network coding for distributed storage systems ieee transactions on information theory, september...

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Network Coding for Distributed Storage Systems

IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010

Alexandros G. Dimakis

Brighten Godfrey

Yunnan Wu

Martin J. Wainwright

Kannan Ramchandran

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Outline

ه Introductionه Backgroundه Analysisه Evaluationه Conclusion

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Introduction

ه Distributed storage systems provide reliable access to data through redundancy spread over individually unreliable nodes.

ه Storing data in distributed storage systemsه the encoded data are spread across nodes.ه require less redundancy than replication.ه replace stored data periodically.

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Introduction

ه Key issue in distributed storage systems.ه repair bandwidthه storage space

ه How to generate encoded data in a distributed way as little data as possible ?

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MDS Codes

ه A common practice to repair from a single node failure for an erasure coded system.1. a new node to reconstruct the whole encoded data object.

2. then, generate just one encoded block.

ه Maximum Distance Separable (MDS) code.ه (n, k)-MDS propertyه recover original file by any k set of encoded data.

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MDS Codes

File divide

M/k

M/k

M/k

M/k

encodestore at n nodes

MDS encode

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Introduction

ه Redundancy must be continually refreshed as nodes fail in distributed storage systems.ه large data transfers across the network.

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Introduction

ه The erasure codes can be repaired without communicating the whole data object.

ه (4, 2)-MSR example when node is fail.ه generate smaller parity packets of their data.ه forward them to the newcomer.ه the newcomer mix packets to generate two new packets.

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0.5

0.5

0.5

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Introduction

ه This paper identifies that there is a optimal tradeoff curve between storage and repair bandwidth.ه smaller storage space => less redundancy => more repair

bandwidth

ه This paper calls codes that lie on this optimal tradeoff curve regenerating codes.

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Introduction

ه Minimum-Storage Regenerating (MSR) codes.ه can be efficiently repaired.

ه Minimum-Bandwidth Regenerating (MBR) codes.ه storage node stores slightly more than M/k .ه the repair bandwidth can be reduced.

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Outline

ه Introductionه Backgroundه Analysisه Evaluationه Conclusion

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Erasure Codes

ه Classical coding theory focuses on the tradeoff between redundancy and error tolerance.

ه In terms of the redundancy-reliability tradeoff, the Maximum Distance Separable (MDS) codes are optimal.ه the most well-known is Reed-Solomon codes.

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Network Coding

ه Network coding allows ه the intermediate nodes to generate output data by encoding

previously received input data.ه information to be “mixed” at intermediate nodes.

ه This paper investigates the application of network coding for the repair problem in distributed storage.ه tradeoff between storage and repair network bandwidth

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Distributed Storage Systems

ه Erasure codes could reduce bandwidth use by an order of magnitude compared with replication.

ه Hybrid strategy: ه one special storage node maintains one full replica.ه multiple erasure encoded data.ه transfer only M / k bytes for a new encoded data by replica node.ه there is the problem when replica data lost.

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Outline

ه Introductionه Backgroundه Analysisه Evaluationه Conclusion

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Information Flow Graph

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Storage-Bandwidth Tradeoff

ه The normal redundancy we want to maintain requires active storage nodesه each storing α bitsه β bits each from any d surviving nodesه total repair bandwidth is γ = d β

ه For each set of parameters (n, k, d, α, γ), there is a family of information flow graphs, each of which corresponds to a particular evolution of node failures / repairs.

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Storage-Bandwidth Tradeoff

ه Denote this family of directed acyclic graphs by

ه (4, 2, 3, 1 Mb, 1.5 Mb) is feasible.

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Storage-Bandwidth Tradeoff

ه Theorem 1 : For any α ≥ α*(n, k, d, γ), the points are feasible.

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Theorem Proof (1/4)

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Theorem Proof (2/4)

ه .

ه .

ه .

ه .

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Theorem Proof (3/4)

ه .

ه .

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Theorem Proof (4/4)

ه .

ه .

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Storage-Bandwidth Tradeoff

ه Code repair can be achieved if and only if the underlying information flow graph has sufficiently large min-cuts.

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Storage-Bandwidth Tradeoff

ه Optimal tradeoff curve between storage α and repair bandwidth γه (γ = 1, α = 0.2) (γ = 1, α = 0.1)

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Special Cases (1/2)

ه Minimum-Storage Regenerating (MSR) Codes

ه .

ه .

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Special Cases (2/2)

ه Minimum-Bandwidth Regenerating (MBR) Codes

ه .

ه .

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Outline

ه Introductionه Backgroundه Analysisه Evaluation

ه Node Dynamics and Objectivesه Modelه Quantitative Results

ه Conclusion

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Node Dynamics and Objectives (1/2)

ه A permanent failureه the permanent departure of a node from the systemه a disk failure resulting in loss of the data stored on the node

ه A transient failureه node rebootه temporary network disconnection

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Node Dynamics and Objectives (2/2)

ه A file is availableه it can be reconstructed from the data stored on currently available

nodes.

ه A file is durabilityه after permanent node failures, it may be available at some point in

the future.

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Model (1/5)

ه The model has two key parameters, f and a.ه a fraction f of the nodes storing file data fail permanently per unit

time.ه at any given time, the node storing data is available with some

probability a.

ه The expected availability and maintenance bandwidth of various redundancy schemes can be computed to maintain a file of M bytes.

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Model (2/5)

ه Replicationه redundancy R replicasه store total R M bytesه replace f R M bytes per unit timeه the file is unavailable if no replica is available

ى probability

ه Ideal Erasure Codesه n = k R, redundancy R n / kه transfer just M / k bytes each packetه replace f R M bytes per unit timeه unavailability probability

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Model (3/5)

ه Hybridه n = k (R− 1)ه store total R M bytesه transfer f R M bytes per unit timeه The file is unavailable if the replica is unavailable and fewer than

k erasure-coded packets are availableى probability

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Model (4/5)

ه Minimum-Storage Regenerating Codesه store total R M bytesه redundancy R n / kه replace f R M bytes per unit timeه extra amount of informationه unavailability

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Model (5/5)

ه Minimum-Bandwidth Regenerating Codesه store total M n bytesه redundancy R n / kه replace f M n bytes per unit timeه extra amount of informationه unavailability

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Estimating f and a

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Quantitative Results (1/2)

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Quantitative Results (2/2)

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Quantitative Comparison

ه Comparison With Hybridه Disadvantage : asymmetric design

ه MBR codesه Disadvantage :

ى reconstruct the entire file, requires communication with n1 nodesى if the reading frequency of a file is sufficiently high and k is sufficiently small,

this inefficiency could become unacceptable.

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Outline

ه Introductionه Backgroundه Analysisه Evaluationه Conclusion

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Conclusion

ه This paper presented a general theoretic framework that can determine the information.ه communicate to repair failures in encoded systems.ه identify a tradeoff between storage and repair bandwidth.

ه One potential application area for the proposed regenerating codes is distributed archival storage or backup.ه regenerating codes potentially can offer desirable tradeoffs in

terms of redundancy, reliability, and repair bandwidth.