1. Beer Therapy Madhu Ralf Vote early, vote often
At Oberwolfach in 2003, Ralf Kötter and Madhu Sudan had a week long beer drinking competition.
Who do you think won?
Ralf
Madhu
Vote early, vote often

2. Local Algorithms & Error-correction
Madhu Sudan
MIT

3. Beer Therapy Madhu Ralf Information Computation
At Oberwolfach in 2003, Ralf Kötter and Madhu Sudan had a week long beer drinking competition.
Who do you think won?
Ralf
Madhu
Information
Computation

4. Dedicated to Ralf Kötter
Dear friend to many …
Wise beyond his age
Happy spirit
… I’ll miss him dearly.
… I already do.

5. Local Error-Correction
Prelude
Algorithmic Problems in Coding Theory
New Paradigm in Algorithms
The Marriage: Local Error-Detection & Correction
February 11, 2009
Local Error-Correction

6. 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 .
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Local Error-Correction

7. 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.
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Local Error-Correction

8. Sub-linear time algorithms
Initiated in late eighties in context of
Program checking [BlumKannan,BlumLubyRubinfeld]
Interactive Proofs/PCPs [BabaiFortnowLund]
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!
February 11, 2009
Local Error-Correction

9. 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).
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Local Error-Correction

10. Local Error-Correction
Rest of this talk
Definitions of LDCs and LTCs
Quick description of known results
The first result: Hadamard codes
Some basic constructions
Recent constructions of LDCs.
[Yekhanin,Raghavendra,Efremenko]
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Local Error-Correction

11. Local Error-Correction
Definitions
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Local Error-Correction

12. Local Error-Correction
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 .
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Local Error-Correction

13. Local Error-Correction
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
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Local Error-Correction

14. 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]
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Local Error-Correction

15. Local Error-Correction

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].
February 11, 2009
Local Error-Correction

16. Local Error-Correction
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]
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Local Error-Correction

17. Local Error-Correction
Motivations
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Local Error-Correction

18. Local decoding: Average-case vs. worst-caseS 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
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Local Error-Correction

19. 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).
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Local Error-Correction

20. 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)?
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Local Error-Correction

21. Local Error-Correction
Some LDCs and LTCs
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Local Error-Correction

22. Local Error-Correction
Hadamard (1st Order RM) Codes
Message:
Encoding:
Parameters:
Locality: ( C o e c i n t s f ) L a r F u m k 2 . e v a l u t i o n s f L F k 2 . k b i t m e s a g ! 2 - c o d w r L ( x ) = + y 2 - o c a l D e d b [ F k r / E i s ] 3 T t B u m R n f
February 11, 2009
Local Error-Correction

23. Hadamard (1st Order RM) Codes
Conclusions:
There exist infinite families of codes
With constant locality (for testing and correcting).
February 11, 2009
Local Error-Correction

24. Local Error-Correction
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 .
February 11, 2009
Local Error-Correction

25. 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
February 11, 2009
Local Error-Correction

26. Local Error-Correction
Polynomial Codes
Many parameters:
Many tradeoffs possible: m , t F L o c a l i t y q w h n = e x p ( k 1 ) L o c a l i t y ( g k ) 2 w h n = 4 L o c a l i t y p k w h n = O ( ) .
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Local Error-Correction

27. Are Polynomial Codes (Roughly) Best?
No! [Ambainis97] [GoldreichS.00] …
No!! [Beimel,Ishai,Kushilevitz,Raymond]
Really … Seriously … No!!!!
[Yekhanin07,Raghavendra08,Efremenko09]
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Local Error-Correction

28. Recent LDCs [Yekhanin ‘07, Raghavendra ‘08, Efremenko ‘09]
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Local Error-Correction

29. Local Error-Correction
The recent result:
Fix q = 3; n = ??? (as function of $k$)
Till 2007:
[Yekhanin ’07]:
[Raghavendra ’08]:
[Efremenko ’09]: n e x p ( k 1 = 5 ) o - b i a r y . n e x p ( k ) b i a r y . n e x p ( k : 1 ) b i a r y . n e x p ( l o g k ) b i a r y .
February 11, 2009
Local Error-Correction

30. Local Error-Correction
Essence of the idea:
Build “good” combinatorial matrix over
Embed in multiplicative subgroup of
Get locally decodable code over Z m Z m F F
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Local Error-Correction

31. “Good” Combinatorial matrix[ [
0 …
… 0 …
… 0 …
… … …
A = k n m a t r i x o v e Z
arbitrary Z e r o s n d i a g l N o n - z e r d i a g l
Columns closed under addition
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Local Error-Correction

32. Local Error-Correction
Embedding into field L e t A = [ a i j ] b \ g o d v r Z m L e t ! = p r i m v h o f u n y F . L e t G = [ ! a i j ] . T h e o r m [ Y k a n i , R g v d E f ] : G t s q u y L D C F .
Highly non-intuitive!
February 11, 2009
Local Error-Correction

33. Local Error-Correction
Improvements A = [ a i j ] ; G ! . O - d i a g o n l e t r s f A m S ) G j + 1 q u y L D C .
(Suffices for [Efremenko]) ! S z e r o s f t - p a l y n m i v F ) G g q u L D C .
(Critical to [Yekhanin])
February 11, 2009
Local Error-Correction

34. Local Error-Correction
“Good” Matrices?
[Yekhanin]:
Picked m prime.
Hand-constructed matrix.
Achieved
Optimal if m prime!
Managed to make S large with t=3.
[Efremenko]
m composite!
Achieves
([Beigel,Barrington,Rudich];[Grolmusz])
Optimal? n = e x p ( k 1 j S ) j S = 3 a n d e x p ( l o g k )
February 11, 2009
Local Error-Correction

35. Limits to Local Decodability: Katz-Trevisan) n = k 1 + ( q .
q queries
Technique:
Recall D(x) computes C(x) whp for all x.
Can assume (with some modifications) that query pattern uniform for any fixed x.
Can find many random strings such that their query sets are disjoint.
In such case, random subset of n^{1-1/q} coordinates of codeword contain at least one query set, for most x.
Yields desired bound.
February 11, 2009
Local Error-Correction

36. Local Error-Correction
Some general results
Sparse, High-Distance Codes:
Are Locally Decodable and Testable
[KaufmanLitsyn, KaufmanS]
2-transitive codes of small dual-distance:
Are Locally Decodable
[Alon,Kaufman,Krivelevich,Litsyn,Ron]
Linear-invariant codes of small dual-distance:
Are also Locally Testable
[KaufmanS]
February 11, 2009
Local Error-Correction

37. Local Error-Correction
Summary
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 Rate(C) = 0
[Katz-Trevisan]
February 11, 2009
Local Error-Correction

38. Local Error-Correction
Questions
Can LTC replace RS (on your hard disks)?
Is a significant rate-loss necessary?
Lower bounds?
Better error models?
Simple/General near optimal constructions?
Other applications to mathematics/computation? (PCPs necessary/sufficient)?
Lower bounds for LDCs?/Better constructions?
February 11, 2009
Local Error-Correction

39. Local Error-Correction
Thank You!
February 11, 2009
Local Error-Correction