Exact Regenerating Codes on Hierarchical
Codes
Ernst Biersack
Eurecom
France
Joint work and Zhen Huang
Outline
:: Introduction and motivation
:: Hierarchical Codes
:: Regenerating Codes
:: Combining Hierarchical Codes and
Regenerating Codes
:: Conclusion
Motivation: Elements of a P2P backup system
Performance metrics:
Storage efficiency: how much redundant information do you store?
From Julian Monteiro
3
Motivation: Network Bandwidth is a scarce resource
Our first objective is to find erasure codes that consume less communication
bandwidth, i.e. have better efficiency factor ρ
- Network communication bandwidth cannot be “put aside” for later use
A second objective should be to adopt repair policies that provide a
smooth utilization of the communication bandwidth
4
Hierarchical Codes
Regenerating Codes
ER-Hierarchical Codes
Linear Codes: Overview
- A particular way to build erasure codes is linear codes
original
fragments
o1
c1,1 Linear combination
o2
c1,2
c1,3
c1,4
o3
o4
P = CO
O=C-1P
p1
parity fragment
pi   ci , j o j
j
[c1,1 c1,2 c1,3 c1,4]
[c2,1 c2,2 c2,3 c2,4]
[c3,1 c3,2 c3,3 c3,4]
[c4,1 c4,2 c4,3 c4,4]
If C is invertible, i.e. the coefficient vectors are
linearly independent,
we can reconstruct the original fragments.
If coefficients are chosen randomly in GF(216), the
matrix is invertible with a very high probability.
p1
p2
p3
p4
p5
p6
Hierarchical codes: Idea
let us try to change the way the code is built:
o1
p1
4+3 Traditional Erasure Code
o2
p3
p4
o3
p5
p6
o4
Probability of failure
p2
100%
100%
80%
80%
60%
60%
40%
40%
20%
20%
0%
0%
1
2
3
4
5
6
# Unavailable Fragments
o1
4+3 Hierarchical Code
7
p2
o2
p3
p4
o3
1
2
3
4
5
6
7
o4
There are sets of 4 parity fragments
that are not sufficient to reconstruct
the original file.
7
p5
p6
# Unavailable fragments
p7
traditional
erasure code
p1
p7
Hierarchical
code
Hierarchical codes : Repair degree
The repair degree determines the efficiency factor ρ
4+3 Hierarchical Code
Probability of cost/failure
4+3 Traditional Erasure Code
100%
100%
90%
90%
80%
80%
70%
70%
60%
60%
50%
Failure
50%
40%
ρ =4
40%
30%
ρ =2
30%
20%
10%
20%
10%
0%
0%
1
2
3
4
5
6
7
1
# Unavailable Fragments
2
3
4
5
# Unavailable fragments
8
6
7
Hierarchical codes: Recursive Construction
HC-(k,h)
•k original blocks
•h redundant blocks
9
Hierarchical codes: Theory
10
Hierarchical codes: Repair
What if p_1 and p_3 are lost?
•Use p_2 , 1 out of {p_7, p_8} and 1 out {p_4, p_5, p_6}
need 3 blocks
What if p_1, p_2, and p_3 are lost?
•Use …..
need ???? blocks
In HC, the earlier we repair the repair is often “cheaper”
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64+64 hierarchical codes: Reliability vs Cost
100%
Cost=2
80%
Cost=4
Cost=8
60%
Cost=16
40%
Cost=32
20%
Cost=64
Failure
0%
Probability of cost/failure
Probability of cost/failure
Two possible instances of a 64+64 hierarchical code
100%
80%
60%
40%
20%
0%
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64
# Unavailable Fragments
# Unavailable Fragments
- Lower repair cost comes at the prices of reduced reliability
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Hierarchical Codes
Regenerating Codes
ER-Hierarchical Codes
Regenerating Codes: Idea
What happens if…
upon a repair we contact
more than k peers?
p1
p2
p5
p’4
d>k
Every peer stores a parity block larger
(or equal) than the usual parity
fragment (i.e. 1/k of the file size)?
o1
o2
o3
o4
p7
b1
|block|≥|file|/k
p8
Regenerating codes (by G. Dimakis) give the answer:
the repair communication requirements are much smaller.
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Regenerating codes: Performance
- regenerating codes are controlled by two additional parameters beyond k and h
:: d the repair degree
k ≤ d ≤ k+h-1
:: i the block expansion index
0 ≤ i ≤ k-1
- if we consider a regenerating code with k=32 and h=32:
classical erasure codes
Block size stretch
1.8
1
Additional space
1.6
i=31
1.4
i=22
i=15
1.2
i=7
1
i=0
0.8
repair-down reduction
2
Reduced communications bandwidth
0.1
0.01
32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62
d
32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62
MBR: Minimum-Bandwidth Regenerating
MSR: Minimum-Storage Regenerating
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d
Regenerating codes: Performance
- k=32 and h=32 and a stored file of 1MB:
1
Additional space
repair-down reduction
Block size stretch
2
1.8
1.6
i=31
1.4
i=22
i=15
1.2
i=7
1
i=0
0.8
Reduced communications bandwidth
0.1
0.01
32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62
d
32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62
Communication is impressively reduced with
small amount of extra storage.
d
code
d
i
repairdown
storage
Classical erasure code
32
0
1 MB
2 MB
“ extreme“ regenerating code
63
30
42.47 KB
2.61 MB
“reasonable” erasure code
40
7
84,62 KB
2.11 MB
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Regenerating codes: A new dimension in the trade-off
Communication
RC(k,h,d,i,)
•k original pieces
•h additional pieces
•d repair degree
•i block expansion factor
Storage
Replication
Regenerating codes can be seen as a generalization of replication and RSE
that allow to more flexibly trade off communication and storage
requirements.
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Regenerating codes: Want to know more
See http://csi.usc.edu/~dimakis/StorageWiki/doku.php
A wiki on Coding for Distributed Storage maintained by Alexandros G. Dimakis
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Hierarchical Codes
Regenerating Codes
ER-Hierarchical Codes
ER-Hierarchical Codes
• Can we combine Hierarchical codes and Regenerating Codes?
• Yes:
ER-Hierarchical Codes combine concepts of Hierarchical Codes and
Regenerating Codes, namely that
• most parity blocks are linear combinations of only a small subset of all
original blocks and that
• a storage block consists of α fragments, while a repair block has only β
fragments, with , β < α
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ER-Hierarchical Codes: Construction
• How to transform Hierarchical code into ER-Hierarchical Code?
21
ER-Hierarchical Codes: Construction
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ER-Hierarchical Codes: Repair
• In HC we would need to
download 4 blocks of size 1 each
•  4 units of traffic
• In ER-HC we now download 5
fragments of size ½ each
•  2.5 units of traffic
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ER-Hierarchical Codes: Traffic reduction (analysis)
• ER-HC reduces the traffic by more than
• 85% as compared to RSE and Regenerating Codes
• 40% compared to Hierarchical codes
Reg Code is MSR with d=k+1
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ER-Hierarchical Codes: Repair Strategies
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ER-Hierarchical Codes: Performance (simulation)
In HC and ER-HC , the
earlier we repair the
“cheaper” the repair;
is not the case for RG
and RSE
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Conclusion
- Have presented some new codes that
-greatly reduce the communications overhead
-Regenerating codes apply principles of network coding to distributed
storage and allow to trade off storage space for communications
bandwidth
-As compared to RSE codes
-Regenerating codes increase the repair degree (number of nodes
that must be contacted for repair) but significantly reduce the
amount of data downloaded from each node
-Hierarchical codes significantly reduce the repair degree while
keeping the amount of data transferred by each node the same (as
RSE)
-Combining Regenerating Codes and Hierarchical Codes makes us win
at both fronts
-Reduces repair degree and the amount of data transmitted by each
node
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Future work
• Further exploit the possibilities offered by ER-Hierarchical Codes
•Study the relationship between coding and repair policies for systems with churn
•Reactive repair results in repair burst
•Proactive repair has smoother repair traffic but does unnecessary repairs. If
repairs are cheap, as they are for ER-HC, proactive repair becomes much
more attractive since the “earlier we repair”, the cheaper a repair
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Thanks
Questions?