1. Cooperative Recovery of Distributed Storage Systems from Multiple Losses with Network Coding Yuchong Hu, Yinlong Xu, Xiaozhao Wang, Cheng Zhan and Pei Li IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 28, NO. 2, FEBRUARY 2010

2. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) MCR Transmission And Coding Schemes Conclusion 2

3. Introduction The recovery from multiple node failures in distributed storage systems. We design a mutually cooperative recovery (MCR) mechanism for multiple node failures.

4. Introduction Dimakis et al.[8][10] prove that the file reconstruction problem in distributed storage systems is equivalent to the multicasting problem. Two symmetric mechanisms for maintaining redundancy – Minimum-storage regenerating (MSR) codes – Minimum-bandwidth regenerating (MBR) codes 4

5. Introduction In MCR all the new nodes repair the lost data cooperatively and simultaneously. Will prove that there exists a random linear coding with the minimal maintenance bandwidth in MCR while keeping the strong-MDS Property. 5

6. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) MCR Transmission And Coding Schemes Conclusion

7. Problem Statement An identical storage capability. Communications between any two nodes are symmetric in a distributed storage system. Original file is (n, k) MDS encoded The n encoded fragments are stored evenly at n nodes chosen from the system. When r nodes become unavailable, the system chooses another r nodes to repair 7

8. Problem Statement 8

9. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) – Model based on MCR – Assumptions in MCR – Information flow graph G(n, k, r, β) – Lower bound of maintenance bandwidth – Comparisons of MCR, MSR and MBR MCR Transmission And Coding Schemes Conclusion

10. Mutually Cooperative Recovery(MCR) Our study bases on the assumption that all the new nodes can mutually cooperatively complete their fragment reconstructions. DSN(n, k, r). 10

11. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) – Model based on MCR – Assumptions in MCR – Information flow graph G(n, k, r, β) – Lower bound of maintenance bandwidth – Comparisons of MCR, MSR and MBR MCR Transmission And Coding Schemes Conclusion

12. Model based on MCR The repair process of our mutually cooperative recovery in DSN(n, k, r) is specified as follows. 12

13. Model based on MCR 13

14. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) – Model based on MCR – Assumptions in MCR – Information flow graph G(n, k, r, β) – Lower bound of maintenance bandwidth – Comparisons of MCR, MSR and MBR MCR Transmission And Coding Schemes Conclusion

15. Assumptions in MCR Assuming that βi,j and β´j,j is the same as β. The total bandwidth overhead for the recovery is 15

16. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) – Model based on MCR – Assumptions in MCR – Information flow graph G(n, k, r, β) – Lower bound of maintenance bandwidth – Comparisons of MCR, MSR and MBR MCR Transmission And Coding Schemes Conclusion

17. Information flow graph G(n, k, r, β) An information flow graph G(n, k, r, β), a similar idea in [8]. Nodes in G(n, k, r, β) Edges in G(n, k, r, β) 17

18. Information flow graph G(n, k, r, β) 18

19. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) – Model based on MCR – Assumptions in MCR – Information flow graph G(n, k, r, β) – Lower bound of maintenance bandwidth – Comparisons of MCR, MSR and MBR MCR Transmission And Coding Schemes Conclusion

20. Lower bound of maintenance bandwidth Finding the lower bound of β by studying the capacity of the min-cut of G(n, k, r, β). To keep the (n, k) MDS property in DSN(n, k, r) – each of the capacities of min-cuts in all the possible information flow graphs must be ≥ the original file size M bytes. 20

21. Lower bound of maintenance bandwidth Lemma 1 – To keep (n, k) MDS property in DSN(n, k, r), min- cuts separating V S from D in all possible information flow graphs G(n, k, r, β) must be not smaller than M bytes. 21

22. Lower bound of maintenance bandwidth 22

23. Lower bound of maintenance bandwidth 23

24. Lower bound of maintenance bandwidth 24

25. Lower bound of maintenance bandwidth 25

26. Lower bound of maintenance bandwidth 26

27. Lower bound of maintenance bandwidth 27

28. Lower bound of maintenance bandwidth 28

29. Lower bound of maintenance bandwidth 29

30. Lower bound of maintenance bandwidth 30

31. Lower bound of maintenance bandwidth 31

32. Lower bound of maintenance bandwidth 32

33. Lower bound of maintenance bandwidth 33

34. Lower bound of maintenance bandwidth Lemma 3: – there exists a random linear network coding scheme guaranteeing that D can reconstruct the original file for any connection choice. – with a probability that can be driven arbitrarily to 1 by increasing the field size of F. 34

35. Lower bound of maintenance bandwidth Theorem 1: – the (n, k) MDS property is still kept after the recovery if β is not smaller than M/[(n − k)k]. Proof by Lemma 2 and Lemma 3. 35

36. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) – Model based on MCR – Assumptions in MCR – Information flow graph G(n, k, r, β) – Lower bound of maintenance bandwidth – Comparisons of MCR, MSR and MBR MCR Transmission And Coding Schemes Conclusion

37. Comparisons of MCR, MSR and MBR MCR – Each node stores α = M/k bytes. – And r nodes become unavailable. – Each of r new node downloads β bytes from each of any d available nodes. The storage cost = (M/k)*n Maintenance bandwidth = [(n − 1)/(n − k)]*(M/k)*r 37

38. Comparisons of MCR, MSR and MBR 38

39. Comparisons of MCR, MSR and MBR Compared with MSR – maintenance bandwidth : 22%. – storage cost : same. Compared with MBR – maintenance bandwidth : same. – storage cost : 23%. 39

40. Comparisons of MCR, MSR and MBR Compared with MSR – maintenance bandwidth : 23%. – storage cost : same. Compared with MBR – maintenance bandwidth : 11%. – storage cost : 23%. 40

41. Comparisons of MCR, MSR and MBR We can conclude that MCR has a better performance in the storage cost and maintenance bandwidth in multi-loss recovery of distributed storage systems compared with other non-cooperative recovery mechanisms. 41

42. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) MCR Transmission And Coding Schemes – Transmission scheme in MCR – Coding scheme in MCR Conclusion

43. MCR Transmission And Coding Schemes Theorem 1 gives a lower bound of maintenance bandwidth with β = M/[k(n − k)]. Constructing a recovery transmission scheme in MCR based on β = M/[k(n − k)] and a linear coding scheme based on Strong-MDS code. 43

44. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) MCR Transmission And Coding Schemes – Transmission scheme in MCR – Coding scheme in MCR Conclusion

45. Transmission scheme in MCR 45

46. Transmission scheme in MCR 46

47. Transmission scheme in MCR 47

48. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) MCR Transmission And Coding Schemes – Transmission scheme in MCR – Coding scheme in MCR Conclusion

49. Coding scheme in MCR 49

50. Coding scheme in MCR Coding scheme in MCR via linear coding is based on the following Lemma 4. Lemma 4 (Schwartz-Zippel Theorem)[11] Theorem 2 to show that if a file in a distributed storage system is Strong-MDS encoded, it will still satisfy Strong-MDS Property after multiloss recovery in MCR. 50 [11] R. Motwani and P. Raghavan, Randomized Algorithms. Cambridge University Press, 1995.

51. Coding scheme in MCR 51

52. Coding scheme in MCR 52

53. Coding scheme in MCR 53

54. Coding scheme in MCR By lemma 4 and lemma 5, Theorem 2 holds. – A file in a distributed storage system with MCR transmission scheme will keep (n, k) strong-MDS property with a probability that can be driven arbitrarily to 1 by increasing the field size of F. – Original file can be reconstructed after multiple losses with a probability that can be driven arbitrarily to 1 by increasing the field size of F. 54

55. Outline Introduction Problem Statement Mutually Cooperative Recovery(MCR) MCR Transmission And Coding Schemes Conclusion

56. This paper gave the tight lower bound of maintenance bandwidth. designed a MCR transmission scheme matching the tight lower bound. design a MCR coding scheme, presented a strong- MDS code and showed the existence of a random linear strong-MDS code with a sufficient large finite field. 56

57. Conclusion The decoding complexity of our random linear coding scheme is expensive The future work is to design a determinate coding algorithm to reduce the decoding complexity. 57