1. Coding and Scheduling for Erasures and Broadcast RamkiGummadi

2. Overview • Ratelesscodes in network applications • Efficient Repair in Storage Problems • Systematic Rateless Codes • Broadcasting with Side Information • Broadcasting over multiple hops • Role of Coding in Wireless Erasure Networks • Control of a Broadcast Server • Fixed Costs for Server • Online Constraint on Server

3. Overview • Ratelesscodes in network applications • Efficient Repair in Storage Problems • Systematic Rateless Codes • Broadcasting with Side Information • Broadcasting over multiple hops • Role of Coding in Wireless Erasure Networks • Control of a Broadcast Server • Fixed Costs for Server • Online Constraint on Server

4. A Dynamic Storage System 1 2….. k

5. A Dynamic Storage System 1 2….. k

6. A Dynamic Storage System 1 2….. k • Repair Complexity: # of purples per repair (avg) • Overhead: smallest dsuch that any k(1+d) sufficient

7. Fountain Codes for Storage? 1 2….. k • Low (En/De)-coding Complexity • Low Overhead • Rateless

8. Fountain Codes for Storage? Not possible to repair even one failurewithout dealing with the whole block!

9. Fountain Codes for Storage? 1 2….. k • Low (En/De)-coding Complexity • Low Overhead • Rateless • Repair Complexity

10. Augmented LT Code Ω 1 2….. k …. 1 3 2 k

11. Augmented LT Code Repair Algorithm …. 1 3 2 k Ω

12. Augmented LT Code Repair Algorithm …. 1 3 2 k ? ?

13. Repair of Fixed Symbols Fixed Rateless

14. Repair of Fixed Symbols

15. Repair of Fixed Symbols

16. Repair of Fixed Symbols

17. Repair of Fixed Symbols

18. Repair of Fixed Symbols: Step 1 # of symbol operations in repair = degree of code symbol processed = 2

19. Repair of Fixed Symbols

20. Repair of Fixed Symbols

21. Repair of Fixed Symbols

22. Repair of Fixed Symbols

23. Repair of Fixed Symbols: Step 2 # of symbol operations in repair = degree of code symbol processed = 3

24. Repair of Fixed Symbols

25. Augmented LT Code Repair Algorithm …. 1 3 2 k ? • Repair Complexity: (1+ε)Ω’(1) • Reduced from θ(k) to θ(1), in exchange for overhead increase from ε to 1+ε ?

26. Augmented LT Code Repair Algorithm …. 1 3 2 k ? • Repair Complexity: (1+ε)Ω’(1) • Reduced from θ(k) to θ(1), in exchange for overhead increase from ε to 1+ε • Next Goal: Improve overhead while keeping Repair complexity order optimal ?

27. Augmented Raptor Codes m1 m2 … mk

28. Augmented Raptor Codes m1 m2 … mk Rate (1+ε) “precode” s1 s2 … sk(1+ε)

29. Augmented Raptor Codes m1 m2 … mk Rate (1+ε) “precode” s1 s2 … sk(1+ε) Ω Object of optimization s1 s2sk(1+ε) ….

30. Overhead Optimization Consider an arbitrary set of k(1+δ) symbols s1 s2 sk(1+ε) ….

31. Overhead Optimization • Consider an arbitrary set of k(1+δ) symbols • Fraction α from fountain part s1 sk(1+ε) k(1+δ)(1-α) k(1+δ)α • Need to recover at least k for precode to take over

32. Overhead Optimization • Consider an arbitrary set of k(1+δ) symbols • Fraction α from fountain part Parameters α(arbitrary) δ(to minimize) ε(to design) s1 sk(1+ε) k(1+δ)(1-α) k(1+δ)α • Need to recover at least k for precode to take over

33. Background: Degree design • r : # code symbols • Ω : Degree distn xt # degree 1 packets t 1 Fraction Decoded [Darling and Norris, 2005]

34. Recovery Constraint δ :minimize ε :design α: adversarial

35. Optimal Overhead By fixing M and Ω we get achievable ‘profiles’

36. Optimal Overhead By fixing M and Ω we get achievable ‘profiles’

37. Optimal Overhead By fixing M and Ω we get achievable ‘profiles’

38. Thm: Achievable Profile

39. Some optimized profiles

40. Systematic Raptor Codes m1 m2 … mk • Matrix Multiplication • Θ(k) per symbol y1 y2… yk Raptor Code Systematic Version

41. Systematic Rateless Codes m1 m2 … mk Systematicprecode s1 s2 … sk(1+ε) • Θ(1) per symbol Ω s1 s2sk(1+ε) ….

42. Overview • Ratelesscodes in network applications • Efficient Repair in Storage Problems • Systematic Rateless Codes • Broadcasting with Side Information • Broadcasting over multiple hops • Role of Coding in Wireless Erasure Networks • Control of a Broadcast Server • Fixed Costs for Server • Online Constraint on Server

43. Coding in Networks • Wireline: • - Multicast/ multiple unicast • - Erasures: As FEC • Wireless: • - Multicast/ multiple unicast • - Erasures: As FEC • - Local Broadcast

44. Wireless Erasure Unicast • Broadcast from i Z with probability c(i,Z)

45. Backpressure Policy for local broadcast D

46. Backpressure Policy for local broadcast D

47. Backpressure Policy for local broadcast D

48. Backpressure Policy for local broadcast • Theorem: Backpressure achieves the mincut • Caveat: requires extensive coordination for every broadcast (which network coding can avoid) • Next Goal: Limitations of distributed routing D

49. Formalizing a constraint on distributed Routing D p

50. Formalizing a constraint on distributed Routing r1(p)=1 1 p r2(p)=1 2 D p 3 r3(p)=0