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Erasure coding

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Erasure coding

Group J’

Yetian Huang

Yang Liu

Yi He

- Get Started with Erasure coding
- Network coding for Distributed Storage System (DSS)
- Introduction the Fundamentals of Network Coding
- Performance Results and Analysis

- A New Erasure Coding Method in Windows Azure Storage
- Conventional Reed-Solomon (RS) code
- Improved Local Reconstruction Coding

- Novel Erasure Codes for Big Data
- Introduction of Locally Repairable Code(LRC)
- Performance evaluation on Amazon EC2 and X's cluster

- Term from Information Theory
- Used for controlling errors
- Add redundancy
- Complex mathematical function

- Numerous disk failures
- Redundancy is necessary
- 3-replication

- Erasure Coding
- Storage saving

Network Coding for Distributed Storage System

Introduction of Erasure Code

Performance Results and Analysis

- Data Security
- High Failure Tolerance
- High Availability
- High Reliable
- Prevent Data Loss

- Low Redundancy Level
- Low Storage Cost
- Low Bandwidth Cost
- Low Coding Complexity

Replication

A

A

File or Data Object

A

B

- Reliable
- High Redundancy Level
- High Storage Cost

B

- Any k out of the n code word symbols are sufficient to recover the original message.
- Optimal erasure codes are maximum distance separablecodes (MDS codes).

- Low Bandwidth Cost

Each storage node is storing two fragments that are linear binary combinations of the original data fragments𝐀𝟏, 𝐀𝟐, 𝐁𝟏, 𝐁𝟐. In this example, the total stored size is M = 4 fragments. Observe that any k = 2 out of the n = 4 storage nodes contain enough information to recover all the data (adopted from [1])

Assume that the first node in the previous storage system failed. it is obtain exact repair by communicating three fragments.

N=4

A

N=3

K=2

A

B

File or Data Object

A

B

A+B

A+2B

A+B

(3,2) MDS Code

(Single parity)

Raid 5 Mode

(4,2) MDS Code

Raid 6 Mode

(4,2) MDS

Erasure Code

Replication

A

A

File or Data Object

A

B

B

A

A+B

B

Low Redundancy Level

Low Storage Cost

B

A+2B

- Regenerating Code
- Repairing lost encoded fragments from existing encoded fragments.
- A new class of erasure code.
- Reduce repair bandwidth.
- Increase number of surviving node connected.

- MDS code:
- High complexity.
- Uses a random linear network coding.

- “repair-by-transfer regenerating code” :
- Less complexity.
- Its process is addition of two packets using bit-wise exclusive OR (XOR).

(4,2) Regenerating Code

A New Erasure Coding Method in Windows Azure Storage

Conventional Reed-Solomon (RS) code

Improved Local Reconstruction Code

- used to correct errors
- RS Encoder: takes a block of digital data and adds extra "redundant" bits
- RS Decoder: processes each block and attempts to correct errors and recover the original data

- RS(n, k):
- k data symbols of s bits each
- n symbol codeword.
- n-k parity symbols of s bits each
- Correct up to t symbols errors, where 2t = n-k.

- Example:
- RS(255,223)
- n=255, k=223, s=8, 2t=32, t=16

- RS (6, 3) Code
- 6 fragments, and 3 parties
- GFS II in Google

- RS (10, 4)
- Facebook HDFS-RAID

- Consideration
- Storage overhead
- The less, the better.

- Reliability
- At least higher than 3-replication
- Fault Tolerance

- Repair Performance
- Lower bandwidth consumption
- Fewer the number of disk I/Os
- Decoding latency

- Storage overhead

Overhead=# of nodes / # of fragments

- Increase the number of fragments
- 6+3 => 12 + 3…

- But reliability decreased…
- Need more redundancy

- Finally
- 12+4
- 1.5x -> 1.33x

- Reconstruction cost
the number of fragments required to reconstruct an unavailable data fragment

- Could we achieve
- Reconstruction cost = 6
- Overhead = 1.33x

- We found
- All failures are equally considered in conventional EC.
- Actually Probability (1 failure) >> Probability (2 failures)

Local Reconstruction Code

- Group fragments and reconstruct locally
12 fragments, 2 local parties and 2 global parties

- Overhead: (12+2+2) / 12 = 1.33x
- reconstruction cost = 6 fragments.

- Definition:
- k – # of data fragments
- l – # of groups (or local parities)
- r – # of global parities
- n - # of nodes, i.e. n = k + l + r
Pattern: LRC(k, l, r)

- Arbitrary 3 failures
- All the information-theoretically decodable 4 failure patterns (86% of all 4 failure patterns)
Satisfy Maximally Recoverable (MR) property:

- decode any failure pattern which is information-theoretically decodable.

3 parities cannot recover 4 failures

- Decodable

- Factors
- Reliability (MTTF)
- Threshold: the MTTF of 3-replication

- Reconstruction cost
- Storage overhead

- Reliability (MTTF)
- Reduce trade-off
space to 2D

LRC vs. RS code

Novel Erasure Codes for Big Data

Introduction of Locally Repairable Code(LRC)

Performance evaluation on Amazon EC2 and X's cluster

Definition

A randomized an explicit family of codes that have logarithmic locality on all coded blocks and distance that is asymptotically equal to that of an MDS code. We call such codes (k; n-k; r) Locally Repairable Codes (LRCs)

X1 X2 X3 X4 X5

A STRIP

5 File blocks

C2

C3

C4

C1

C5

A Local Parity Block

S1

X1C1+X2C2+X3C3+X4C4+X5C5 = S1

44

If a single block failure!

For example, if block X3 is lost!

45

A strip

(10 file blocks)

RS Code

(4 parity blocks)

X1C1+X2C2+X3C3+X4C4+X5C5=S1

X6C6+X7C7+X8C8+X9C9+X10C10=S2

1.

We set C5’=C6’=1

2.

Set S1+S2+S3=0

3.

Set P1C1’+P2C2’+P3C3’+P4C4’=S3

If p2 failure:

47

- They have different storage space.
- When the parity block fails, Locally Repairable Code could cost less to repair.
- Faster for single block failure. Efficiency: Local parity > RS parity block.

Repair Duration is simply calculated as the time interval between the starting time of the first repair job and the ending time of the last repair job.

HDFS Bytes Read corresponds to the total amount of data read by the jobs initiated for repair.

Network traffic represents the total amount of data sent from nodes in the cluster

Evaluation on Amazon EC2

Be encoding to one stripe with 14 and 16 size in HDFS-RS and HDFS-Xorbas.

HDFS Bytes Read

Network Traffic

Repair Duration

Evaluation on X’s clusters

Since block repair depends only on blocks of the same stripe, using larger files that would yield more than one stripe would not affect

our results. An experiment involving arbitrary file sizes use X’s clusters.

- Cluster network traffic.

- Cluster Disk Bytes Read.

- Cluster average CPU utilization.

- Replication method with erasure coding makes the storage more efficiency.
- Introduced a new set of erasure codes called Local Reconstruction Codes that reduces reconstruction cost while still keeping the storage overhead low.
- Designed and implemented a new code based on RS Code, which is more practically relevant in large-scale cloud storage systems.

- AwassadaPhutathum, Implementing Distributed Storage Systems by Network Coding and Considering Complexity of Decoding, Master's Thesis, XR-EE-KT 2012:001, KTH Royal Institute of Technology, Stockholm, Sweden, August 2012
- A. G. Dimakis, P. B. Godfrey, Y. Wu, M. J. Wainwright, and K. Ramchandran, “Network coding for distributed storage systems," IEEE Trans. on Info. Theory, Sep. 2010. Preliminary version appeared in Infocom 2007.
- Erasure code - Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Erasure_code

- C. Huang, H. Simitci, Y. Xu, A. Ogus, B. Calder, P. Gopalan, J. Li, S. Yekhanin, “Erasure coding in Windows Azure storage,” USENIX ATC, June 2012.
- B. Calder et al., “Windows Azure Storage: A Highly Available Cloud Storage Service with Strong Consistency,” ACM SOSP, 2011.
- I. S. Reed and G. Solomon, “Polynomial Codes over Certain Finite Fields”, J. SIAM, 8(10), 300-304, 1960.

- XORing Elephants: Novel Erasure Codes for Big Data
- MosharafChowdhury, MateiZaharia, Justin Ma, Michael I. Jordan, and IonStoica.Managing data transfers in computer cluster with orchestra.SIGCOMMComput, Commun. Rev, 41:98-109, August2011
- O. Khan, R. Burns, J. Plank, W. Pierce, and C. Huang. Rethinking erasure codes for cloud file systems: Minimizing i/o for recovery and degraded reads. In Usenix Conference on File and Storage Technologies (FAST), 2012

Thank you!