Modified Generalized Integrated Interleaved Codes for Local Erasure Recovery
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1 Modified Generalized Integrated Interleaved Codes for Local Erasure Recovery Xinmiao Zhang Dept. of Electrical and Computer Engineering The Ohio State University
2 Outline Traditional failure recovery schemes for distributed storage system Locally recoverable erasures codes Generalized integrated interleaved (GII) codes for local erasure recovery Modified GII codes achieving locality improvement Comparisons and conclusions
3 Failure Recovery for Distributed Storage Mobile HPC Cloud Big Data Apps Network fabric A distributed storage system has many storage nodes, and redundancy is needed to tolerate and recover from failures The latency and network traffic overhead of recovering failed data packets or storage nodes largely affect the overall system performance
4 Conventional Failure Recovery Schemes Triple replication Erasure coded Network Fabric Network Fabric Low repair overhead Very high redundancy k symbols k data disks Encoder n-k parity disks n symbols failures = erasures in coding terminology indices of tt tt erased symbols kk un-erased symbols Decoder ttt recovered symbols An nn, kk code can recover at most tt = nn kk erasures; codes meeting this bound are called maximum distance separable (MDS) codes Traditional erasure codes need kk symbols to recover from any failures; have large repair overheads
5 Codes with Reduced Network Traffic for Failure Recovery Distributed storage needs coded failure recovery schemes accessing much less than kk symbols (possible if actual erasure number < tt: the erasure correction capability) Lower network traffic Reduced recovery latency Better data availability Minimum storage regenerating codes Array MDS codes Read from a larger number of nodes but fewer symbols from each node Locally recoverable (LRC) erasure codes Access fewer storage nodes More redundancy is added to achieve locality recover codeword codeword recover
6 Generalized Integrated Interleaved (GII) Codes Parities for individual interleaves c 0 c 1 c 2 c 3 cc 0 cc 1 cc 2 Parities shared by interleaves Nesting matrix GG Correction capability cc 0 cc 1 cc vv 1 cc 0, cc 1,, cc mm 1
7 Decoding of GII Codes Parities for individual interleaves c 0 c 1 c 2 c 3 cc 0 cc 1 cc 2 Parities shared by interleaves Interleave syndromes Syndrome conversion matrix nested syndromes ll 1, ll 2,, ll bb : indices of interleaves with more than tt 0 erasures (exceptional interleaves) tt syndromes are needed to correct tt erasures Higher-order syndromes for the interleaves are generated from the nested syndromes Syndrome conversion matrix is always invertible Each nested syndrome is generated by utilizing all interleaves Need all the interleaves if any of them has more than tt 0 erasures!
8 Modified GII Codes Previous GII shared parities Modified GII tt vv tt vv 1 tt vv 2 tt 1 Less powerful nestings involve fewer interleaves Form the syndrome conversion matrix using the bottom rows of GGG as much as possible The selected nestings should have sufficient correction capability The selected nestings should cover every exceptional interleave Consecutive nestings are used to simplify the selection
9 Invertibility of Syndrome Conversion Matrix The columns of the syndrome conversion matrix correspond to the exceptional interleaves All-zero columns can be avoided, but the syndrome conversion matrix may still have zero entries The syndrome conversion matrix is invertible if the number of interleaves does not exceed the values in the following table for given number of nested codewords (vv) and finite field order
10 Correction Capability and Locality Comparisons Modified GII codes preserve the same correction capability as the original GII codes for most practical settings Modified GII codes require fewer interleaves to utilize the shared parities when there are fewer extra erasures to correct Have very small implementation overhead compared to the GII codes Achieve good tradeoff on locality and correction capability
11 Conclusions Modified GII codes substantially improve the locality for erasure correction over prior GII codes Modified GII codes do not bring any correction capability degradation for most practical settings Modified GII codes achieve good tradeoff on the locality and correction capability compared to other LRC codes Further locality improvement can be achieved by multi-layer integrated interleaved codes
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