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Erasure Codes for Reading and Writing PowerPoint Presentation

Erasure Codes for Reading and Writing

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### Thank you for your attention!

Agenda

- Erasure (Resilient) Codes in storage networks
- The Read-Write-Coding-System
- A Lower Bound and Perfect Codes
- Requirements and Techniques

Erasure (Resilient) Coding

- n-symbol message x with symbols from alphabet
- m-symbol encoding y with symbols from (m > n)
- erasure coding provides mapping: n!m such that
- reading any n·r < m symbols of y are sufficient for recovery
- (mostly: r = n )optimal for reading)

- advantages:
- bm-rcerasures can be tolerated
- storage overhead is a factor of

- Generally, erasure codes are used to guarantee information recovery for data transmission over unreliable channels (RS-, Turbo-, LT-Codes, …)
- Lots of research in code properties such as
- scalability
- encoding/decoding speed-up
- rateless-ness

- Attractive also to storage networks: downloads (P2P) and fault-tolerance

coding

Erasure Codes for Storage (Area) Networks

- SANs require
- high system availability
- disks fail or be blocked (probability $ size)

- efficient modification handling
- Slow devices ) expensive I/O-operations

- Properties:
- a fixed set E of existing errors can be considered at encoding time
- E can have changed to E‘ at decoding time
- Additional requirements to erasure codes:
- tolerate some certain number of erasures
- ensure modification of codeword even if erasures occur
- consider E at encoding time and E‘ at decoding time

Network

The Read-Write-Coding-System

- An (n, r, w, m)b-Read-Write-Coding System (RWC) is defined as follows:
- The base b : b-symbol alphabet b as the set of all used items
- n 1 blocks of information x1, …, xnb
- mn code blocks y1, …, ym b
- any nrmcode words sufficient to read the information
- any nwmcode words sufficient to change the information by 1, …, n
- (In the language of Coding Theory) : given m, n, r, w, our RW-Codes provide:
- a (linear) code of dimension n and block length m such that for n·r,w·m:
- the minimum distance of the code is at least m-r+1
- any two codewords y1, y2 are within a distance of at most w from another
- distance(x, y):=|{1· i·m: xiyi }|

m, r, w

n

coding

A Lower Bound for RW-Codes

n

m

|S| = WR {n, n-1}

w

r

Theorem: For r+w<n+mand any base b there does not exist any (n, r, w, m)b-RWC

system !

Proof:

- We know: nr,wm
- Assume: r = w = nmn+1
- Write and subsequent read

Index Sets (W, R):

- |W| = w

- |R| = r

Assume: |S| = n

- there are bn possible change vectors to be encoded by `write` into S; only basis
for reading with r= n (notice: R\S code words remain unchanged)

Assume: |S| <n = n-1

at most bn-1 possible change vectors for S can be encoded by `write`

´read´ will produce faulty output

Codes at Lower Bound: Perfect Codes

- In the best case (n, r, w, m)b-RWC have parameters r + w = n + m (perfect Codes)

- Unfortunately, perfect RWC do not always exist !!
- E.g. there is no (1, 2, 2, 3)2-RWC but there exists a (1, 2, 2, 3)3-RWC !

- But: all perfect RW-Codes exist if the alphabet is sufficiently large !

- Notice to RAID:
- Definition of parity RAID (RAID 4/5) corresponds to an (n, n, n+1, n+1)2-RWC
- From the lower bounds it follows: there is no (n, n, n, n+1)2-RWC
- ) there is no RAID-system with improved access properties !

The Model: Operations

- Given:
- X=x1,…, xnthe n-symbol information vectorover a finite alphabet .
- Y=y1,…, yn the m-symbol code over
- b=||.
- P(M) : the power set of M, Pk(M):={S2 P(M): |S|=k}
- Define [m]:={1,…,m}
- An (n, r, w, m)b-RWC-system consists of the following operations:
- Inital state:X02n, Y02m
- Read function: f: Pr([m]) £r!m
- Write function: g: Pr([m]) £r£Pw([m]) £n!w
- Differential write function: : Pw([m]) £n!w

Initialization: Compute the Encoding Y0

- RW-Codes are closely related to Reed-Solomon-Codes !

Given (in general):

- the information vector X= x1, …, xnb
- the encoded vector Y= y1, …, ynb
- internal variables V = v1, …, vk for k =m-w = r-n, with no particular information
- set of functions M=M1,…,Mn for encoding
- Compute yi from X and V by function Mi ; define Mi as linear combination of X and V
- yi = Mi(x1,…,xn,v1,…,vk) = j=1nxjMi,j + l=1k vlMi,l
- ( Define M as some m£r matrix; Mi as rows. It follows: M(XV = Y )

The Matrix Approach: (n, r, w, m)b-RWC

Consider:

- the information vector X= x1, …, xnb
- the encoded vector Y= y1, …, ymb
- internal slack variables V = v1, …, vk for k =m-w = r-n
- Further:
- an mr generator matrix M: Mi,jb
- the submatrix (Mi,j)i [m], j{n+1, …, r} is called the variable matrix

=

Efficient Encoding: b = F[b] (Finite Fields)

- RWC requires efficient arithmetic on elements of b for encoding
- )set b = F[b] (finite field with b elements (formerly: GF(b)))

- b = pn for some prime number p and integer n) F[pn] always exists
- Computation of binary words of length v: b = 2v, F[2v] = {0,…,2v-1}
- Features:
- F[b] is closed under addition, multiplication
- )exact computation on field elements )not more than v bits for representiation of results

- Addition, subtraction via XOR (avoids rounding, no carryover)
- Multiplication, division via mapping tables (analogous to logarithm tables for real numbers)
- T : table mapping an integer to its logarithm in F[2v]
- IT: table mapping an integer to its inverse logarithm in F[2v]
- )multiplication, division by
- adding/subtracting the logs
- taking the inverse log

The Vandermonde Matrix

- Consider M as m£r Vandermonde matrix Mi,j = ji-1:
- X, Y, V 2 F[b]
- Mi,j2 F[b] and all elements are different
- The Vandermonde matrix is non-singular ) invertible
- Any k‘ £k‘ submatix M‘ is also invertible

=

Consider: each device i in the SAN corresponds to a row of M and element yi

Reading (or Recovery)

- Read: Given any r code entries from Y, compute X
- Rearrange rows of M and Y such that first r entries of Y are available
- (any r rows of M are linear independent in a Vandermonde matrix)

- M!M‘ and Y!Y‘
- The first r rows of M‘ describe an invertible r£r matrix M‘‘
- X is computed by: (X | V)T = (M‘‘)-1Y

(X | V)

M

Y

r

M‘

Y‘

m

Differential Write

- Given:
- The change vector =1,…,n and w code entries from Y
- X‘ = X +is new information vector ) change X without reading
- entries (XOR)
- Compute the difference for the w code entries of Y

- Further:
- Only choices w < r make sense
- Rearrange m£r matrix M and Y as follows: y1,…,yw (denote M‘ and Y‘)
- k = r-n (slack vector V)

Differential Write (con‘t)

n+1…r

n

- Define following sub-matrices:
- MÃ" = (M‘i,j)i2[w], j2[n]
- M"! = (M’i,j)i2[w], j2{n+1,…,r}
- MÃ# = (M’i,j)i2{w+1,…,m}, j2[n]
- M#! = (M’i,j)i2{w+1,…,m}, j2{n+1,…,r}

w

M"!

MÃ"

M#!

MÃ#

w+1…m

- M#!is k£k = m-w£r-n matrix) M#! invertible
- The vector Y can then be updated by a vector =1,…,w:
- = ((MÃ") – (M"!)(M#!)-1(MÃ#)) ¢

Differential Write: Proof

- Use:
- Vector = 1,…,kthe change of vector V
- Vector = 1,…,w the change of vector Y

- X’ =X+
- V’ =V+
- Y’ =Y+

Correctness follows by combining:

+

M

+ M

=

= M

= M

This equation is equivalent to:

(M#!) + (MÃ#) = 0,

(MÃ") + (M"!) =

Since is given, is obained as follows:

= (M#!)-1(-MÃ#) ¢

M"!

MÃ"

M#!

MÃ#

Heinz Nixdorf Institute

& Computer Science Institute

University of Paderborn

Fürstenallee 11

33102 Paderborn, Germany

Tel.: +49 (0) 52 51/60 64 51

Fax: +49 (0) 52 51/62 64 82

E-Mail: [email protected]

http://www.upb.de/cs/ag-madh

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