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Erasure coding. Group J’ Yetian Huang Yang Liu Yi He. Overview. 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

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Erasure coding

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

What is erasure coding
What is Erasure Coding?

  • Term from Information Theory

    • Used for controlling errors

    • Add redundancy

    • Complex mathematical function

Erasure coding for distributed storage
Erasure Coding for Distributed Storage

  • Numerous disk failures

    • Redundancy is necessary

    • 3-replication

  • Erasure Coding

    • Storage saving

Network coding for distributed storage system

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 code
Replication Code




File or Data Object



  • Reliable

  • High Redundancy Level

  • High Storage Cost


Erasure coding1
Erasure Coding

  • 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).

Erasure coding2
Erasure Coding

  • Low Bandwidth Cost

Example of erasure code
Example of Erasure Code

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])

Example of erasure code1
Example of Erasure Code

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

Erasure coding mds
Erasure Coding: MDS







File or Data Object






(3,2) MDS Code

(Single parity)

Raid 5 Mode

(4,2) MDS Code

Raid 6 Mode

Erasure coding vs replica
Erasure Coding VS. Replica

(4,2) MDS

Erasure Code




File or Data Object







Low Redundancy Level

Low Storage Cost



I mprovement of mds
Improvement of MDS

  • 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 and regenerating code
MDS and Regenerating Code

  • 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).

Performance results
Performance Results

Performance results cont
Performance Results (Cont.)

(4,2) Regenerating Code

Performance results cont1
Performance Results (Cont.)

Performance results cont2
Performance Results (Cont.)

A new erasure coding method in windows azure storage

A New Erasure Coding Method in Windows Azure Storage

Conventional Reed-Solomon (RS) code

Improved Local Reconstruction Code

Reed solomon code
Reed-Solomon 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 code definition
RS Code Definition

  • 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

Reed solomon code in dss
Reed-Solomon Code in DSS

  • RS (6, 3) Code

    • 6 fragments, and 3 parties

    • GFS II in Google

  • RS (10, 4)

    • Facebook HDFS-RAID

Which erasure codes are best
Which Erasure Codes are best?

  • 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


Overhead=# of nodes / # of fragments

Further reduce the overhead
Further reduce the overhead?

  • Increase the number of fragments

    • 6+3 => 12 + 3…

  • But reliability decreased…

    • Need more redundancy

  • Finally

    • 12+4

    • 1.5x -> 1.33x

When failure occurs
When failure occurs…

  • Reconstruction cost

    the number of fragments required to reconstruct an unavailable data fragment

Erasure coding

  • 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

Local reconstruction code
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)

Fault tolerance
Fault Tolerance

  • 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.

4 failures
4 failures

3 parities cannot recover 4 failures

4 failures1
4 failures

  • Decodable

Code parameter selection
Code Parameter Selection

  • Factors

    • Reliability (MTTF)

      • Threshold: the MTTF of 3-replication

    • Reconstruction cost

    • Storage overhead

  • Reduce trade-off

    space to 2D

Novel erasure codes for big data

Novel Erasure Codes for Big Data

Introduction of Locally Repairable Code(LRC)

Performance evaluation on Amazon EC2 and X's cluster

Erasure coding


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)


Introduce locality to rs
Introduce “locality” to RS

X1 X2 X3 X4 X5


5 File blocks






A Local Parity Block


X1C1+X2C2+X3C3+X4C4+X5C5 = S1


How locally repairable codes work
How Locally Repairable Codes Work

If a single block failure!

For example, if block X3 is lost!


How locally repairable codes work1
How Locally Repairable Codes Work

A strip

(10 file blocks)

RS Code

(4 parity blocks)



How to keep the parity blocks satisfy
How to keep the parity blocks satisfy


We set C5’=C6’=1


Set S1+S2+S3=0


Set P1C1’+P2C2’+P3C3’+P4C4’=S3

If p2 failure:


Locally repairable code vs local reconstruction code
Locally Repairable Code vsLocal Reconstruction Code

  • 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.

Performance evaluation on amazon ec2 and x s cluster
Performance evaluation on Amazon EC2 and X's cluster

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

Erasure coding

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

Erasure coding

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

  • 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