![Page 1: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/1.jpg)
1
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
![Page 2: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/2.jpg)
2
Outline
ه Introductionه Backgroundه Analysisه Evaluationه Conclusion
![Page 3: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/3.jpg)
3
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.
![Page 4: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/4.jpg)
4
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 ?
![Page 5: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/5.jpg)
5
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.
![Page 6: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/6.jpg)
6
MDS Codes
File divide
M/k
M/k
M/k
M/k
encodestore at n nodes
MDS encode
![Page 7: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/7.jpg)
7
Introduction
ه Redundancy must be continually refreshed as nodes fail in distributed storage systems.ه large data transfers across the network.
![Page 8: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/8.jpg)
8
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.
0.50.50.50.5
0.5
0.5
0.5
![Page 9: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/9.jpg)
9
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.
![Page 10: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/10.jpg)
10
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.
![Page 11: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/11.jpg)
11
Outline
ه Introductionه Backgroundه Analysisه Evaluationه Conclusion
![Page 12: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/12.jpg)
12
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.
![Page 13: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/13.jpg)
13
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
![Page 14: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/14.jpg)
14
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.
![Page 15: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/15.jpg)
15
Outline
ه Introductionه Backgroundه Analysisه Evaluationه Conclusion
![Page 16: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/16.jpg)
16
Information Flow Graph
![Page 17: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/17.jpg)
17
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.
![Page 18: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/18.jpg)
18
Storage-Bandwidth Tradeoff
ه Denote this family of directed acyclic graphs by
ه (4, 2, 3, 1 Mb, 1.5 Mb) is feasible.
![Page 19: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/19.jpg)
19
Storage-Bandwidth Tradeoff
ه Theorem 1 : For any α ≥ α*(n, k, d, γ), the points are feasible.
![Page 20: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/20.jpg)
20
Theorem Proof (1/4)
![Page 21: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/21.jpg)
21
Theorem Proof (2/4)
ه .
ه .
ه .
ه .
![Page 22: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/22.jpg)
22
Theorem Proof (3/4)
ه .
ه .
![Page 23: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/23.jpg)
23
Theorem Proof (4/4)
ه .
ه .
![Page 24: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/24.jpg)
24
Storage-Bandwidth Tradeoff
ه Code repair can be achieved if and only if the underlying information flow graph has sufficiently large min-cuts.
![Page 25: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/25.jpg)
25
Storage-Bandwidth Tradeoff
ه Optimal tradeoff curve between storage α and repair bandwidth γه (γ = 1, α = 0.2) (γ = 1, α = 0.1)
![Page 26: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/26.jpg)
26
Special Cases (1/2)
ه Minimum-Storage Regenerating (MSR) Codes
ه .
ه .
![Page 27: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/27.jpg)
27
Special Cases (2/2)
ه Minimum-Bandwidth Regenerating (MBR) Codes
ه .
ه .
![Page 28: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/28.jpg)
28
Outline
ه Introductionه Backgroundه Analysisه Evaluation
ه Node Dynamics and Objectivesه Modelه Quantitative Results
ه Conclusion
![Page 29: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/29.jpg)
29
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
![Page 30: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/30.jpg)
30
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.
![Page 31: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/31.jpg)
31
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.
![Page 32: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/32.jpg)
32
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
![Page 33: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/33.jpg)
33
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
![Page 34: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/34.jpg)
34
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
![Page 35: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/35.jpg)
35
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
![Page 36: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/36.jpg)
36
Estimating f and a
![Page 37: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/37.jpg)
37
Quantitative Results (1/2)
![Page 38: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/38.jpg)
38
Quantitative Results (2/2)
![Page 39: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/39.jpg)
39
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.
![Page 40: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/40.jpg)
40
Outline
ه Introductionه Backgroundه Analysisه Evaluationه Conclusion
![Page 41: Network Coding for Distributed Storage Systems IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 2010 Alexandros G. Dimakis Brighten Godfrey Yunnan Wu](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649dc65503460f94aba908/html5/thumbnails/41.jpg)
41
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.