1. A Fast Repair Code Based on Regular Graphs for Distributed Storage Systems Yan Wang, East China Jiao Tong University Xin Wang, Fudan University 1 12/11/2013

2. Outline  Introduction  Related work  The code framework  Performance analysis  Conclusion 2 12/11/2013

3. Introduction  Introducing a class of distributed storage codes, the fast repair codes (FRC).  Based on regular graphs.  Simple lookup and exact repair.  minimum repair bandwidth and low disk I/O overhead. 3 12/11/2013

4. Outline  Introduction  Related work  The code framework  Performance analysis  Conclusion 4 12/11/2013

5. Related work  Minimum storage regenerating (MSR).  Minimum bandwidth regenerating (MBR).  (n, k, f)-SRC code.  Twin-MDS codes.  Uncoded repair property and fractional repetition codes.  Self-repairing code. 5 12/11/2013

6. Related work ─ Regular graph 12/11/2013 6

7. Related work ─ Twin-MDS codes 7 12/11/2013

8. Related work ─ Uncoded repair property and fractional repetition codes  Using an outer MDS code followed by an inner repetition code.  Exact repair for the minimum bandwidth regime.  Can totally tolerate ρ − 1 node failures.  ρ = 2, achieved based on regular graph.  ρ > 2, achieved based on Steiner system. 8 12/11/2013

9. Related work ─ Self-repairing code  Low complexity and bandwidth consumption.  Repair one block, often two blocks are enough.  Only encoding operation is XOR.  However, their code does not satisfy the (n, k)-MDS property.  Not any k storage node can reconstruct the data file. 9 12/11/2013

10.  [1] “Network coding for distributed storage system,” in Proc.IEEE Int. Conf. on Computer Commun, May 2007  [5] “Simple regenerating codes,” in arXiV, Aug 2011, Aug. 2011.  [6] “Enabling node repair in any erasure code for distributed storage,” in ISIT, Sep. 2011.  [7] “Fractional repetition codes for repair in distributed storage systems,” in Proc. Allerton Conf., Sep. 2010.  [9] “Self-repairing homomorphic codes for distributed storage systems,” in INFOCOM, 2011 Proceedings IEEE, april 2011, pp. 1215 –1223. Related work ─ References 12/11/2013 10

11. Outline  Introduction  Related work  The code framework  Code construction  Construction of regular graph  Example  Performance analysis  Conclusion 11 12/11/2013

12. The code framework ─ Code construction  The data file to be stored in n distributed storage nodes.  A series of 0-1 bits of length M.  (n’, k’)-MDS code.  Partition the file into k’ blocks of equal size.  Encode the file into n’ coded blocks.  Choose a d-regular graph G(V,E).  Deploy the n’ coded blocks to n storage nodes.  n nodes, each node corresponds to a storage node and has d neighbors.  Each edge as a coded block, each node stores d coded blocks.  n’ = nd/2. 12 12/11/2013

13. The code framework ─ Code construction  When a node fails, we select a newcomer to perform exact repair.  As each block is stored in two nodes.  Can be done by retrieving coded block one from each neighbor node in the regular graph.  Need to access several storage nodes to get no less than k’ distinct coded blocks.  The number of distinct coded blocks stored in k storage nodes equals to the number of edges covered by the k nodes in the regular graph. 13 12/11/2013

14.  Let Rc(G, k) denote the minimum number of edges covered by any k nodes in a regular graph G.  k’ ≤ Rc(G, k).  Can often get k’ distinct blocks from accessing less than k storage nodes.  Because we choose the storage nodes randomly while Rc(G, k) considers the minimum distinct blocks we can get from k storage nodes. The code framework ─ Code construction 12/11/2013 14

15. The code framework ─Construction of regular graph 15 12/11/2013

16. The code framework ─Example 12/11/2013 16

17.  Consider a DSS with (n = 10, k = 4, d = 3).  choose n’ = 15, as there are 15 edges in the regular graph.  K’ = 8, as any 4 nodes can cover 8 distinct edges at least.  Use a (15, 8)-MDS code to encode the file into 15 coded blocks.  Each storage node stores 3 blocks corresponding the 3 adjacent edges in graph. The code framework ─Example 12/11/2013 17

18. Outline  Introduction  Related work  The code framework  Performance analysis  Coding rate  Other aspects  Trade strict MDS property for better average performance  Conclusion 18 12/11/2013

19. Performance analysis ─Coding rate  Store 2n’ = nd blocks in all, while the file is of size k’ blocks.  choose an (nd/2, k’)-MDS code.  K’ is no more than Rc(G, k).  Maximizing the coding rate = maximizing Rc(G, k).  However, it is a challenging problem. 19 12/11/2013

20. Performance analysis ─Coding rate  Proposition 1: For general d-regular graph, Rc(G, k) ≥ dk/2.  Proof: As each node has d neighbors and each edge is counted in dk at most twice.  Therefore, let k’ = Rc(G, k), and the coding rate of the FRC code is no less than k/2n. 20 12/11/2013

21. When a node fails:  We retrieve one block from each of its d neighbor nodes to exactly restore the data.  The total repair bandwidth is the same as the storage per node  The number of disk access is d.  Only look-up and replications are performed.  the lowest coding complexity for repairing.  the lowest repair bandwidth as well. Performance analysis ─Other aspects 12/11/2013 21

22.  Not access a fixed number of storage nodes.  Keep accessing new storage nodes until we collect enough distinct coded blocks.  no constraint on k’ and thus can start with any given coding rate. Performance analysis ─ Trade strict MDS property for better average performance 12/11/2013 22

23. Performance analysis ─ Trade strict MDS property for better average performance 12/11/2013 23

24. Performance analysis ─ Trade strict MDS property for better average performance 12/11/2013 24

25. Performance analysis ─ Trade strict MDS property for better average performance 12/11/2013 25

26. Performance analysis ─ Trade strict MDS property for better average performance 12/11/2013 26

27. Outline  Introduction  Related work  The code framework  Performance analysis  Conclusion 27 12/11/2013

28. Conclusion  FRC codes minimizes bandwidth consumption and coding complexity in the repairing process.  Analytical results show that the FRC codes outperform the others in terms of low repair complexity and disk I/O overhead 28 12/11/2013

29.  The challenging issue is the relatively small coding rates  Considering acceptable as a trade-off for the simple repairing process  As future research  It is challenging to find a class of regular graphs with large coding rates Conclusion 12/11/2013 29