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Correcting Localized Deletions Using Guess & Check Codeseceweb1.rutgers.edu/~csi/Allerton_GC.pdf ·...
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Correcting Localized Deletions Using Guess & Check Codes Salim El Rouayheb Joint work with Serge Kas Hanna and Hieu Nguyen Rutgers University 55th Annual Allerton Conference on Communication, Control, and Computing
Transcript of Correcting Localized Deletions Using Guess & Check Codeseceweb1.rutgers.edu/~csi/Allerton_GC.pdf ·...
CorrectingLocalized DeletionsUsingGuess&CheckCodes
SalimElRouayheb
JointworkwithSergeKas HannaandHieu Nguyen
RutgersUniversity
55thAnnualAllertonConferenceonCommunication, Control,andComputing
Motivation
2
• Ourmotivation:filesynchronization,E.g.Dropbox
Alice Bob
10101010…
• Recentapplication:DNA-basedstorage
• DeletionswerefirststudiedbyVarshamov-Tenengolts (‘65)andLevenshtein (‘66)
110010…
• Deletions:10101010 110010Transmitted Received
LocalizedDeletions
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• Motivation:filesynchronization,E.g.Dropbox
Localizededits
PreviousWorkonDeletions
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Ø Unrestricteddeletions
Ø Bursty deletions
• Filesynchronization:[Maetal.‘11]
• Codeconstructions: [Levenshtein ‘67],[Chengetal.14],[Schoeny etal‘17]
Existenceofcodesforlocalizedmodelw=3,4
• Informationtheoreticapproach:[Gallager ’61],[Dobrushin ‘67];lowerandupperboundsonthecapacity:[Mitzenmacher andDrinea ‘06],[Diggavi etal.‘07],[Kanoria andMontanari ‘13],[Venkataramanan etal.’13]…
• Codeconstructionsandfundamentallimits:[Varshamov andTenenglots‘65],[Levenshtein ‘66],[SchulmanandZuckerman ‘99],[Helberg andFerreira ‘02],[Cullina andKiyavash ‘14],[Gabrys etal.‘16],[Brankensieketal.’16],[Thomas etal.’17]…
• Recentfilesynchronizationalgorithms:[Yazdi andDolecek ‘14],[Venkataramanan etal.‘15],[Salaetal.’17] …
ModelandContribution
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! = 101010111000100100010110101110000 01010001100111111000010
Ø # ≤ % deletions localizedinawindowofsize%windowofsize%
# deletions(inred)
Ø Ourassumptions:1)positionsofthedeletions areindependent ofthecodeword;2)informationmessage isuniform iid
Ø Contribution:Explicitcodeswithdeterministicpolynomialtimeencodinganddecodingthatcancorrectlocalizeddeletionswhp
• Logarithmicredundancy:& − ( = ) log ( +% + 1
• Polynomialtimeencodinganddecoding• Asymptoticallyvanishingprobabilityofdecoding failure• Canbegeneralized tomultiplewindows
Guess&Check(GC)Codes
Ø Hardproblemfor% = &
Ø [Schoeny etal.’17]:existenceofcodesfor% = 3,4
GCCodesExample
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• Encodingthemessageoflengthk=16: 1110000000110001
1110000000110001GF(17)
14 0 3 1 (6,4)MDS 14 0 3 1 1 0
Ø Guess1:
• Decoding
111000000000112 0 1?
5 12 0 1
2MDSparities
Ø Guess2: 14 3 0 1
Ø Guess3: 14 0 3 1
Ø Guess4: 14 0 0 4
1110 000000001
1110000000001
1110000000001
0
Decodedusing1st parity
Checkwith2nd parity
1110000000001
• Assumethatthedeletions (inred)affectonlyonesystematicblock
16bits 13bits
111000000011 0001
GF(17)
[Kas HannaandElRouayhebISIT17’]
Deletionsoccurinoneofthesewindows
GeneralizingtoAnyWindowPosition
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Ø Guess1:
• Decoding,windowofsizelogkcanaffectatmost2consecutive blocks
11000001100013 1?
14 4 3 1
Ø Guess2: 12 7 2 1
Ø Guess3: 12 1 11 15
1100000110001
1100000110001
12
Decodedusing1st &2nd parity
Checkwith3rd parity
• Sameencodingwithoneextraparity
1110010000110001GF(17)
(7,4)MDS 14 4 3 1 5 83MDSparities
12logkbits
1110010000110001 1100000110001
• Assumethat3deletions (inred)affect systematicbits,w=logk=4bits
16bits 13bits
window
RecoveringtheMDSparities
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• Buffer:wzeros +asingleone
Ø Howtorecover theMDSparitysymbolsatthedecoder?
Ø Trivialsolution:repeat theparitybits
systematicbits
(# + 1) repetitioncode
Ø Better solution:insertabuffer betweensystematicandMDSparitybits
• Ifparitybitsgetaffected ->simplyoutputthefirst( bits.ElseapplyGuess&Checkdecoding
• Deletionscannotaffect bothsystematicandparitybitssimultaneously
111000000011000100010000MDSparities
systematicbits
11100000001100010000000000001111 0… 0
systematicbits
111000000011000100001 00010000MDSparitiesbuffer
w+1bits
Whendoesdecodingfail?
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• Encodingthemessageoflengthk=16: 1100010001110010
1100010001110010GF(17)
14 4 7 2 (6,4)MDS 14 4 7 2 8 13
Ø Guess1:
• Decoding
00100011100104 7 2?
14 4 7 2
2MDSparities
Ø Guess2: 2 14 7 2
Ø Guess3: 2 3 1 2
Ø Guess4: 2 3 9 11
0010001110010
001000111 0010
0010001110010
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Decodedusing1st parity
Checkwith2nd parity
logkbits
0010001110010
• Assumethatthedeletions (inred)affectonlyonesystematicblock
16bits 13bits
1100 010001110010
GF(17)
DecodingFailure
Simulations– DecodingFailure
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( (messagelengthinbits)
Prob
abilityofFailure
4 = 106 iterations
• Simulationresults for:% = log ( , # = log ( − 1 , c = 3,4MDSparities
MainResults
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Sketch of Theorem 2 (8 > : windows):
Ø Encoding complexity is ;(( log (), Decoding complexity is ;((<=>)
Ø Probability of decoding failure: Pr A ≤ (<(B=C)DE
Ø Redundancy: )(F% + 1) log (
Theorem 1 (One window): Guess & Check (GC) codes can correct inpolynomial time up to # ≤ % = G(log () localized deletions, whereH log ( < % < H + 1 log ( for some integer H ≥ 0. Let ) > H+ 2 be aconstant integer.
Ø Encoding complexity is ;(( log (), Decoding complexity is ;((L/ log ()Ø Probability of decoding failure: Pr A ≤ (B=CDE/ log (
Ø Redundancy: ) log ( + %+ 1
Sketch of Theorem 3 [ISIT ‘17] (Unrestricted deletions):Ø Redundancy: )(# + 1) log (
Ø Encoding complexity is ;(( log (), Decoding complexity is ;((N=>/ logN ()Ø Probability of decoding failure: Pr A = ;((>NDE/ logN ()
TestGCCodesOnline
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Ø C++&PythoncodesareavailableonGitHub
GitHub repository:https://github.com/serge-k-hanna/GC
Ø TestthecodesonlineusingtheJupyter notebook
Goto:https://try.jupyter.org/
Upload thenotebook filesfromhttp://eceweb1.rutgers.edu/csi/software.html
Ø Formoredetails:http://eceweb1.rutgers.edu/csi/software.html
Simulations– RunningTime
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C++:Earlyterminationwithprecomputing
Python:Allcaseswithoutprecomputing
Python:Allcaseswithprecomputing
Python:Earlyterminationwithprecomputing
DecodingFailures:WhatHappened.
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• Exampleforonedeletion:16-bitmessage0000100011110110
Ø (6,4)MDSencodingoverGF(17): 0 8 15 6 12 52parities
Ø Suppose14th bitgetsdeleted,decoding:v Guess1: 8 4 7 10
v Guess4: 0 8 15 6
Guesses1&4satisfythe2parities
• Probabilityofdecoding failureforagivenstring:combinatorialproblemthatdependsonthestringanddeletionposition
v Guess2:
v Guess3:8 4 7 10
8 4 7 10
• Decodingfailure:morethanonepossibleguess,different decodedstrings
• Proofapproach:assumemessage isuniformiid,averageoverallpossiblemessages
DecodingFailure– 1Deletion15
Setofalltransmittedk-bitstrings
B B
A
SetofallGCdecoderoutputs
Setsatisfyingall) parities
AssumeWLOGthatGuess 1iscorrect,observetheoutputofdecoderatwrong GuessO ≠ 1
DecodingfailsifdecodedstringisinA
Lemma:atmost2differenttransmitted sequences canleadtothesamedecodedstring inanyGuessO ≠ 1
Setsatisfyingfirstparity
QR decodingfailureinguessO ≠ 1 = QR(decodedstringisin])
···
·
··
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Fixed:- Guess- Deletion- Firstparity
ProofofPr(F)forOneDeletion
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B B
A
k : length of message
Yi : string decoded in Guess i
c : number of parities
A : set satisfying all c parities
B : set satisfying first parity
Unionbound
Lemma
Subspacecardinality
q : field size
Claim
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Ø Claim1(onedeletion):atmost2different transmittedstringscanleadtothesamedecodedstringinanywrongguess
Setofalltransmittedk-bitstrings
B B
SetofallGCdecoderoutputs
Setsatisfyingfirstparity
·
···
Fixed:- Guess- Deletionposition- Firstparity
Ø Claim2(# deletions): aconstant numberofdifferent transmittedstringscanleadtothesamedecodedstringinanywrongguess
Claim- Example
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^: = 0000000000000000
^_ = 0010000000000010
000000000000000
000000000000010
Ø Claim1(onedeletion):atmost2different transmittedstringscanleadtothesamedecodedstringinanywrongguess
0000000000000000
Ø Example
Ø 3rd bitdeleted;Guess:deletionoccurred in4th block
0 0 0 0 0parity
` 0 0 ` 0
EncodinginGF(16)
Ø ^L = 1111111111111111 and^a = 0010000000000000
Ø Twoconditions:(1)Symmetryconstraint;(2)Algebraic linearconstraint
Decodeb:
b_Received
Claim
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Ø Suppose3rd bitisdeleted, guess:deletionoccurred in4th block
8b1 + 4b2 + 2b4 + b5
b1b2b4b5 b6b7b8b9 b10b11b12b13 b14b15b16
8b6 + 4b7 + 2b8 + b9 8b10 + 4b11 + 2b12 + b13 ?
Symbol1 Symbol2 Symbol3 Erasure
Ø Howmanydifferent messagescanleadtosamedecodedstring?
GF(17)
Ø Symmetryconstraint: samedecodedstring⇒ samebitvaluesatpositionsofsymbols1,2and3
Ø Bitswhichcanbedifferent: and(deleted bit)b14, b15, b16 b3
Ø Algebraicconstraint:erasure isdecodedusingfirstparity
4b14 + 2(b3 + b15) + b16 = p1
b14, b15, b16, b3 2 GF (2)
p1 2 GF (17)
Equationhasatmost2solutions
ApplicationtoFileSynchronization
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• Interactive synchronizationalgorithmby[Venkataramanan etal.’15]Ø Isolatesingledeletions,useVTcodesØ Modification:isolate# orfewer deletions,useGCcodes
• Gain:(1)lesscommunicationrounds,(2)lowercommunicationcost
Upto75%improvement
Upto15%improvement
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N=1000iterations
Summary
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• Guess&CheckCodes forlocalizeddeletions
Ø Explicitcodeconstruction withlogarithmicredundancy
Ø Deterministicpolynomialtimeencodinganddecoding
Ø Asymptoticallyvanishingprobabilityofdecoding failure
Ø Forsingleormultiplewindows
• Openproblems• Capacityofdeletionchannelwithlocalizeddeletions?• Codes foradversarial localizeddeletions• Andofcourse for“unrestricted” deletioncapacityandcodesarestill
openproblems
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