Mario Vodisek 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Erasure...

Mario Vodisek 1

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

Erasure Codes for Reading and Writing

Mario Vodisek ( joint work with AG Schindelhauer)

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Algorithms and ComplexityAgenda

• Erasure (Resilient) Codes in storage networks

• The Read-Write-Coding-System

- A Lower Bound and Perfect Codes- Requirements and Techniques

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• 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-rc erasures 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

Erasure (Resilient) Coding

coding

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Algorithms and ComplexityErasure 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

Network

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Algorithms and ComplexityThe 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, …, xn b

• m n code blocks y1, …, ym b

• any n r m code words sufficient to read the information

• any n w m code 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: xi yi }|

coding

m, r, w

n

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Algorithms and ComplexityA Lower Bound for RW-Codes

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

system !

• We know: n r,w m• Assume: r = w = n m n+1

• Write and subsequent read

n

m

Proof: w

rIndex Sets (W, R):• |W| = w• |R| = r

|S| = W R {n, n-1}

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

´read´ will produce faulty output

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Algorithms and ComplexityCodes 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 !

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Algorithms and ComplexityThe Model: Operations

Given:

• X=x1,…, xn the n-symbol information vector over a finite alphabet .

• Y=y1,…, yn the m-symbol code over

• b=||.

• P(M) : the power set of M, Pk(M):={S 2 P(M): |S|=k}

• Define [m]:={1,…,m}

An (n, r, w, m)b-RWC-system consists of the following operations:

• Inital state: X0 2 n, Y0 2 m

• Read function: f: Pr([m]) £ r ! m

• Write function: g: Pr([m]) £ r £ Pw([m]) £ n ! w

• Differential write function: : Pw([m]) £ n ! w

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Algorithms and ComplexityInitialization: Compute the Encoding Y0

Given (in general):

• the information vector X = x1, …, xn b

• the encoded vector Y = y1, …, yn b

• 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=1

n xj Mi,j + l=1k vl Mi,l

( Define M as some m £ r matrix; Mi as rows. It follows: M(XV = Y )

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

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Algorithms and ComplexityThe Matrix Approach: (n, r, w, m)b-RWC

Consider:

• the information vector X = x1, …, xn b

• the encoded vector Y = y1, …, ym b

• internal slack variables V = v1, …, vk for k = m-w = r-n

Further:

• an m r generator matrix M: Mi,j b

• the submatrix (Mi,j)i [m], j {n+1, …, r} is called the variable matrix

=

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Algorithms and ComplexityEfficient 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

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Algorithms and ComplexityThe Vandermonde Matrix

Consider M as m £ r Vandermonde matrix Mi,j = ji-1:

• X, Y, V 2 F[b]

• Mi,j 2 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

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Algorithms and ComplexityReading (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‘‘)-1YM (X | V)Y

r

m

M‘Y‘

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Algorithms and Complexity 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)

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Algorithms and ComplexityDifferential Write (con‘t)

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}

MÃ" M"!

MÃ# M#!

w

n

w+1…m

n+1…r

• M#! is k £ k = m-w £ r-n matrix ) M#! invertible

• The vector Y can then be updated by a vector = ,…,w:

= ((MÃ") – (M"!)(M#!)-1(MÃ#)) ¢

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Algorithms and ComplexityDifferential Write: Proof

Use:

• Vector = 1,…,k the change of vector V

• Vector = 1,…,w the change of vector Y

MÃ" M"!

MÃ# M#!

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#) ¢

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Heinz Nixdorf Institute& Computer Science InstituteUniversity of PaderbornFürstenallee 1133102 Paderborn, Germany

Tel.: +49 (0) 52 51/60 64 51Fax: +49 (0) 52 51/62 64 82E-Mail: [email protected]://www.upb.de/cs/ag-madh

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Mario Vodisek 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Erasure...

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