Shifted Codes Sachin Agarwal Deutsch Telekom A.G., Laboratories Ernst-Reuter-Platz 7 10587 Berlin Germany Joint work with Andrew Hagedorn and Ari Trachtenberg

Shifted CodesSachin AgarwalDeutsch Telekom A.G., LaboratoriesErnst-Reuter-Platz 7 10587 BerlinGermany

Joint work with Andrew Hagedorn and Ari Trachtenberg at Boston University

Best viewed on-screen in slide-show mode

2S. Agarwal, [email protected], January 2008

Outline

1. Motivation & Problem Definition

2. Backgrounda. Rateless Codes

b. Digital Fountain Codes

3. Shifted Codesa. Motivation – Inefficiency of LT codes

b. Construction of Shifted Codes

c. Analysis – Communication and Computation Complexity

4. Experimental Comparisona. LT vs. Shifted Codes

b. Constraint Sensors – Deployment on TMotes

5. Discussion and Round-up


Outline











Partial Information

Transmission Channel with Erasures

Transmitter Receiver

Input symbols Received Symbols


Partial Information





Partial Information





Partial Information





Partial Information





Partial Information





Partial Information

Multiple Receivers may have different erasures

Transmitter

Receiver 1

Receiver 2

Receiver 3

Given the situation of multiple receivers having partial information, how can all of

them be updated to full information efficiently, and over a broadcast channel?


Partial InformationAnother Example

Multiple mobile devices may have out-dated information

a. Mobile databases

b. Sensor network information aggregation

c. RSS updates for devices

Broadcaster

Mobile device 1

Mobile device 2

Mobile device 3

Latest version of information


Problem Definition

Given an encoding host with k input symbols and a decoding host with n out of the k input symbols, the goal is to efficiently determine the remaining k-n input symbols at the decoding host.

The encoding host has no information of which k-n input symbols are missing at the decoding host.

Different decoding hosts may be missing different input symbols

Efficiency1.Communication complexity – Information transmitted from the encoding host to the decoding host should be close in size to the transmission size of the missing k-n input symbols

2.Computational complexity – The algorithm must be computationally tractable


Information Theoretic Lower Bound

Known ResultAt a minimum, the encoding host would have to send only a little less than the exact contents of the missing input symbols to the decoding host.

Intuition

Decoding host is missing k-n input symbols

Special case of set reconciliation

b

nkbnkC

)lg()(

k – Number of input symbols

n – Number of symbols known a priori at the decoding host

b – Field size of each symbol


Outline











Rateless Codes

Definition“A class of erasure codes with the property that a potentially limitless sequence of encoding symbols can be generated from a given set of source symbols such that the original source symbols can be recovered from any subset of the encoding symbols of size equal to or only slightly larger than the number of source symbols. ”

Wikipedia.org

Examples1. Random Linear Codes

2. LT Codes

3. Raptor Codes

4. Shifted Codes

5. …


Rateless Codes - EncodingUsed for content distribution over error-prone channels

Random choice of edges based on a probability density function

At least k Encoded Symbolsk input symbols

1 =A+B

2 =B

3 =A+B+C

4 =A+C

A

B

C


Rateless Codes - DecodingUsed for content distribution over error-prone channels

At least k Encoded Symbols

1 =A+B

2 =B

3 =A+B+C

4 =A+C

k input symbols

SolveGaussian Elimination, Belief Propagation

System of Linear Equations

Irrespective of which encoded symbols are lost in the communication channel, as long as sufficient encoded symbols are received, the decoding can retrieve all the k input symbols

A

B

C


Decoding Using Belief Propagation

Decoded k Input Symbols

k+ Encoded Symbols

Decoding host

Redundant!

Decode

Input Symbols


Digital Fountain CodesLT Codes1.Class of rateless erasure codes

invented by Michael Luby1

2.Computationally practical (as compared to Random Linear Codes)

3.Fast decoding algorithm based on Belief propagation instead of Gaussian Elimination

4.Form the outer code for Raptor Codes3, which have linear decoding computational complexity

5.Designed for the case when no input symbols are available at the Decoding host initially

Asymptotic Properties2

Expected number of encoded symbols required for successful decoding

Expected decoding computational complexity

k: number of input symbols2Assuming a constant probability of failure

)ln( 2 kkOk

1Michael Luby, “LT codes,” in The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 271–282.3Amin Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52, no. 6, 2006, pp. 2551–2567.

)ln( kkO


Digital Fountain CodesLT Codes’ Robust Soliton Probability Distribution

Robust Soliton Probability Distribution k,

Probability of an encoded symbol with degree d is k(d)

Property of releasing degree 1 symbols at a controlled, near-constant rate throughout the decoding process

0 200 400 600 800 1000-6

-5

-4

-3

-2

-1

0

Degree

log 1

0(P

roba

bilit

y)

LT Code (Robust Soliton)

LT code distribution, with parameters k = 1000, c = 0.01, = 0.5.


Outline











Inefficiency of LT Codes for our Problem

k+ Encoded Symbols

Decoding host

Decode

Input Symbols

n out of k input symbols are known a priori at the decoding host

Many redundant encoded symbols


Inefficiency of LT Codes for our ProblemThe number of these redundant encoded symbols grows with the ratio of input symbols known at the decoder (n) to the total input symbols (k)

If n input symbols are known a priori, then an additional LT-encoded symbol will provide no new information to the decoding host with probability

…which quickly approaches 1 as n → k

d

i

k

dk ik

ind

01

)(


Intuitive Fix

n known input symbols serve the function of degree 1 encoded symbols, disproportionately skewing the degree distribution for LT encoding

We thus propose to shift the Robust Soliton distribution to the right in order to compensate for the additional functionally degree 1 symbols

Questions

1) How?

2) By how much?

0 200 400 600 800 1000-6

-5

-4

-3

-2

-1

0

Degree

log 1

0(P

roba

bilit

y)

LT Code (Robust Soliton)


Shifted Code Construction

Definition

The shifted robust soliton distribution is given by

Intuition

n known input symbols at the decoding host reduce the degree of each encoding symbols by an expected fraction

j

kn

iij nknk

1roundfor )(0)(,

kn

1

1


Shifted Code Distribution

0 200 400 600 800 1000-6

-5

-4

-3

-2

-1

0

Degree

log 1

0(P

roba

bilit

y)

LT Code (Robust Soliton)Shifted Code

LT code distribution and proposed Shifted code distribution, with parameters k = 1000, c = 0.01, = 0.5. The number of known input symbols at the decoding host is set to n = 900 for the Shifted code distribution. The probabilities of the occurrence of encoded symbols of some degrees is 0 with the shifted code distribution.


Shifted Code – Communication ComplexityLemma IV.2 A decoder that knows n of k input symbols needs

encoding symbols under the shifted distribution to decode all k input symbols with probability at least 1−.

ProofWe have k-n input symbols comprising the encoded symbols after the n known input symbols are removed from the decoding graph. The expresson follows from Luby‘s analysis.

nknkOnkm 2ln)(


Shifted Code – Average Degree of Encoded Symbol

Lemma IV.3 The average degree of an encoding node under the k,n distribution is given by

ProofThe proof follows from the definitions, since a node with degree d in the μk distribution will correspond to a node with degree roughly

in the shifted code distribution.From Luby‘s analysis,the expresson for the average degree of an LT encoded symbol is

)ln( nk

nk

kO

kn

d

1

)(ln kO


Shifted Codes – Computational ComplexityLemma IV.4*

For a fixed , the expected number of edges R removed from the decoding graph upon knowledge of n input symbols at the decoding host is given by

R = O (n ln(k − n))Theorem IV.5

For a fixed probability of decoding failure , the number of operations needed to decode using a shifted code is

O (k ln(k − n)) Proof

Summing Lemma IV.4 and the computational complexity of (LT) decoding for the unknown k-n input symbols

*Proof described in: S. Agarwal, A. Hagedorn and A. Trachtenberg, “Rateless Codes Under Partial Information”, Information Theory and Applications Workshop, UCSD, San Diego, 2008


Outline











0 100 200 300 400 500 600 700 800 900 10000

200

400

600

800

1000

1200

Known input symbols (n)

Req

uire

d en

codi

ng s

ymbo

ls a

t D

ecod

ing

host

Without Invention

With Invention

Benefit

For k = 1000, n = 900, the decoding host needs to download about 700 encoded symbols using conventional LT codes. But using shifted codes, only about 180 encoded symbols are required

Experimental ComparisonLT Codes vs. Shifted Codes

The experiment was repeated 100 times and the error-bars of the standard deviation are also plotted in the graph.

LTShifted Code

Y-axis

Number of encoded symbols required at the mobile device to obtain the whole data-set

X-axis

Number of input symbols n available a priori at the mobile device


Experimental ComparisonConstraint Sensors – Deployment on TMotes

100 200 300 4000

0.5

1

1.5

2

2.5

3

Number of Input Symbols

Tim

e to

Enc

ode

(s)

LT (Robust Soliton)Shifted Code distribution

Total time to Encode(Measure of computational complexity)

100 200 300 4000

2

4

6

8

10

12

Number Input Symbols

Tim

e T

o D

ecod

e (s

)

LT (Robust Soliton)Shifted Code distribution

Total time to Decode(Measure of computational complexity)


More Data: Communication Savings

-200 0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

n, number of known input symbols at decoding host

Req

uire

d en

code

d sy

mbo

ls f

or s

ucce

ssfu

l \ d

ecod

ing

k=1000 input symbols, 20 randomized trials

LT Robust Soliton

Shifted Code


More Data: Communication Savings Normalized

100 200 300 400 500 600 700 800 9000

0.2

0.4

0.6

0.8

1


Enc

oded

sym

bols

req

uire

d, n

orm

aliz

ed w

ith L

T-R

S


LT Robust Soliton

Shifted Code


More Data: Time Savings, Normalized

100 200 300 400 500 600 700 800 9000

0.2

0.4

0.6

0.8

1


Tim

e ta

ken

to d

ecod

e, n

orm

aliz

ed w

ith L

T-R

S


LT Robust Soliton

Shifted Code


Distribution ShiftingWhen the estimate of n at the Encoding Host is not accurate

1

0

)()(0)(k

kp ipj j

k

i

1roundfor

The Theta distribution shifting decodes input symbols much more quickly than the standard LT codes.


Outline











Many Applications

1. Broadcasting coded updates to synchronize databases2. Adapting LT codes when partial information has been

delivereda. Continuous shifting of the distributionb. Using the partial information in case of unsuccessful decoding

(when only some of the input symbols were decoded)

3. Efficient erasure correction when channel characteristics are already known

a. For example, input symbols can be first sent as plain-text, and then depending on the estimate of number of lost input symbols, shifted-coded symbols can be transmitted

4. Heterogeneous channel data delivery5. Application in gossip protocols, particularly in later

iterations 6. Sensor networks - data aggregation, routing

information, etc.7. Restoring storage media that are partially erased…


Conclusions & Future-work

Conclusions

a. Generalization of LT Code when some of the input symbols are already available at the decoding host

b. Many applications

Future Work

a. By adopting Raptor Code concepts (inner code), Shifted codes can be made more efficient

b. Analytical expressions for Distribution Shifting

c. Application specific shifted codes design

d. “Shifting” other rateless codes


Further Reading

1. S. Agarwal, A. Hagedorn and A. Trachtenberg, “Rateless Codes Under Partial Information”, Information Theory and Applications Workshop, UCSD, San Diego, 2008

2. S. Agarwal (Deutsche Telekom A.G.), “Method and System for Constructing and Decoding Rateless Codes with Partial Information”, European Patent Application EP 07 023 243.4

3. Michael Luby, “LT codes,” in The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 271–282.

4. Amin Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52, no. 6, 2006, pp. 2551–2567.

Documents

Shifted Codes Sachin Agarwal Deutsch Telekom A.G., Laboratories Ernst-Reuter-Platz 7 10587 Berlin Germany Joint work with Andrew Hagedorn and Ari Trachtenberg