Recent Developments in Coding for Distributed Storage

Transcription

1 Recent Developments in Coding for Distributed Storage P. Vijay Kumar Professor, Department of Electrical Communication Engineering Indian Institute of Science, Bangalore Adjunct Professor, EE-Systems University of Southern California Workshop on Emerging Wireless Networks Institute for Pure and Applied Mathematics University of California, Los Angeles Feb. 6-10, 2017

2 Thanks to Christina Fragouli and Suhas Diggavi and for the kind invite... 2 / 43

3 Collaborators Past and Present Research Collaborators Alexander Barg, Ithak Tamo S. Narayanamurthy, R. Kumar, S. Husein and Siddhartha Nandi (NetApp Inc., India) Past and Present Students Narayanamoorthy Prakash (Post-Doc, MIT) Lalitha Vadlamani (Faculty, IIITH) Gaurav Agrawal (now pursuing PhD at UCLA) K. P. Prasanth (Qualcomm, India) Birenjith Sasidharan (PhD student, to graduate shortly) S. B. Balaji, Nikhil M. Krishnan, Myna Vajha (PhD students) Ganesh Kini, Bhagyashree Puranik, R. Vinayak (Master s students) 3 / 43

4 Outline Codes with Local and Sequential Erasure Recovery Improving Decoding Performance Through Locality Codes with Hierarchical Locality A Recent High-Rate Regenerating Code Construction (briefly) 4 / 43

5 Inefficiency of Linear Node Repair in an MDS Code An obvious approach: Connect to any 2 nodes, A Disk 1 B New disk 1 Reconstruct entire data file, Reconstruct data stored in the node A B B Disk 2 A+B Disk 3 A+B A+θB Disk 4 (4, 2) MDS code Used in RAID 6 But downloading 2 units of data to revive a node that stores 1 units of data is wasteful! 5 / 43

6 A Second Example: Facebook s Code 1 1 Node Node Node 3 [14, 10] MDS code Node 4 Node 5 Node 6 Node 7 Node 8 Has the any 10 out of 14 property Used in Facebook data centers Node 9 Node 10 D. Borthakur, R. Schmit, R. Vadali, S. Chen, and P. Kling. HDFS RAID. Tech P1 Node 11 talk. Yahoo Developer Network, Nov. P2 Node P3 Node 13 P4 Node 14 6 / 43

7 The Facebook Code is Inefficient in Number of Helper Nodes Needed 1 1 Node Node 2 Node 3 Node 4 Node 5 Node 6 Node 7 Node 8 Node 9 Needs to connect to 10 nodes to repair a failed node This calls for interrupting operations in 10 nodes (apart from downloading the entire data file) Node is the repair degree P1 Node 11 P2 P3 Node 12 Node 13 Are there better options? P4 Node 14 7 / 43

8 Adding a Branch (or two) to Coding Theory Regenera'ng(Codes( Codes(with((Locality( Regenera'ng(codes(reduce(repair(bandwidth( Codes(with(locality(reduce(repair(degree( A. G. Dimakis, P. B. Godfrey, Y. Wu, M. Wainwright, and K. Ramchandran, Network Coding for Distributed Storage Systems, IEEE Trans. Inform. Th., Sep P. Gopalan, C. Huang, H. Simitci, and S. Yekhanin, On the Locality of Codeword Symbols, IEEE Trans. Inf. Theory, Nov Image: 8 / 43

9 Codes with Locality to Minimize Reduce Repair Degree (Example of Windows Azure Storage) MicrosoH+Azure+Code+ X 1+ X 2+ X 3+ X 4+ X 5+ X 6+ X 7+ Y 1+ Y 2+ Y 3+ Y 4+ Y 5+ Y 6+ Y 7+ P 1+ XPcode+ P X+ YPcode+ P Y+ P 2+ Comparison:+In+terms+of+reliability+of+data+and+number+of+helper+nodes+contacted+for+ node+repair,+the+two+codes+are+comparable.+ + +The+overheads+are+quite+different,+29%+for+the+Azure+code+versus+43%+for+the+RS+code This+difference+has+reportedly+saved+MicrosoH+millions+of+dollars!++ Comparison: In terms of reliability and number of helper nodes contacted for node repair, the two codes are comparable. The overheads however are quite different, 1.29 for the Azure code versus 1.5 for the RS code. This difference has reportedly saved Microsoft millions of dollars. Reed$Solomon* X 1* X 2* X 3* X 4* X 5* X 6* P 1* P 2* P 3* Codeword* Y 1+ Y 2+ Y 3+ Y 4+ Y 5+ Y 6+ Y 7+ P 1+ P 2+ P Y+ (any*6*of*9*can*be*used*to*recover*the*codeword)* Huang, Simitci, Xu, Ogus, Calder, Gopalan, Li, Yekhanin, Erasure Coding in Windows Azure Storage, USENIX, Boston, MA, ReedPSolomon+Code+ 9 / 43

10 Codes with Locality For Multiple Erasures Some approaches: Sequential Approach - Recovery from at most t erasures. There exists at least one sequence of erased code symbol recovery at every step. Availability - For every code symbol there are multiple mutually disjoint recovery sets. Selectable Recovery - Every erased symbol has a parity check that does not involve any other erased symbol. Codes with Sequential Recovery Codes with Selectable Recovery Stronger Local Codes Codes with Availability Codes with co-operative local recovery 10 / 43

11 Codes with Locality for Sequential Recovery Definition An [n, k] code over a field F q is defined as a code with sequential recovery from t erasures having locality r if for any set of s t erased symbols, {c σ1,..., c σs }, there exists a codeword h in the dual of the code, of Hamming weight r + 1, such that supp(h) {σ 1,..., σ s } = 1. We denote the above defined codes as (n, k, r, t) seq codes. N. Prakash, V. Lalitha, P. Vijay Kumar, Codes with Locality for Two Erasures, ISIT S. B. Balaji, Ganesh R. Kini and PVK, A Tight Rate Bound and a Matching Construction for Locally Recoverable Codes with Sequential Recovery From Any Number of Multiple Erasures, arxiv: v6 [cs.it] 9 Feb / 43

12 Comparison of Sequential Recovery and Availability 12 / 43

13 Upper Bound on Rate Under Sequential, Local Recovery Theorem Rate Bound: Let C be an (n, k, r, t) seq code over a field F q. Let r 3. Then where s = t+1 2. k n k n r t 2 r t for t an even integer, (1) t 2 1 i=0 r i r s r s + 2 s 1 i=1 r for t an odd integer, (2) i / 43

14 Proof By Inferring Form of Parity-Check Matrix H = D 0 A D 1 A D 2 A D A t D t 2 2 A t D t C. D 14 / 43

15 Regular Graph Construction (t = 2, r = 3) Edges correspond to symbols Nodes correspond to parity-checks 15 / 43

16 Product Code for (t = 3, r = 4) c 11 c 12 c 13 c 14 c 15 c 21 c 22 c 23 c 24 c 25 c 31 c 32 c 33 c 34 c 35 c 41 c 42 c 43 c 44 c 45 c 51 c 52 c 53 c 54 c / 43

17 A Binary Rate-Optimal Code (t = 4, r = 3) 17 / 43

18 Locality and Decoding (BCH Codes) 18 / 43

19 Desired Null Spectrum of the BCH Code (zeros of the generator polynomial) distance zeros 19 / 43

20 Desired Null Spectrum + Conjugacy distance zeros distance zeros (cyc. cosets) 20 / 43

21 Desired Null Spectrum for Locality = Locality Plus Conjugacy locality zeros Itzhak Tamo, Alexander Barg, Sreechakra Goparaju, Robert Calderbank, Cyclic LRC Codes, binary LRC codes, and upper bounds on the distance of cyclic codes, ISIT / 43

22 Final Null Spectrum distance zeros distance zeros (cyc. cosets) locality zeros 22 / 43

23 Belief Propagation (BP+OSD) of a BCH Code Assumption: Improve beliefs through one round of belief propagation. x1 x2 xi xi+1 xr+1.. log log (x1 = 0 y1) (x1 = 1 y1) (xr+1 = 0 yr+1) (xr+1 = 1 yr+1) x1 x2 xi xr xr+1.. log (xr+1 = 0 y1, y2..yr) (xr+1 = 1 y1, y2..yr) 23 / 43

24 Quick-Look ML Decoder 24 / 43

25 Ordered Statistics Decoding Hard decision decoding based on most reliable bits (Order-0) Sort the symbols of the received vector in descending order of reliability. Hard decision decode the first k independent bits (Most Reliable Independent, MRI) bits and obtain the n bit codeword. Order-l reprocessing For 0 i l, change each of the i bits of the k MRI bits to obtain m = l ( k ) i=0 i message vectors. Find the m codewords corresponding to the m information vectors. Pick the codeword closest to the received vector. Marc P. C. Fossorier and Shu Lin Soft-Decision Decoding of Linear Block Codes Based on Ordered Statistics, IT-Trans, Sep / 43

26 Performance of Locality-aware OSD We consider two codes in binary input AWGN channel: C 1 : [255, 206] BCH code doesn t have any locality. C 2 : [255, 192] BCH-like constructed by adding locality r = 16 in C 1. M. Nikhil Krishnan, Bhagyashree Puranik, PVK, Itzhak Tamo, and Alexander Barg, A Study on the Impact of Locality in the Decoding of Binary Cyclic Codes, to be uploaded to arxiv. 26 / 43

27 Scheme-1: Locality-aware Ordered Statistics Decoding In the next slide, we consider decoding a type-1 Doubly Transitive Invariant Code that is majority logic decodable, with parameters [255, 175], having availability, t = 16 and locality, r = / 43

28 Performance of Locality-Aware OSD on an Availability Code Comparison of OSD, BP+OSD, and majority logic decoding of a [255, 175] type-i DTI code, locality r = 15, availability t = 16 in binary input AWGN channel. 28 / 43

29 Hierarchical Locality 29 / 43

30 Windows Azure Uses Information Locality, How About All-Symbol Locality? X 1 X 2 X 3 X 4 X 5 Y 1 Y 2 Y 3 Y 4 Y 5 P G1 P G2 P X P Y (global pari9es) (local codeword) (local codeword) (information-symbol locality) (local codeword) P Z P X X 1 X 2 X 3 X 4 Y 1 Y 2 Y 3 Y 4 P Y (local codeword) (local codeword) 30 / 43

31 The Tamo-Barg Approach to All-Symbol Locality) f(p a ) f(p b ) f(p c ) select subset of polynomials : (f (P 1 ),, f (P n )), with deg(f ) (k 1) such that: given point P a there exist other points fitted by a lower-degree polynomial such as a line, which can be used for correction There is also a Chinese Remainder Theorem interpretation Itzhak Tamo, Alexander Barg, A family of optimal locally recoverable codes, IEEE Transactions on Information Theory. Aug / 43

32 Chinese Remainder Theorem Sun Zi Suanjing Master Sun s Mathematical Manual Problem 26, Volume 3 (estimated to be published AD) reads: There are certain things whose number is unknown. A number is repeatedly divided by 3, the remainder is 2; divided by 5, the remainder is 3; and by 7, the remainder is 2. What will the number be? The problem can be expressed as x 2 (mod 3) 3 (mod 5) 2 (mod 7). Sun Zi solved the problem as we do, giving (mod 3) (mod 5) (mod 7) x =? x = 2(70) + 3(21) + 2(15) (mod 105). Shen Kangsheng, Historical Development of the Chinese Remainder Theorem. Wikipedia, Chinese Remainder Theorem. 32 / 43

33 Chinese Remainder Theorem One interpretation of the motivation1 1 Figure from A Mechanical Proof of the Chinese Remainder Theorem, David M. Russinoff, Advanced Micro Devices.. Advanced Micro Devices, Inc / 43

34 Codes with Hierarchical Locality Birenjith Sasidharan, Gaurav Kumar Agarwal, PVK, Codes With Hierarchical Locality, ISIT / 43

35 Codes with Locality do not Scale [24,14,7]' [4,3,2]' [4,3,2]' [4,3,2]' [4,3,2]' [4,3,2]' [4,3,2]' If the local code is overwhelmed, then one has to appeal to the overall code which means contacting all 14 nodes for node repair. So how does one ensure scalability 2? 2 Question posed during University Melbourne talk. 35 / 43

36 Codes with Hierarchical Locality Codes with hierarchical locality ensure scalability by providing an intermediate layer of codes these can help when the when the local code at the bottom, fails. 36 / 43

37 Hierarchical Construction Uses Groups and Subgroups to Partition Coordinates Code lengths must satisfy divisibility condition n 2 n 1 n Here Subgroup chain H 2 H 1 H 0 2 With sizes given by 4, 12, / 43

38 Chinese-Remainder-Theorem-View of Hierarchical Codes Impact in higher layers of restricting polynomials in bottom layer to have degree 2, rather than the customary / 43

39 A Final Refinement for Strengthening Middle and Top-Level Codes Restricting degree of polynomials in middle and top layers strengthens the corresponding codes. 39 / 43

40 Regenerating Codes 40 / 43

41 Regenerating Codes - Formal Definition Parameters: ( (n, k, d), (α, β), B, F q ) 1 2 α 1 2 β β 1 k α α Data Collector 3 β k+1 d+1 n n α capacity nodes α capacity nodes Data to be recovered by connecting to any k of n nodes Nodes to be repaired by connecting to any d nodes, downloading β symbols from each node; (dβ << file size B ) 41 / 43

42 High-Rate MSR Code x y x y Z = (0,0,0) Z A1 A2 Z = (0,0,0) Z B1 B2 Z = (1,1,1) Z = (1,1,1) Min Ye, Alexander Barg, Explicit constructions of optimal-access MDS codes with nearly optimal sub-packetization, arxiv: v1, 27 May Birenjith Sasidharan, Myna Vajha, P. Vijay Kumar, An Explicit, Coupled-Layer Construction of a High-Rate MSR Code with Low Sub-Packetization Level, Small Field Size and d < (n 1), arxiv: v1, 25 Jan / 43

43 Thanks! 43 / 43