On Linearity: A Taxonomy of Linear Network Codes

1. On Linearity: A Taxonomy of Linear Network Codes Sidharth Jaggi, Michelle Effros Tracey Ho, Muriel Medard Acknowledgement-Ralf Koetter

2. Network Coding http://tesla.csl.uiuc.edu/~koetter/NWC/Bibliography.html • 72 papers… • R. Ahlswede, N. Cai, S.-Y. R. Li and R. W. Yeung, "Network information flow," IEEE Trans. on Information Theory, vol. 46, pp. 1204-1216, 2000. • S.-Y. R. Li, R. W. Yeung, and N. Cai. "Linear network coding". IEEE Transactions on Information Theory,  Feburary, 2003 • *linear* Є 10 paper titles…, *algebraic* Є 5 paper titles… • # of papers in which linearity seems to play an integral part > 42

3. Linear Network Coding • Li et al. - “Sufficiently large field” • Koetter et al., … – “Algebraic” b0 b1 bm-1

4. Linear Network Coding • Li et al. - “Sufficiently large field” • Koetter et al., … – “Algebraic” • Jaggi et al., Ho et al., … - “Block” b0 b1 bm-1

5. Linear Network Coding • Li et al. - “Sufficiently large field” • Koetter et al., … – “Algebraic” • Jaggi et al., Ho et al., … - “Block” • Erez et al., Fragouli et al, … - “Convolutional” b0 b1

6. Linear Network Coding • Li et al. - “Sufficiently large field” • Koetter et al., … – “Algebraic” • Jaggi et al., Ho et al., … - “Block” • Erez et al., Fragouli et al, … - “Convolutional” • Dougherty et al., some networks need non-linear codes A B C

7. Outline B • Inter-relationships • Global reduction Feasibility ? ? A C ? ? A B

8. Outline B • Inter-relationships • Global reduction Feasibility • Local reduction Distributed single design ? ? A C ? A B ? A B

9. Outline B • Inter-relationships • Global reduction Feasibility • Local reduction Distributed single design • I/O ≠ ? ? A C ? A B A B y y’ x x

10. Outline B • Inter-relationships • Global reduction Feasibility • Local reduction Distributed single design • I/O ≠ • I/O = Different notions of linearity co-exist in network ? ? A C ? A B A B y y x x

11. Outline B • Inter-relationships • Global reduction Feasibility • Local reduction Distributed single design • I/O ≠ • I/O = Different notions of linearity co-exist in network ? ? A C ? A B Multicast Vs. General

12. Outline B • Inter-relationships • Global reduction Feasibility • Local reduction Distributed single design • I/O ≠ • I/O = Different notions of linearity co-exist in network • Complexity ? ? A C ? A B Multicast Vs. General

13. Outline B • Inter-relationships • Global reduction Feasibility • Local reduction Distributed single design • I/O ≠ • I/O = Different notions of linearity co-exist in network • Complexity • Unified Framework ? ? A C ? B Multicast Vs. General C A

14. Inter-relationships B ? ? A ? C

15. Inter-relationships G General Global A Algebraic B Block C Convolutional B Block Algebraic? Convolutional? G ? G ? A C

16. Inter-relationships G General Global A Algebraic B Block C Convolutional Does not exist Lehman et al/Ho et al example networks B “Switching” possible only for block codes G G A C

17. Inter-relationships M Multicast G General Global A Algebraic B Block C Convolutional Does not exist Algebraic, Block and Convolutional multicast codes exist B Algebraic Convolutional Block M M G G A C M

18. Inter-relationships M Multicast G General Global a Acyclic A Algebraic B Block C Convolutional Does not exist Only true for acyclic networks B Algebraic Convolutional Block M M a a G G A C M a

19. Algebraic versus Block M Multicast G General Global a Acyclic A Algebraic B Block C Convolutional Does not exist B M a G A

20. Algebraic versus Block M Multicast G General Global Local I/O ≠ a Acyclic A Algebraic B Block C Convolutional Does not exist B B A ? M a G ? A [β] β

21. Algebraic versus Block S n2 links . . . . . . n1 links R

22. Algebraic versus Block (β1)n1= (β2)n2=1 [β1]=[ ] 1 0 0 0 S S [β2]=[ ] 0 0 0 1 β1 β2 [β1] [β2] β1 β2 [β1] [β2] β2 [β2] β1 [β1] β2 [β2] β1 [β1] n2 links n2 links . . . . . . . . . . . . n1 links n1 links β2 [β2] β1 [β1] β1 β2 [β1] [β2] “Switching” possible only for block codes R R “Destructive interference” 0

23. Algebraic versus Block M Multicast G General Global Local I/O ≠ a Acyclic A Algebraic B Block C Convolutional Does not exist B ? M a M a G A

24. Algebraic versus Block M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist Addition is identical in both. One can choose [β]s which mimic β multiplication B G A B M a M a G A β [β] y y x x

25. Algebraic versus Block M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist Gives algorithm for block codes, if one has already designed algebraic codes. B G M a M a G A

26. Algebraic versus Convolutional M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist A C M a

27. Algebraic versus Convolutional M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist “Destructive interference” M a A C M a

28. Algebraic versus Block M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist Alternative to Erez et al. M a A C M a M a

29. Algebraic versus Convolutional Convolutional Algebraic Addition Addition Multiplication Multiplication

30. Algebraic versus Convolutional M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist M a A C M a M a

31. Algebraic versus Convolutional M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist Algebraic/block transfer function Convolutional transfer function (FIR) ≠ 0 0 . . . . . . 0 0 M a A C M a M a G

32. Algebraic versus Convolutional M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist Algebraic/Block transfer function Convolutional transfer function (IIR) ≠ 0 . . . . . . 0 0 M a A C M a M a G

33. Algebraic versus Convolutional M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist ? G M a A C M a M a G a

34. Block versus Convolutional M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist B G M a C

35. Block versus Convolutional M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist Block transfer function Convolutional transfer function ≠ B G G M a C

36. Block versus Convolutional M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist B G G M a C Fudge factor

37. Block versus Convolutional M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist Є epsilon rate loss B G Є G G M a C Fudge factor

38. Block versus Convolutional n n n Convolutional Sources n-d n-d n-d For all n>d (decoding delay), total throughput = C(n-d) Sinks

39. Block versus Convolutional Time-domain transfer matrix n Convolutional d 0 n 0 For all n>d (decoding delay), total throughput = C(n-d) Rate C(1-d/n)

40. Block versus Convolutional M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist Є epsilon rate loss B G Є G G M a C

41. Big picture M Multicast G General Global Local I/O ≠ Local I/O = a Acyclic A Algebraic B Block C Convolutional Does not exist Є epsilon rate loss B G G G Є G M a G M M a a G ? M a A C M a M a G

42. Complexity – Acyclic networks . . . intermediate nodes Ditto block, even non-linear codes (JCJ,TE,LL) . . . receivers Minimum block length m ≈0.5(log(|T|)) m>0.25(log(|T|)) (FIR) m>0.125(log(|T|)) (IIR)

43. Unified Framework Algebraic 101001101010… 010101110101… 001010110010… . . . Block Convolutional D??? What other features besides linearity? • Causality 2. L-stationarity 3. Finite number of memory elements

44. Unified Framework Finite number of memory elements L-stationarity Causality Filter banks F(z) l l

45. Conclusions • Reductions between different notions of linearity • Which notion is strongest? • Single algorithm design. • Co-existence of different types of linearity in a network. • Complexity • Delay elements are necessary for some cyclic networks. • Even for acyclic networks, convolutional codes can have shorter block-lengths. • Unified framework • Filter banks most general “reasonable” form of linearity.

47. Clip art acknowledgements • http://clear.msu.edu:16080/dennie/clipart/ • http://particleadventure.org/particleadventure/frameless/weak.html • http://www.finalfantasy.8m.net/pics/ff3/the%20end.gif

48. All the following slides contain material from a previous draft, which might be useful, but will not show up in the Allerton presentation.

49. Unified Framework Implementational interpretation Algebraic interpretation Filter banks Transfer function at any node matrix of rational functions F(z) l l

50. Complexity Distinct linear combinations s (1, 0) (1, 1) (1, 2) … (1, q-1) (0, 1) q+1 . . . intermediate nodes . . . receivers Therefore q≥n Therefore minimum block length m = log(q) ≥log(n)≈0.5(log(|T|))