Exact Regenerating Codes on Hierarchical Codes

Ernst BiersackEurecomFrance

Joint work and Zhen Huang

Outline

:: Introduction and motivation:: Hierarchical Codes:: Regenerating Codes:: Combining Hierarchical Codes and

Regenerating Codes:: Conclusion

3

Motivation: Elements of a P2P backup system

Performance metrics:Storage efficiency: how much redundant information do you store?

From Julian Monteiro

4

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

Hierarchical Codes

Regenerating Codes

ER-Hierarchical Codes

Linear Codes: Overview

- A particular way to build erasure codes is linear codes

o1

o2

o3

o4

original fragments

p1

p2

p3

p4

[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]

P = CO O=C-1P

If C is invertible, i.e. the coefficient vectors are linearly independent,we can reconstruct the original fragments.

p1

c1,1

j

jjii ocp ,

c1,2

c1,4

c1,3parity fragment

Linear combination

p5

p6

If coefficients are chosen randomly in GF(216), the matrix is invertible with a very high probability.

7

Hierarchical codes: Idea

let us try to change the way the code is built:

o1

o2

o3

o4

p1

p2

p3

p4

p5

p6

p7

There are sets of 4 parity fragments that are not sufficient to reconstruct the original file.

1 2 3 4 5 6 70%

10%20%30%40%50%60%70%80%90%

100%

4+3 Traditional Erasure Code

# Unavailable Fragments

Prob

abili

ty o

f fai

lure

1 2 3 4 5 6 70%

10%20%30%40%50%60%70%80%90%

100%

4+3 Hierarchical Code

# Unavailable fragments

1 2 3 4 5 6 70%

10%20%30%40%50%60%70%80%90%

100%

4+3 Traditional Erasure Code

# Unavailable Fragments

Prob

abili

ty o

f fai

lure

1 2 3 4 5 6 70%

10%20%30%40%50%60%70%80%90%

100%

4+3 Hierarchical Code

# Unavailable fragments

o1

o2

o3

o4

p1

p2

p3

p4

p5

p6

p7

traditional erasure code

Hierarchical code

8

Hierarchical codes : Repair degree

1 2 3 4 5 6 70%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

4+3 Traditional Erasure Code

# Unavailable Fragments

Prob

abili

ty o

f cos

t/fa

ilure

Failure

1 2 3 4 5 6 70%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

4+3 Traditional Erasure Code

# Unavailable Fragments

Prob

abili

ty o

f cos

t/fa

ilure

ρ =4

1 2 3 4 5 6 70%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

4+3 Hierarchical Code

# Unavailable fragments

1 2 3 4 5 6 70%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

4+3 Hierarchical Code

# Unavailable fragments

1 2 3 4 5 6 70%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

4+3 Hierarchical Code

# Unavailable fragments

ρ =2

The repair degree determines the efficiency factor ρ

9

Hierarchical codes: Recursive Construction

HC-(k,h)•k original blocks•h redundant blocks

10

Hierarchical codes: Theory

11

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”

12

64+64 hierarchical codes: Reliability vs Cost

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 610%

10%20%30%40%50%60%70%80%90%

100% Cost=2

Cost=4

Cost=8

Cost=16

Cost=32

Cost=64

Failure

# Unavailable Fragments

Prob

abili

ty o

f cos

t/fa

ilure

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 610%

10%20%30%40%50%60%70%80%90%

100%

# Unavailable Fragments

Prob

abili

ty o

f cos

t/fa

ilure

Two possible instances of a 64+64 hierarchical code

- Lower repair cost comes at the prices of reduced reliability

ER-Hierarchical Codes

Regenerating Codes

Hierarchical Codes

14

Regenerating Codes: Idea

What happens if…

Regenerating codes (by G. Dimakis) give the answer: the repair communication requirements are much smaller.

upon a repair we contact more than k peers?

p1

p2

p5

p7

p’4 d>k

p8

Every peer stores a parity block larger (or equal) than the usual parity fragment (i.e. 1/k of the file size)?

o1o2o3o4

|block|≥|file|/k

b1

15

Regenerating codes: Performance

- regenerating codes are controlled by two additional parameters beyond k and h:: d the repair degree

:: i the block expansion index k ≤ d ≤ k+h-10 ≤ i ≤ k-1

- if we consider a regenerating code with k=32 and h=32:

32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 620.8

1

1.2

1.4

1.6

1.8

2

i=31i=22i=15i=7i=0

d

Bloc

k si

ze s

tret

ch

Additional space

32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 620.01

0.1

1

d

repa

ir-d

own

redu

ction

Reduced communications bandwidth

classical erasure codes

MBR: Minimum-Bandwidth Regenerating MSR: Minimum-Storage Regenerating

16

Regenerating codes: Performance - k=32 and h=32 and a stored file of 1MB:

32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 620.8

1

1.2

1.4

1.6

1.8

2

i=31i=22i=15i=7i=0

d

Bloc

k s

ize

stre

tch

32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 620.01

0.1

1

d

repa

ir-d

own

redu

ction

Reduced communications bandwidth

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

Communication is impressively reduced with small amount of extra storage.

Additional space

17

Regenerating codes: A new dimension in the trade-off

Storage

Communication

Erasure Codes

Replication

Regenerating codes can be seen as a generalization of replication and RSE that allow to more flexibly trade off communication and storage requirements.

RC(k,h,d,i,)•k original pieces•h additional pieces•d repair degree•i block expansion factor

18

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

Hierarchical Codes

Regenerating Codes

ER-Hierarchical Codes

20

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

21

ER-Hierarchical Codes: Construction

• How to transform Hierarchical code into ER-Hierarchical Code?

22

ER-Hierarchical Codes: Construction

23

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

24

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

25

ER-Hierarchical Codes: Repair Strategies

26

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

27

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

28

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

-

Thanks Questions?