the middle, Precoding-Based NA o PBNA Feasibility of PBNA o Conclusion 56 Outline

57. 57 Conclusion

58. 58 Open Questions

59. 59 http://odysseas.calit2.uci.edu/doku.php/public:publication Thank you ! Questions ?