Update and Repair Efficient Codes for Distributed Storage
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1 Update and Repair Efficient Codes for Distributed Storage Ankit Singh Rawat Wireless Networking and Communications Group (WNCG) The University of Texas at Austin December 18, 2013 A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
2 Distributed Storage System (DSS) Data Source A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
3 Data reconstruction A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
4 Node failure A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
5 Node failure Need to introduce redundancy into the system. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
6 Replication vs. coding a a a b b b a b a + b 2a + b 3-replication (4, 2) Erasure coding A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
7 Replication vs. coding a a a b b b a b a + b 2a + b 3-replication (4, 2) Erasure coding Traditional MDS codes are optimal for storage vs. reliability trade-off. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
8 Replication vs. coding a a a b b b a b a + b 2a + b 3-replication (4, 2) Erasure coding Traditional MDS codes are optimal for storage vs. reliability trade-off. Reliability is not the only metric of interest: I Overhead of updating data. I Node repair overhead. I Availability. I... A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
9 Replication vs. coding a a a b b b a b a + b 2a + b 3-replication (4, 2) Erasure coding Traditional MDS codes are optimal for storage vs. reliability trade-off. Reliability is not the only metric of interest: I Overhead of updating data. I Node repair overhead. I Availability. I... May need to move away from MDS codes. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
10 Overhead of updating data (Update Complexity) Information to be stored is almost never static. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
11 Overhead of updating data (Update Complexity) Information to be stored is almost never static. Update complexity: maximum number of encoded symbols that must be updated when any single information symbol is changed. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
12 Example: (7, 4) Hamming code x 1 x 1, x 2, x 3, x 4 Source Node x 2 x 3 x 4 x 2 + x 3 + x 4 x 1 + x 3 + x 4 2 y = x 1 x 2 x 3 x x 1 + x 2 + x 4 A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
13 Example: (7, 4) Hamming code x 1 x 1, x 2, x 3, x 4 Source Node x 2 x 3 x 4 x 2 + x 3 + x 4 x 1 + x 3 + x 4 2 y = x 1 x 2 x 3 x x 1 + x 2 + x 4 A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
14 Example: (7, 4) Hamming code x 1 x 1, x 2, x 3, x 4 Source Node x 2 x 3 x 4 x 2 + x 3 + x 4 x 1 + x 3 + x 4 x 1 + x 2 + x 4 2 y = x 1 x 2 x 3 x Update complexity minimum distance. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
15 Update Complexity Information to be stored is almost never static. Update complexity: maximum number of encoded symbols that must be updated when any single information symbol is changed. Updating data consumes bandwidth and energy. Design codes with small update complexity. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
16 Update Efficiency Constant update complexity. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
17 Update Efficiency Constant update complexity. Linear update complexity. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
18 Update Efficiency Constant update complexity. Linear update complexity. I Low update complexity MDS codes for storage: F e.g. X-code [XuBruck], EVENODD code [BlaumBradyBruckMenon]. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
19 Update Efficiency Constant update complexity. Linear update complexity. I Low update complexity MDS codes for storage: F e.g. X-code [XuBruck], EVENODD code [BlaumBradyBruckMenon]. Coding schemes with sub-linear update complexity. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
20 Kolchin generator (KG) codes Random ensemble of codes with logarithmic update complexity. P. Anthapadmanabhan, E. Soljanin and S. Vishwanath, in Allerton A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
21 Kolchin generator (KG) codes Random ensemble of codes with logarithmic update complexity. Generator matrix: 2 G = log n P(G i,j = 1) =O n n k n P. Anthapadmanabhan, E. Soljanin and S. Vishwanath, in Allerton A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
22 Kolchin generator (KG) codes Random ensemble of codes with logarithmic update complexity. Generator matrix: 2 G = log n P(G i,j = 1) =O n n k n With high probability, every column has O(log n) non-zero entries. P. Anthapadmanabhan, E. Soljanin and S. Vishwanath, in Allerton A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
23 Failure tolerance of KG codes Minimum distance of KG codes is at most O(log n). A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
24 Failure tolerance of KG codes Minimum distance of KG codes is at most O(log n). Random erasure model: each node answer with probability p. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
25 Failure tolerance of KG codes Minimum distance of KG codes is at most O(log n). Random erasure model: each node answer with probability p. W.h.p., arandom set of k n (1 + ) encoded symbols is sufficient to reconstruct data. I Can tolerate erasure probability p < n kn n. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
26 Node repair Replacement node A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
27 Efficient node repair Important to perform efficient node repairs. Multiple metrics to measure repair efficiency. Repair bandwidth: amount of data downloaded during a node repair. Locality: number of remaining nodes contacted during a node repair. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
28 Efficient node repair Important to perform efficient node repairs. Multiple metrics to measure repair efficiency. Repair bandwidth: amount of data downloaded during a node repair. I A. S. Rawat, A. Bhowmick, S. Vishwanath and E. Soljanin, in ISIT Locality: number of remaining nodes contacted during a node repair. I M. Asteris and A. Dimakis, in ISIT I Modification of KG codes appraoch. I A. Mazumdar, V. Chandar and G. Wornell, in ITA A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
29 Repairable fountain codes M. Asteris and A. Dimakis, in ISIT A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
30 Repairable fountain codes x 1 x 2 x 3 x kn M. Asteris and A. Dimakis, in ISIT A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
31 Repairable fountain codes x 1 x 2 x 1 x 3 x 4 x 3 x kn x kn M. Asteris and A. Dimakis, in ISIT A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
32 Repairable fountain codes x 1 x 2 x 1 x 3 x 4 x 3 x kn x kn n k n parity symbols Each parity node is obtained by randomly throwing (log n) edges. M. Asteris and A. Dimakis, in ISIT A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
33 Repairable fountain codes. x 1 x 2 x 1 x 3 x 4 x 3 x kn x kn n k n parity symbols Each node has locality (log n). W.h.p., update complexity: O(log n). I Maximum load of throwing O(kn log k n ) balls into k n bins. Failure tolerance: I Any random set of kn (1 + ) encoded symbols are sufficient for data reconstruction. M. Asteris and A. Dimakis, in ISIT A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
34 Modification of KG codes approach A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
35 Modification of KG codes approach x 1 x 2 x 3 x kn A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
36 Modification of KG codes approach x 1 x 1 x 2 x 3 x 4 x 3 x kn x kn A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
37 Modification of KG codes approach x 1 x 1 x 2 x 3 x 4 x 3 x kn x kn n k n parity symbols For parity nodes, each edge is present w. p. ( log n n ). A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
38 Modification KG codes approach. x 1 x 1 x 2 x 3 x 4 x 3 x kn x kn n k n parity symbols W.h.p., each node has locality (log n). W.h.p, update complexity: O(log n). Failure tolerance: I Any random set of kn (1 + ) encoded symbols are sufficient for data reconstruction. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
39 Discussion and open questions Explicit constructions for update efficient codes. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
40 Discussion and open questions Explicit constructions for update efficient codes. Study of update efficient codes under general failure model. I B. Kanukurthi, N. Chandran and R. Ostrovsky, in TCC 2014 I Construction of update efficient locally decodable codes for prefix hamming metrics. A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
41 Locality and availability A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
42 Locality and availability c 1 A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
43 Locality and availability c 1 A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
44 Locality and availability c 1 c 1 A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
45 Locality and availability c 1 c 1 A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
46 Locality and availability c 1 c 1 A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
47 Locality and availability c 1 c 1 c 1 c 1 3 disjoint repair groups ) 3-availability. A. Rawat, D. Papailiopoulos, A. Dimakis, and S. Vishwanath, in Allerton A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
48 Locality and availability c 1 c 1 c 1 c 1 3 disjoint repair groups ) 3-availability. In general, t availability with r locality. A. Rawat, D. Papailiopoulos, A. Dimakis, and S. Vishwanath, in Allerton A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
49 Locality and availability c 1 c 1 c 1 c 1 3 disjoint repair groups ) 3-availability. In general, t availability with r locality. Open question: how to address general t request patterns. I t requests for c1. I t/2 requests for c1, and t/2 requests for c 2. I t/3 requests for each of c1, c 2 and c 3. A. Rawat, D. Papailiopoulos, A. Dimakis, and S. Vishwanath, in Allerton A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
50 Thank you! A. S. Rawat (UT Austin) Update Efficiency December 18, / 24
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