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Local Error-Detection and Error-correction
Madhu Sudan MIT December 17, 2007 Coding & Sublinear time
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Algorithmic Problems in Coding Theory
Code: Encoding: Error-detection ( Testing): Error-correction (Decoding): E : k ! n ; I m a g e ( ) = C R , o r l i z d s t c . F i x C o d e a n s c t E : k ! . G v m 2 , p u ( ) - G i v e n x 2 , d c f 9 m k s . t = E ( ) G i v e n x 2 , d c f 9 m k s . t ( E ) ; G i v e n x 2 , c o m p u t k h a z s ( E ) ; r d . December 17, 2007 Coding & Sublinear time
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Sublinear time algorithmics
Given can it be “computed” in time? Answer 1: Clearly NO, since that is the time it takes to even read the input/write the output f : ; 1 g k ! n o ( k ; n ) x - o r a c l e j f f ( x ) i w h e r f ( x ) i f ( x ) x i Answer 2: YES, if we are willing to Present input implicitly (by an oracle). Represent output implicitly Compute function on approximation to input. Extends to computing relations as well. December 17, 2007 Coding & Sublinear time
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Sub-linear time algorithms
Initiated in late eighties in context of Program checking Interactive Proofs/PCPs Now successful in many more contexts Property testing/Graph-theoretic algorithms Sorting/Searching Statistics/Entropy computations (High-dim.) Computational geometry Many initial results are coding-theoretic! December 17, 2007 Coding & Sublinear time
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Sub-linear time algorithms & Coding
Encoding: Not reasonable to expect in sub-linear time. Testing? Decoding? – Can be done in sublinear time. In fact many initial results do so! Codes that admit efficient … … testing: Locally Testable Codes (LTCs) … decoding: Locally Decodable Codes (LDCs). December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Rest of this talk Definitions of LDCs and LTCs Quick description of known results Some basic constructions (Time permitting) Yekhanin’s construction of LDCs. December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Definitions December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Locally Decodable Code Code: C : k ! n i s ( q ; ) - L o c a l y D e d b f 9 r . t g v 2 [ ] w m = , n w D ( i ) r e a d s q n o m p t f w u . l 2 = 3 W h a t i f > ( C ) = 2 ? M g n e d o r p l s u ` c w . December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Locally List-Decodable Code Code: C i s ( ; ` ) - l t d e c o a b f 8 w 2 n , # r . m C i s ( q ; ` ) - l o c a y t d e b f 9 D r . g v n 2 [ k ] j w m 1 : n w D ( i ; j ) r e a d s q n o m p t f w u . l 2 = 3 December 17, 2007 Coding & Sublinear time
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History of definitions
Constructions predate formal definitions [Goldreich-Levin ’89]. [Beaver-Feigenbaum ’90, Lipton ’91]. [Blum-Luby-Rubinfeld ’90]. Hints at definition (in particular, interpretation in the context of error-correcting codes): [Babai-Fortnow-Levin-Szegedy ’91]. Formal definitions [S.-Trevisan-Vadhan ’99] (local list-decoding). [Katz-Trevisan ’00] December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Locally Testable Codes Code: C n i s ( q ; ) - L o c a l y T e t b f 9 r . n w T r e a d s q ( n ) o m p i t : I f w 2 C c . 1 - , h j = “Weak” definition: hinted at in [BFLS], explicit in [RS’96, Arora’94, Spielman’94, FS’95]. December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Strong Locally Testable Codes Code: C n i s ( q ; ) - L o c a l y T e t b f 9 r . n w T r e a d s q ( n ) o m p i t : I f w 2 C c . 1 F v y , j ; “Strong” Definition: [Goldreich-S. ’02] December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Motivations December 17, 2007 Coding & Sublinear time
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Motivations for Local decoding
S u p o s e C N i l c a y - d b f r = 2 n . ( F t h m w , j g ) c 2 C a n b e v i w d s f u t o : ; 1 g ! . L o c a l d e i n g ) m p u t ( x f r v y , s . R - w [ S T V ] A l t e r n a i p o : C m u c ( x ) w h - v g . L d s I H [ B F ] , P f R G K S December 17, 2007 Coding & Sublinear time
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Motivation for Local-testing
No generic applications known. However, Interesting phenomenon on its own. Intangible connection to Probabilistically Checkable Proofs (PCPs). Potentially good approach to understanding limitations of PCPs (though all resulting work has led to improvements). December 17, 2007 Coding & Sublinear time
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Contrast between decoding and testing
Decoding: Property of words near codewords. Testing: Property of words far from code. Decoding: Motivations happy with n = quasi-poly(k), and q = poly log n. Lower bounds show q = O(1) and n = nearly-linear(k) impossible. Testing: Better tradeoffs possible! Likely more useful in practice. Even conceivable: n = O(k) with q = O(1)? December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Some LDCs and LTCs December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Codes via Multivariate Polynomials Message: Encoding: Parameters: c o e i n t s f d g , m - v a r p l y P F P F m (Reed Muller code) e v a l u t i o n s f P F m . k ( t = m ) , n j F . December 17, 2007 Coding & Sublinear time
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Basic insight to locality
Local Decoding: Local Testing: m - v a r i t e p o l y n f d g s c < . ( ) u b P i c k s u b p a e t h r o g n x f , d . Q y m l q = j F ; T ( ) ! V e r i f y s t c d o p a g . S m l x December 17, 2007 Coding & Sublinear time
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Summary of Constructions
Polynomial Codes: (Locally decodable and testable) Polynomial Codes + Composition/Concatenation: Codes based on “Algebraic Designs” [Yekhanin] L o c a l i t y q w h n = e x p ( k 1 ) L o c a l T e s t b i y w h q = O ( 1 ) n d ~ k = k ( l o g ) c . L o c a l D e d b i t y w h n = x p ( k 1 q ) L o c a l D e d b i t y w h q = 3 n x p ( k ) December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
[Yekhanin ’07]’s LDCs December 17, 2007 Coding & Sublinear time
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Recall: Combinatorial Designs
Families of Sets: Restrictions on Intersections: E.g., Basic Question: S 1 ; : k , T . i f m g i vs. i: j S i \ T e v n . ( L a r g e ) ( S m a l ) i vs. j: j S i \ T o d . ( S m a l ) ( L a r g e ) H o w l a r g e c n k b ? ( A s a f u n c t i o m ? ) T y p i c a l n s w e r k = ( m ) December 17, 2007 Coding & Sublinear time
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[Yekhanin]’s Algebraic Designs
Families of Vectors: Restrictions on Inner Products: Basic Question: u 1 ; : k , v . i 2 F m p p s m a l r i e h u i ; v = h u i ; v = h u i ; v j 6 = h u i ; v j 2 S 6 3 ( p ; S ) - d e s i g n B a s i c p - d e g n H o w l a r g e c n k b ? A t m o s j S ! C a n w e c h i v ? m p 1 December 17, 2007 Coding & Sublinear time
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[Yekhanin]’s Algebraic Designs
Families of Vectors: Restrictions on Inner Products: Basic Question: u 1 ; : k , v . i 2 F m p p s m a l r i e h u i ; v = h u i ; v = h u i ; v j 6 = h u i ; v j 2 S 6 3 ( p ; S ) - d e s i g n B a s i c p - d e g n H o w l a r g e c n k b ? A t m o s j S ! C a n w e c h i v ? m p 1 December 17, 2007 Coding & Sublinear time
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[Y’07] Algebraic designs and LDCs
m a 1 : B s i c p - d g n w t h k v o r F ) q u y ( b D C k = m p 1 ) n e x ( (Matches some of the early constructions) L e m a 2 : 9 q = ( p ; S ) s . t - d i g n w h k v c o r F u y D C b q ( p ; S ) - A l g e b r a i c n s o f F . December 17, 2007 Coding & Sublinear time
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[Y’07] Algebraic designs and LDCs
m a 2 : 9 q = ( p ; S ) s . t - d i g n w h k v c o r F u y D C b q ( p ; S ) - a l g e b r i c n s o f F . D e n i t o : S s q - a l g b r c y f 9 p m h ( x ) 2 F [ ] = 1 . d j v ( O n e o f t w q u i v a l d s ) December 17, 2007 Coding & Sublinear time
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[Y’07] Algebraic designs and LDCs
m a 2 : 9 q = ( p ; S ) s . t - d i g n w h k v c o r F u y D C b q ( p ; S ) - a l g e b r i c n s o f F . E x a m p l e : = 1 2 7 ; S f 4 8 6 3 g S i s 3 - a l g e b r c y n m 7 l o n g ( p ; S ) - d e s i x t ! ) 3 - q u e r y L D C m a p i n g k b t s o x ( 1 = 7 December 17, 2007 Coding & Sublinear time
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[Y’07] Algebraic designs and LDCs
m a 2 : 9 q = ( p ; S ) s . t - d i g n w h k v c o r F u y D C b q ( p ; S ) - a l g e b r i c n s o f F . L e m a 3 : p = 2 t 1 ) S f ; 4 g i s - l b r c y n . L e m a 4 : S u l t i p c v s b g r o f F ) 9 ( ; - d n h j . T h e o r m : 9 3 - q u y L D C a p i n g k b t s x ( 1 ) . December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Proofs? Disclaimer: Proof of Lemma 2, Lemma 3 too long to fit here. (Many context switches, but elementary.) Will only attempt to show Lemmas 1 and 4. December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Basic designs and LDCs G i v e n u 1 ; : k G = 2 6 4 . 1 h u i ; x 3 7 5 u i x December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Basic designs and LDCs 2 6 4 . 1 h u i ; x 3 7 5 m 1 ; : k message … codeword … y s . t h u i ; 6 = y + 2 v i y + ( p 1 ) v i y + v i Report parity of locations December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Basic designs and LDCs 2 6 4 . 1 h u i ; x 3 7 5 = 2 6 4 . ? 3 7 5 i t h r o w - a l n e s . + + + o t h e r w s - p 1 n . … codeword December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Proof of Lemma 4 Construction of Basic p-designs: Construction of (p,S)-designs for S multiplicative: i $ s e t o f z x a c l y p 1 u i = c h a r t e s v o f . v i = c h a r t e s o f m p l n . h u i ; v = j \ 2 f 1 : p g T a k e u i ; v s b o U s e ~ u i ; v = p j S t h n o r w f . December 17, 2007 Coding & Sublinear time
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Coding & Sublinear time
Conclusions Local algorithms in error-detection/correction lead to interesting new questions. Non-trivial progress so far. Limits largely unknown O(1)-query LDCs must have R(C) = 0 [Katz-Trevisan] December 17, 2007 Coding & Sublinear time
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