Reviews work of Koetter et al. and Dimakis et al. 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.

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

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)
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

24. 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

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
x2
x3
x4
y1
y2
x5
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
x2
x3
x4
y1
y2
x5
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 un-
coded 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 un-
coded copy of original file available

33. 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)

34. 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

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

36. 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

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
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

42. 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

43. 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

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

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

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. 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