Network Coding for Distributed Storage Systems* Presented by Jayant Apte ASPITRG 7/9/13 & 7/11/13

Transcription

1 Network Coding for Distributed Storage Systems* Presented by Jayant Apte ASPITRG 7/9/13 & 7/11/13 *Dimakis, A.G.; Godfrey, P.B.; Wu, Y.; Wainwright, M.J.; Ramchandran, K. "Network Coding for Distributed Storage Systems", Information Theory, IEEE Transactions on, On page(s): Volume: 56, Issue: 9, Sept. 2010

2 Outline Part 1 Single Source Multi-cast Linear Network Coding Part 2 The repair problem Reduction of repair problem to single source multicast network Family of single source multi-cast networks arising from the reduction A lower bound on min-cuts(i.e. An upper bound on max-flow and hence coding capacity of network) Minimization of storage bandwidth subject to this lower bound

3 Some background on single source multi-cast network coding *Koetter, R.; Medard, M., "An algebraic approach to network coding," Networking, IEEE/ACM Transactions on, vol.11, no.5, pp.782,795, Oct. 2003

4 Some background on single source multi-cast network coding *Koetter, R.; Medard, M., "An algebraic approach to network coding," Networking, IEEE/ACM Transactions on, vol.11, no.5, pp.782,795, Oct. 2003

5 Max-Flow-Min-Cut Theorem

6 Max-Flow-Min-Cut Theorem

7 Max-Flow-Min-Cut Theorem

8 Some background on single source multi-cast network coding *Koetter, R.; Medard, M., "An algebraic approach to network coding," Networking, IEEE/ACM Transactions on, vol.11, no.5, pp.782,795, Oct. 2003

9 Basic Network Model

10 Basic Network Model

11 Local coding coefficients

12 Global coding coefficients

13 Matrix formulation

14 The transfer matrix

15 Proof of Theorem 2

16

17

18 Proof of Theorem 3

19 Some background on single source multi-cast network coding *Koetter, R.; Medard, M., "An algebraic approach to network coding," Networking, IEEE/ACM Transactions on, vol.11, no.5, pp.782,795, Oct. 2003

20 Extension to multicast

21 Part 2- Outline Introduction The repair problem Reduction of repair problem to single source multicast network Family of single source multi-cast networks arising from the reduction A lower bound on min-cuts(i.e. An upper bound on max-flow and hence coding capacity of network) Minimization of storage bandwidth subject to this lower bound

22 Distributed storage We are living in an internet age Demand for large scale data storage has increased significantly Social networks, file and video sharing require seamless storage, access and security for massive amounts of data Storage mediums(viz. hard-drives) are individually unreliable Hence we introduce redundancy via the use of erasure codes to improve reliability

23 A storage code((4,2) MDS) A1 Fragment 1 A2 Disk 1 A1 Kwefgws Jwehfwg SjfJHFJ jhfefog Sikytrd sdjhvkjd A2 B1 B2 B1 B2 Disk 2 A1+B1 A2+B2 Disk 3 Fragment 2 A2+B1 A1+ A2+B2 Disk 4

24 A storage code((4,2) MDS) A1 Fragment 1 A2 Disk 1 A1 Kwefgws Jwehfwg SjfJHFJ jhfefog Sikytrd sdjhvkjd A2 B1 B2 B1 B2 Disk 2 A1+B1 A2+B2 Disk 3 Fragment 2 A2+B1 A1+ A2+B2 Disk 4

25 Part 2- Outline Introduction The repair problem Reduction of repair problem to single source multicast network Family of single source multi-cast networks arising from the reduction A lower bound on min-cuts(i.e. An upper bound on max-flow and hence coding capacity of network) Minimization of storage bandwidth subject to this lower bound

26 Problem Definition Storage nodes are distributed and connected in a network Together they represent some storage code(mds or approximate MDS like LDPC) The issue of repairing a node arises when a storage node of the system fails The still functioning nodes are called active nodes A newcomer node called repair node must connect to a subset of active nodes, obtain information from them and reconstruct the storage code i.e, repair the code The objective is to minimize amount of information transferred in this process

27 Notation

28 The repair problem x1 y1 x2 y2 x3 x5 x4 ( Example: A (4,2) MDS code = repair bandwidth per node )

29 The repair problem Data object (2Mb) is divided into two fragments: y1,y2 (1 Mb each) 4 encoded fragments generated: x1,x2,x3,x4 (1 Mb each) x4 fails, x5, the newcomer needs to communicate with existing nodes and create a new encoded packet Any two out of x1,x2,x3,x5 must suffice to recover original data object

30 The repair problem What(and how much) should x1,x2,x3 communicate to x5 such that are minimized? x1 y1 x2 y2 x3 x5 x4 Example 1: A (4,2) MDS code

31 Variants of the repair problem Exact Repair: Failed blocks are exactly regenerated i.e. newcomer node must reconstruct exact replica of encoded block in the failed node Functional Repair: Newly generated data block need not be exact replica of encoded block on the failed node Exact repair of the systematic part: Only repair the systematic part exactly so there is always a uncoded copy of original file available

32 Variants of the repair problem Exact Repair: Failed blocks are exactly regenerated i.e. newcomer node must reconstruct exact replica of encoded block in the failed node Functional Repair: Newly generated data block need not be exact replica of encoded block on the failed node Exact repair of the systematic part: Only repair the systematic part exactly so there is always a uncoded copy of original file available

33 Functional repair example (Using RLNC) a1 1 b1 2 a1 a2 2 b1 b2 1 a2 a1+b1+a2+b2 b2 File fragments p1=a1+2b1 p2=2a2+b a1+2b1+a2+2b2 1 a1+2b1+3a2+b2 2 1 p1=4a1+5b1+4a2+5b2 1 Encoded repair packets 3a1+2b1+2a2+3b2 Encoded data blocks 1 5a1+7b1+8a2+7b2 6a1+9b1+6a2+6b2 Repair node (Each box is 0.5Mb)

34 Functional repair example (Using RLNC) Flow across this Cut is repair b/w a1 1 b1 2 a1 a2 2 b1 b2 1 a2 a1+b1+a2+b2 b2 File fragments p1=a1+2b1 p2=2a2+b a1+2b1+a2+2b2 1 a1+2b1+3a2+b2 2 1 p1=4a1+5b1+4a2+5b2 1 Encoded repair packets 3a1+2b1+2a2+3b2 Encoded data blocks 1 5a1+7b1+8a2+7b2 6a1+9b1+6a2+6b2 Repair node (Each box is 0.5Mb)

35 An attempt at solution x1 y1 x2 y2 x3 x4 Example 1: A (4,2) MDS code x5

36 An attempt at solution x1 y1 x2 y2 x3 x4 Example 1: A (4,2) MDS code x5 x5 Recovers original data object and creates a new independent linear combination

37 Can we do better than this?

38 Can we do better than this? YES!

39 Part 2- Outline Introduction The repair problem Reduction of repair problem to single source multicast network Family of single source multi-cast networks arising from the reduction A lower bound on min-cuts(i.e. An upper bound on max-flow and hence coding capacity of network) Minimization of storage bandwidth subject to this lower bound

40 Reduction to information flow graph

41 Example x1in S x2in x3in x4in x1out DC x2out x3out x4out x5in x5out Information flow graph corresponding to Example 1: A (4,2) MDS code Node 4 has failed

42 Dynamic nature of information flow graph due to given failure pattern x1in S x2in x3in x4in x1out DC x2out x3out x4out x5in x5out Information flow graph corresponding to Example 1: A (4,2) MDS code Node 4 has failed

43 Family of information flow graphs S x1in x1out x2in x2out x3in x3out x4in x4out x6in x6out DC x5in x5out Information flow graph corresponding to Example 1: A (4,2) MDS code Node 3 also failed say a few minutes later

44 Lemma 1

45 Outline The repair problem Reduction of repair problem to single source multicast network Family of single source multi-cast networks arising from the reduction A lower bound on min-cuts(i.e. An upper bound on max-flow and hence coding capacity of network) Minimization of storage bandwidth subject to this lower bound

46

47 Information flow graph S

48 Information flow graph S

49 Information flow graph S

50 Information flow graph S

51 Information flow graph S

52 Information flow graph S

53 Proof

54

55

56 WLOG

57 Outline The repair problem Reduction of repair problem to single source multicast network Family of single source multi-cast networks arising from the reduction A lower bound on min-cuts(i.e. An upper bound on max-flow and hence coding capacity of network) Minimization of storage bandwidth subject to this lower bound

58 Minimize subject to the lower bound

59 Nature of constraint

60 LHS of constraint as function of

61 LHS of constraint as function of

62 Solution to the optimization

63 Simplification of solution

64 Simplification of solution

65 Solution

66 Minimum repair bandwidth

67 Storage-Bandwidth Tradeoff Relationship between and [1]

68 References [1]Alexandros G. Dimakis, P. Brighten Godfrey, Yunnan Wu, Martin J. Wainwright, and Kannan Ramchandran Network coding for distributed storage systems. IEEE Trans. Inf. Theor. 56, 9 (September 2010), [2]Koetter, R.; Medard, M., "An algebraic approach to network coding," Networking, IEEE/ACM Transactions on, vol.11, no.5, pp.782,795, Oct [3]Tracey Ho and Desmond Lun Network Coding: An Introduction. Cambridge University Press, New York, NY, USA. [4]Dimakis, A.G.; Ramchandran, K.; Wu, Y.; Changho Suh, "A Survey on Network Codes for Distributed Storage," Proceedings of the IEEE, vol.99, no.3, pp.476,489, March 2011