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4 4 first decoding {a i, b i } 4 i=1 using the decoding algorithm of the code of Fig 1a, which then allows for removal of the piggybac ( i=1 ib i + 2 i=1 ia i) from the second instance, maing the remainder identical to the code of Fig 1a Our piggybacing framewor, enhances existing codes by adding piggybacs from one instance onto the other The design of these piggybacs determine the properties of the resulting code In this paper, we provide a few designs of piggybacing and specialize it to existing codes to obtain the four specific classes mentioned above This framewor, while being powerful, is also simple, and easily amenable for code constructions in other settings and scenarios The rest of the paper is organized as follows Section II introduces the general piggybacing framewor Sections III and IV then present code designs and repair-algorithms based on this framewor, special cases of which result in classes 1 and 2 discussed above Section V provides piggybac design which result in low-repair locality along with low data-read and download Section VI provides a comparison of these codes and various other codes in the literature Section VII demonstrates the use of piggybacing to enable efficient parity repair in existing codes that were originally constructed for repair of only the systematic nodes Section VIII draws conclusions Readers interested only in repair locality may sip Sections III and IV, and readers interested only in the mechanism of imbibing efficient parity-repair in existing codes optimized for systematic-repair may sip Sections III, IV, and V without any loss in continuity II THE PIGGYBACKING FRAMEWORK The piggybacing framewor operates on an existing code, which we term the base code The choice of the base code is arbitrary The base code is associated to n encoding functions {f i } n i=1 : it taes the message u as input and encodes it to n coded symbols {f 1 (u),, f n (u)} Node i (1 i n) stores the data f i (u) The piggybacing framewor operates on multiple instances of the base code, and embeds information about one instance into other instances in a specific fashion Consider α instances of the base code The encoded symbols in α instances of the base code are f 1 (a) f 1 (b) f 1 (z) Node n f n (a) f n (b) f n (z) where a,, z are the (independent) messages encoded under these α instances We shall now describe the piggybacing of this code For every i, 2 i α, one can add an arbitrary function of the message symbols of all previous instances {1,, (i 1)} to the data stored under instance i These functions are termed piggybac functions, and the values so added are termed piggybacs Denoting the piggybac functions by g i,j (i {2,, α}, j {1,, n}), the piggybaced code is thus: f 1 (a) f 1 (b) + g 2,1 (a) f 1 (c) + g 3,1 (a,b) f 1 (z) + g α,1 (a,,y) Node n f n (a) f n (b) + g 2,n (a) f 1 (c) + g 3,n (a,b) f n (z) + g α,n (a,,y) The decoding properties (such as the minimum-distance or the MDS nature) of the base code are retained upon piggybacing In particular, the piggybaced code allows for decoding of the entire message from any set of nodes from which the base code allowed decoding To see this, consider any set of nodes from which the message can be recovered in the base code Observe that the first column of the piggybaced code is identical to a single instance of the base code Thus a can be recovered directly using the decoding procedure of the base code The piggybac functions {g 2,i (a)} n i=1 can now be subtracted from the second column The remainder of this column

6 6 III PIGGYBACKING DESIGN 1 In this section, we present our first design of piggybac functions and associated repair algorithms This design allows one to reduce data-read and download during repair while having a small number of substripes For instance, when the number of substripes is is small as 2, we can achieve a 25 to 35% savings during repair of systematic nodes We shall first present the piggybac design for optimizing the repair of systematic nodes, and then move on to the repair of parity nodes A Efficient repair of systematic nodes This design operates on α = 2 instances of the base code We first partition the systematic nodes into r sets, S 1,, S r of equal size (or nearly equal size if is not a multiple of r) For ease of understanding, let us assume that is a multiple of r, which fixes the size of each of these sets as r Then, let S 1 = {1,, r }, S 2 = { r + 1,, 2 r } and so on, with S i = { (i 1) r + 1,, i r } for i = 1,, r Define the following length vectors: q 2 = [ p r,1 p r, 0 0 ] T r q 3 = [ 0 0 p r, r, 0 0 ] T 2 r r q r = [ 0 0 p r, r p r, (r 1) r 0 0 ] T q r+1 = [ 0 0 p r, r p r, ] T Also, let v r = p r q r = [ p r,1 p r, r (r 2) 0 0 p r, r (r 1)+1 p r, ] T Note that each element p i,j is non-zero since the base code is MDS We shall use this property during repair operations The base code is piggybaced in the following manner: a 1 b 1 Node Node +1 Node +2 Node +r a p T 1 a p T 2 a p T r a b pt 1 b pt 2 b + qt 2 a p T r b + q T r a Fig 1b depicts an example of such a piggybacing We shall now perform an invertible transformation of the data stored in node ( + r) In particular, the first symbol of node ( + r) in the code above is replaced with the difference of this symbol from its second symbol, ie, node ( + r) now stores

18 18 code constructions We believe that this framewor has a greater potential, and clever designs of other piggybacing functions and application to other base codes could potentially lead to efficient codes for various other settings as well Further exploration of this rich design space is left as future wor Finally, while this paper presented only achievable schemes for data-read efficiency during repair, determining the optimal repair-efficiency under these settings remains open REFERENCES [1] D Borthaur, HDFS and Erasure Codes (HDFS-RAID), 2009 [Online] Available: [2] D Ford, F Labelle, F Popovici, M Stoely, V Truong, L Barroso, C Grimes, and S Quinlan, Availability in globally distributed storage systems, in Proceedings of the 9th USENIX Symposium on Operating Systems Design and Implementation, 2010 [3] Erasure codes: the foundation of cloud storage, Sep 2010 [Online] Available: [4] K V Rashmi, N B Shah, P V Kumar, and K Ramchandran, Explicit construction of optimal exact regenerating codes for distributed storage, in Proc 47th Annual Allerton Conference on Communication, Control, and Computing, Urbana-Champaign, Sep 2009, pp [5] K V Rashmi, N B Shah, and P V Kumar, Optimal exact-regenerating codes for the MSR and MBR points via a product-matrix construction, IEEE Transactions on Information Theory, vol 57, no 8, pp , Aug 2011 [6] I Tamo, Z Wang, and J Bruc, MDS array codes with optimal rebuilding, in Proc IEEE International Symposium on Information Theory (ISIT), St Petersburg, Jul 2011 [7] V Cadambe, C Huang, and J Li, Permutation code: optimal exact-repair of a single failed node in MDS code based distributed storage systems, in IEEE International Symposium on Information Theory (ISIT), 2011, pp [8] O Khan, R Burns, J Plan, W Pierce, and C Huang, Rethining erasure codes for cloud file systems: minimizing I/O for recovery and degraded reads, in Proc Usenix Conference on File and Storage Technologies (FAST), 2012 [9] Z Wang, I Tamo, and J Bruc, On codes for optimal rebuilding access, in Communication, Control, and Computing (Allerton), th Annual Allerton Conference on, 2011, pp [10] S Jiea, A Kermarrec, N Scouarnec, G Straub, and A Van Kempen, Regenerating codes: A system perspective, arxiv: , 2012 [11] A S Rawat, O O Koyluoglu, N Silberstein, and S Vishwanath, Optimal locally repairable and secure codes for distributed storage systems, arxiv: , 2012 [12] K Shum and Y Hu, Functional-repair-by-transfer regenerating codes, in IEEE International Symposium on Information Theory (ISIT), Cambridge, Jul 2012, pp [13] S El Rouayheb and K Ramchandran, Fractional repetition codes for repair in distributed storage systems, in Allerton Conference on Control, Computing, and Communication, Urbana-Champaign, Sep 2010 [14] O Olmez and A Ramamoorthy, Repairable replication-based storage systems using resolvable designs, arxiv: , 2012 [15] Y S Han, H-T Pai, R Zheng, and P K Varshney, Update-efficient regenerating codes with minimum per-node storage, arxiv: , 2013 [16] B Gastón, J Pujol, and M Villanueva, Quasi-cyclic regenerating codes, arxiv: , 2012 [17] B Sasidharan and P V Kumar, High-rate regenerating codes through layering, arxiv: , 2013 [18] A G Dimais, P B Godfrey, Y Wu, M Wainwright, and K Ramchandran, Networ coding for distributed storage systems, IEEE Transactions on Information Theory, vol 56, no 9, pp , 2010 [19] N B Shah, K V Rashmi, P V Kumar, and K Ramchandran, Explicit codes minimizing repair bandwidth for distributed storage, in Proc IEEE Information Theory Worshop (ITW), Cairo, Jan 2010 [20] F Oggier and A Datta, Self-repairing homomorphic codes for distributed storage systems, in INFOCOM, 2011 Proceedings IEEE, 2011, pp [21] P Gopalan, C Huang, H Simitci, and S Yehanin, On the locality of codeword symbols, IEEE Transactions on Information Theory, Nov 2012 [22] D Papailiopoulos and A Dimais, Locally repairable codes, in Information Theory Proceedings (ISIT), 2012 IEEE International Symposium on IEEE, 2012, pp [23] G M Kamath, N Praash, V Lalitha, and P V Kumar, Codes with local regeneration, arxiv: , 2012 [24] N B Shah, K V Rashmi, P V Kumar, and K Ramchandran, Distributed storage codes with repair-by-transfer and non-achievability of interior points on the storage-bandwidth tradeoff, IEEE Transactions on Information Theory, vol 58, no 3, pp , Mar 2012 [25] Z Wang, A G Dimais, and J Bruc, Rebuilding for array codes in distributed storage systems, in Worshop on the Application of Communication Theory to Emerging Memory Technologies (ACTEMT), Dec 2010 [26] L Xiang, Y Xu, J Lui, and Q Chang, Optimal recovery of single dis failure in RDP code storage systems, in ACM SIGMETRICS, vol 38, no 1, 2010, pp [27] M Blaum, J Brady, J Bruc, and J Menon, EVENODD: An efficient scheme for tolerating double dis failures in RAID architectures, IEEE Transactions on Computers, vol 44, no 2, pp , 1995

19 19 [28] P Corbett, B English, A Goel, T Grcanac, S Kleiman, J Leong, and S Sanar, Row-diagonal parity for double dis failure correction, in Proc 3rd USENIX Conference on File and Storage Technologies (FAST), 2004, pp 1 14 [29] M Blaum, J Bruc, and A Vardy, MDS array codes with independent parity symbols, Information Theory, IEEE Transactions on, vol 42, no 2, pp , 1996 [30] D Papailiopoulos, A Dimais, and V Cadambe, Repair optimal erasure codes through hadamard designs, in Proc 47th Annual Allerton Conference on Communication, Control, and Computing, Sep 2011, pp [31] I Reed and G Solomon, Polynomial codes over certain finite fields, Journal of the Society for Industrial & Applied Mathematics, vol 8, no 2, pp , 1960 APPENDIX Proof of Theorem 1: Let us restrict our attention to only the nodes in set S, and let S denote the size of this set From the description of the piggybacing framewor above, the data stored in instance j (1 j α) under the base code is a function of U j This data can be written as a S -length vector f(u j ) with the elements of this vector corresponding to the data stored in the S nodes in set S On the other hand, the data stored in instance j of the piggybaced code is of the form (f(u j ) + g j (U 1,, U j 1 )) for some arbitrary (vector-valued) functions g Now, I ( ) ( ) {Y i } i S ; U 1,, U α = I {f(u j ) + g j (U 1,, U j 1 )} α j=1 ; U 1,, U α (11) = = = α I l=1 α I l=1 ( {f(u j ) + g j (U 1,, U j 1 )} α j=1 ; U l U 1,, U l 1 ) ( f(u l ), {f(u j ) + g j (U 1,, U j 1 )} α j=l+1 ; U l U 1,, U l 1 ) (12) (13) α I (f(u l ) ; U l U 1,, U l 1 ) (14) l=1 α I (f(u l ) ; U l ) (15) l=1 = I ( {X i } i S ; U 1,, U α ) (16) where the last two equations follow from the fact that the messages U l of different instances l are independent

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