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The former provides an algebraic framework for linear network coding. The latter reduces the so called repair problem to single-source multicast network-coding problem and shows that there is a tradeoff between amount of data stored in a distributed sturage system and amount of data transfer required to repair the system if a node(hard-drive) fails.

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- 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): 4539 – 4551 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. Proof of Theorem 3
- 17. 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
- 18. Extension to multicast
- 19. 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
- 20. 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
- 21. A storage code((4,2) MDS) Kwefgws Jwehfwg SjfJHFJ jhfefog Sikytrd sdjhvkjd A1 A2 B1 B2 A1 A2 B1 B2 A1 +B1 A2 +B2 A2 +B1 A1 + A2 +B2 Fragment 1 Fragment 2 Disk 1 Disk 2 Disk 3 Disk 4
- 22. A storage code((4,2) MDS) Kwefgws Jwehfwg SjfJHFJ jhfefog Sikytrd sdjhvkjd A1 A2 B1 B2 A1 A2 B1 B2 A1 +B1 A2 +B2 A2 +B1 A1 + A2 +B2 Fragment 1 Fragment 2 Disk 1 Disk 2 Disk 3 Disk 4
- 23. 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
- 24. 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
- 25. Notation
- 26. The repair problem x1 x2 x3 x4 y1 y2 x5 Example: A (4,2) MDS code ( = repair bandwidth per node )
- 27. 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
- 28. The repair problem ● What(and how much) should x1 ,x2 ,x3 communicate to x5 such that are minimized? x1 x2 x3 x4 y1 y2 x5 Example 1: A (4,2) MDS code
- 29. 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 un- coded copy of original file available
- 30. 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 un- coded copy of original file available
- 31. Functional repair example (Using RLNC) a1 b1 a2 b2 a1 +b1 +a2 +b2 a1 +2b1 +a2 +2b2 a1 +2b1 +3a2 +b2 3a1 +2b1 +2a2 +3b2 a1 b1 a2 b2 p1=a1 +2b1 p2=2a2 +b2 p1=4a1 +5b1 +4a2 +5b2 5a1 +7b1 +8a2 +7b2 6a1 +9b1 +6a2 +6b2 1 2 2 1 3 1 1 1 1 1 2 2 File fragments Encoded data blocks Encoded repair packets Repair node (Each box is 0.5Mb)
- 32. Functional repair example (Using RLNC) a1 b1 a2 b2 a1 +b1 +a2 +b2 a1 +2b1 +a2 +2b2 a1 +2b1 +3a2 +b2 3a1 +2b1 +2a2 +3b2 a1 b1 a2 b2 p1=a1 +2b1 p2=2a2 +b2 p1=4a1 +5b1 +4a2 +5b2 5a1 +7b1 +8a2 +7b2 6a1 +9b1 +6a2 +6b2 1 2 2 1 3 1 1 1 1 1 2 2 File fragments Encoded data blocks Encoded repair packets Repair node (Each box is 0.5Mb) Flow across this Cut is repair b/w
- 33. An attempt at solution x1 x2 x3 x4 y1 y2 x5 Example 1: A (4,2) MDS code
- 34. An attempt at solution x1 x2 x3 x4 y1 y2 x5 Example 1: A (4,2) MDS code x5 Recovers original data object and creates a new independent linear combination
- 35. Can we do better than this?
- 36. Can we do better than this? YES!
- 37. 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
- 38. Reduction to information flow graph
- 39. Example x1 in x2 in x3 in x4 in x5 in x1 out x2 out x3 out x4 out S x5 out DC Information flow graph corresponding to Example 1: A (4,2) MDS code Node 4 has failed
- 40. Dynamic nature of information flow graph due to given failure pattern x1 in x2 in x3 in x4 in x5 in x1 out x2 out x3 out x4 out S x5 out DC Information flow graph corresponding to Example 1: A (4,2) MDS code Node 4 has failed
- 41. Family of information flow graphs x1 in x2 in x3 in x4 in x5 in x1 out x2 out x3 out x4 out S x5 out DC Information flow graph corresponding to Example 1: A (4,2) MDS code Node 3 also failed say a few minutes later x6 in x6 out
- 42. Lemma 1
- 43. 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
- 44. Information flow graph S
- 45. Information flow graph S
- 46. Information flow graph S
- 47. Information flow graph S
- 48. Information flow graph S
- 49. Information flow graph S
- 50. Proof
- 51. WLOG
- 52. 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
- 53. Minimize subject to the lower bound
- 54. Nature of constraint
- 55. LHS of constraint as function of
- 56. LHS of constraint as function of
- 57. Solution to the optimization
- 58. Simplification of solution
- 59. Simplification of solution
- 60. Solution
- 61. Minimum repair bandwidth
- 62. Storage-Bandwidth Tradeoff Relationship between and [1]
- 63. References ● [1]Alexandros G. Dimakis, P. Brighten Godfrey, Yunnan Wu, Martin J. Wainwright, and Kannan Ramchandran. 2010. Network coding for distributed storage systems. IEEE Trans. Inf. Theor. 56, 9 (September 2010), 4539-4551. ● [2]Koetter, R.; Medard, M., "An algebraic approach to network coding," Networking, IEEE/ACM Transactions on , vol.11, no.5, pp.782,795, Oct. 2003 ● [3]Tracey Ho and Desmond Lun. 2008. 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

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