Correcting Localized Deletions Using Guess & Check Codes

Transcription

1 55th Annual Allerton Conference on Communication, Control, and Computing Correcting Localized Deletions Using Guess & Check Codes Salim El Rouayheb Rutgers University Joint work with Serge Kas Hanna and Hieu Nguyen

2 Motivation Transmitted Received Deletions: Deletions were first studied by Varshamov-Tenengolts ( 65) and Levenshtein ( 66) Our motivation: file synchronization, E.g. Dropbox Alice Bob 0000 Recent application: DNA-based storage 000 2

3 Localized Deletions Motivation: file synchronization, E.g. Dropbox Localized edits 3

4 Previous Work on Deletions Ø Unrestricted deletions Information theoretic approach: [Gallager 6], [Dobrushin 67]; lower and upper bounds on the capacity: [Mitzenmacher and Drinea 06], [Diggavi et al. 07], [Kanoria and Montanari 3], [Venkataramanan et al. 3] Recent file synchronization algorithms: [Yazdi and Dolecek 4], [Venkataramanan et al. 5], [Sala et al. 7] Code constructions and fundamental limits: [Varshamov and Tenenglots 65], [Levenshtein 66], [Schulman and Zuckerman 99], [Helberg and Ferreira 02], [Cullina and Kiyavash 4], [Gabrys et al. 6], [Brankensiek et al. 6], [Thomas et al. 7] Ø Bursty deletions File synchronization: [Ma et al. ] Existence of codes for localized model w=3,4 Code constructions: [Levenshtein 67], [Cheng et al. 4], [Schoeny et al 7] 4

5 Model and Contribution! = Ø # % deletions localized in a window of size % Ø Hard problem for % = & # deletions (in red) window of size % Ø [Schoeny et al. 7]: existence of codes for % = 3,4 Ø Our assumptions: ) positions of the deletions are independent of the codeword; 2) information message is uniform iid Ø Contribution: Explicit codes with deterministic polynomial time encoding and decoding that can correct localized deletions whp Guess & Check (GC) Codes Logarithmic redundancy: & ( = ) log ( + % + Polynomial time encoding and decoding Asymptotically vanishing probability of decoding failure Can be generalized to multiple windows 5

6 GC Codes Example Deletions occur in one of these windows Encoding the message of length k=6: GF(7) (6,4) MDS 2 MDS parities GF(7) Assume that the deletions (in red) affect only one systematic block Decoding? Ø Guess : 6 bits bits Decoded using st parity Ø Guess 3: Ø Guess 4: Ø Guess 2: 0 0 Check with 2nd parity 0 [Kas Hanna and El Rouayheb ISIT 7 ] 6

7 Generalizing to Any Window Position Assume that 3 deletions (in red) affect systematic bits, w=log k=4 bits window bits 6 bits Same encoding with one extra parity log k bits (7,4) MDS MDS parities GF(7) Decoding, window of size log k can affect at most 2 consecutive blocks Ø Guess :? 000 Ø Guess 2: Ø Guess 3: Decoded using st & 2nd parity Check with 3rd parity

8 Recovering the MDS parities Ø How to recover the MDS parity symbols at the decoder? systematic bits Ø Trivial solution: repeat the parity bits systematic bits Buffer: w zeros + a single one MDS parities (# + ) repetition code Ø Better solution: insert a buffer between systematic and MDS parity bits systematic bits buffer w+ bits MDS parities Deletions cannot affect both systematic and parity bits simultaneously If parity bits get affected -> simply output the first ( bits. Else apply Guess & Check decoding 8

9 When does decoding fail? Encoding the message of length k=6: log k bits GF(7) (6,4) MDS 2 MDS parities GF(7) Assume that the deletions (in red) affect only one systematic block bits 6 bits Decoding? Ø Guess : Decoding Ø Guess 2: 000 Failure Decoded using st parity Ø Guess 3: Ø Guess 4: Check with 2nd parity 3 9

10 Simulations Decoding Failure Simulation results for: % = log (, # = log (, c = 3,4 MDS parities Probability of Failure 4 = 0 6 iterations ( (message length in bits) 0

11 Main Results Theorem (One window): Guess & Check (GC) codes can correct in polynomial time up to # % = G(log () localized deletions, where H log ( < % < H + log ( for some integer H 0. Let ) > H + 2 be a constant integer. Ø Redundancy: ) log ( + % + Ø Encoding complexity is ;(( log (), Decoding complexity is ;(( L / log () Ø Probability of decoding failure: Pr A ( B=CDE / log ( Sketch of Theorem 2 (8 > : windows): Ø Redundancy: )(F% + ) log ( Ø Encoding complexity is ;(( log (), Decoding complexity is ;(( <=> ) Ø Probability of decoding failure: Pr A ( <(B=C)DE Sketch of Theorem 3 [ISIT 7] (Unrestricted deletions): Ø Redundancy: )(# + ) log ( Ø Encoding complexity is ;(( log (), Decoding complexity is ;(( N=> / log N () Ø Probability of decoding failure: Pr A = ;(( >NDE / log N ()

12 Test GC Codes Online Ø Test the codes online using the Jupyter notebook Go to : Upload the notebook files from Ø C++ & Python codes are available on GitHub GitHub repository: Ø For more details: 2

13 Simulations Running Time Python: All cases without precomputing Python: All cases with precomputing C++ : Early termination with precomputing Python: Early termination with precomputing 3

14 Decoding Failures: What Happened. Decoding failure: more than one possible guess, different decoded strings Example for one deletion: 6-bit message Ø (6,4) MDS encoding over GF(7): 0 2 parities Ø Suppose 4th bit gets deleted, decoding: v Guess : v Guess 2: v Guess 3: v Guess 4: Guesses & 4 satisfy the 2 parities Probability of decoding failure for a given string: combinatorial problem that depends on the string and deletion position Proof approach: assume message is uniform iid, average over all possible messages 3

15 Decoding Failure Deletion Assume WLOG that Guess is correct, observe the output of decoder at wrong Guess O Set of all transmitted k-bit strings Set of all GC decoder outputs B Fixed: - Guess - Deletion - First parity Decoding fails if decoded string is in A B A Set satisfying first parity Set satisfying all ) parities QR decoding failure in guess O = QR (decoded string is in ]) Lemma: at most 2 different transmitted sequences can lead to the same decoded string in any Guess O 4

16 Proof of Pr(F) for One Deletion Union bound k : length of message Y i :stringdecodedinguessi c : number of parities A : set satisfying all c parities B : set satisfying first parity q :fieldsize Lemma B B A Subspace cardinality 5

17 Claim Ø Claim (one deletion): at most 2 different transmitted strings can lead to the same decoded string in any wrong guess Set of all transmitted k-bit strings Set of all GC decoder outputs B B Set satisfying first parity Fixed: - Guess - Deletion position - First parity Ø Claim 2 (# deletions): a constant number of different transmitted strings can lead to the same decoded string in any wrong guess 7

18 Claim - Example Ø Claim (one deletion): at most 2 different transmitted strings can lead to the same decoded string in any wrong guess Ø Example ^: = ^_ = Encoding in GF(6) parity ` 0 0 ` 0 Ø 3 rd bit deleted; Guess: deletion occurred in 4 th block Received b : b _ Decode Ø ^L = and ^a = Ø Two conditions: () Symmetry constraint; (2) Algebraic linear constraint 7

19 Claim Ø Suppose 3 rd bit is deleted, guess : deletion occurred in 4 th block b b 2 b 4 b 5 b 6 b 7 b 8 b 9 b 0 b b 2 b 3 b 4 b 5 b 6 GF(7) 8b +4b 2 +2b 4 + b 5 8b 6 +4b 7 +2b 8 + b 9 8b 0 +4b +2b 2 + b 3? Symbol Symbol 2 Symbol 3 Erasure Ø How many different messages can lead to same decoded string? Ø Symmetry constraint: same decoded string same bit values at positions of symbols, 2 and 3 Ø Bits which can be different: b 4,b 5,b 6 and b 3 (deleted bit) Ø Algebraic constraint: erasure is decoded using first parity 4b 4 + 2(b 3 + b 5 )+b 6 = p b 4,b 5,b 6,b 3 2 GF (2) p 2 GF (7) Equation has at most 2 solutions 8

20 Application to File Synchronization Interactive synchronization algorithm by [Venkataramanan et al. 5] Ø Isolate single deletions, use VT codes Ø Modification: isolate # or fewer deletions, use GC codes Gain: () less communication rounds, (2) lower communication cost Up to 75% improvement N=000 iterations Up to 5% improvement 20 9

21 Summary Guess & Check Codes for localized deletions Ø For single or multiple windows Ø Explicit code construction with logarithmic redundancy Ø Deterministic polynomial time encoding and decoding Ø Asymptotically vanishing probability of decoding failure Open problems Capacity of deletion channel with localized deletions? Codes for adversarial localized deletions And of course for unrestricted deletion capacity and codes are still open problems 6

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