Download presentation
Presentation is loading. Please wait.
1
of 30 Locality in Coding Theory Madhu Sudan MSR June 16, 2015ISIT: Locality in Coding Theory1
2
of 30 Error-Correcting Codes June 16, 2015ISIT: Locality in Coding Theory2
3
of 30 Locality in Algorithms June 16, 2015ISIT: Locality in Coding Theory3
4
of 30 LTCs and LCCs: Formally June 16, 2015ISIT: Locality in Coding Theory4
5
of 30 Outline of this talk Part 0: Definitions of LTC, LCC Part 1: Elementary construction Part 2: Motivation (historical, current) Part 3: State-of-the-art constructions Part 4 (brief): Towards practicality June 16, 2015ISIT: Locality in Coding Theory5
6
of 30 Part 1: Elementary Construction June 16, 2015ISIT: Locality in Coding Theory6
7
of 30 Main Example: Reed-Muller Codes June 16, 2015ISIT: Locality in Coding Theory7
8
of 30 LDCs and LTCs from Polynomials June 16, 2015ISIT: Locality in Coding Theory8 Analysis non-trivial
9
of 30 Part 2: Motivations June 16, 2015ISIT: Locality in Coding Theory9
10
of 30 Motivation – 1 (“Practical”) How to encode massive data? – Solution I Encode all data in one big chunk Pro: Pr[failure] = exp(-|big chunk|) Con: Recovery time ~ |big chunk| – Solution II Break data into small pieces; encode separately. Pro: Recovery time ~ |small| Con: Pr[failure] = #pieces X Pr[failure of a piece] – Locality (if possible): Best of both Solutions!! June 16, 2015ISIT: Locality in Coding Theory10
11
of 30 Aside: LCCs vs. other Localities Local Reconstruction Codes (LRC): – Recover from few (one? two?) erasures locally. – AND Recover from many errors globally. Regenerating Codes (RgC): – Restricted access pattern for recovery: Partition coordinates and access few symbols per partition. Main Differences: – #errors: LCCs high vs LRC/RgC low – Asymptotic (LCC) vs. Concrete parameters (LRC/RgC) June 16, 2015ISIT: Locality in Coding Theory11
12
of 30 Motivation – 2 (“Theoretical”) June 16, 2015ISIT: Locality in Coding Theory12
13
of 30 Aside: PCPs (1 of 3) June 16, 2015ISIT: Locality in Coding Theory13
14
of 30 PCPs (2 of 3) - Definition June 16, 2015ISIT: Locality in Coding Theory14 W 0 11 V HTHHTH 1.Tosses random coins 2. Determines query locations 3.Reads locations. Accepts/Rejects
15
of 30 PCPs (3 of 3): “Polynomial-speak” June 16, 2015ISIT: Locality in Coding Theory15
16
of 30 Part 3: Recent Progress on LCCs + LTCs June 16, 2015ISIT: Locality in Coding Theory16
17
of 30 Summary of Recent Progress June 16, 2015ISIT: Locality in Coding Theory17
18
of 30 Main References Multiplicity codes [KoppartySarafYekhanin’10] See also – Lifted Codes [GuoKoppartySudan’13] – Expander codes [HemenwayOstrovskyWootters’13] Tensor codes [Viderman ‘11] (see also [GKS’13] ) Above + Alon-Luby composition: [KoppartyMeirRon-ZewiSaraf’15] June 16, 2015ISIT: Locality in Coding Theory18 1 3 2
![of 30 Main References Multiplicity codes [KoppartySarafYekhanin’10] See also – Lifted Codes [GuoKoppartySudan’13] – Expander codes [HemenwayOstrovskyWootters’13] Tensor codes [Viderman ‘11] (see also [GKS’13] ) Above + Alon-Luby composition: [KoppartyMeirRon-ZewiSaraf’15] June 16, 2015ISIT: Locality in Coding Theory](http://images.slideplayer.com/19/5775000/slides/slide_18.jpg "of 30 Main References Multiplicity codes [KoppartySarafYekhanin’10] See also – Lifted Codes [GuoKoppartySudan’13] – Expander codes [HemenwayOstrovskyWootters’13] Tensor codes [Viderman ‘11] (see also [GKS’13] ) Above + Alon-Luby composition: [KoppartyMeirRon-ZewiSaraf’15] June 16, 2015ISIT: Locality in Coding Theory")
19
of 30 Lifted Codes June 16, 2015ISIT: Locality in Coding Theory19
20
of 30 Multiplicity Codes June 16, 2015ISIT: Locality in Coding Theory20
21
of 30 Multiplicity Codes - 2 June 16, 2015ISIT: Locality in Coding Theory21
22
of 30 State-of-the-art as of 2014 June 16, 2015ISIT: Locality in Coding Theory22
23
of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] June 16, 2015ISIT: Locality in Coding Theory23 Short MDS Expander-based Interleaving Message Encoding
![of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] June 16, 2015ISIT: Locality in Coding Theory23 Short MDS Expander-based Interleaving Message Encoding](http://images.slideplayer.com/19/5775000/slides/slide_23.jpg "of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] June 16, 2015ISIT: Locality in Coding Theory23 Short MDS Expander-based Interleaving Message Encoding")
24
of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] ISIT: Locality in Coding Theory24 Short MDS Expander-based Interleaving Message Encoding Rate/Distance of final code ~ Rate of MDS [Meir] Locality ~ Locality of Rate 1 code Proof = Picture
![of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] ISIT: Locality in Coding Theory24 Short MDS Expander-based Interleaving Message Encoding Rate/Distance of final code ~ Rate of MDS [Meir] Locality ~ Locality of Rate 1 code Proof = Picture](http://images.slideplayer.com/19/5775000/slides/slide_24.jpg "of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] ISIT: Locality in Coding Theory24 Short MDS Expander-based Interleaving Message Encoding Rate/Distance of final code ~ Rate of MDS [Meir] Locality ~ Locality of Rate 1 code Proof = Picture")
25
of 30 Subpolynomial Locality June 16, 2015ISIT: Locality in Coding Theory25
26
of 30 Part 4: Conclusions June 16, 2015ISIT: Locality in Coding Theory26
27
of 30 The Locality Advantage June 16, 2015ISIT: Locality in Coding Theory27
28
of 30 Locality in Practice? June 16, 2015ISIT: Locality in Coding Theory28
29
of 30 Conclusion Locality: (moderately) new model Remarkable effects possible Connect to many other questions in combinatorics/computer science Useful as a data storage mechanism? June 16, 2015ISIT: Locality in Coding Theory29
30
of 30 Thank You June 16, 2015ISIT: Locality in Coding Theory30
Similar presentations
![Multiplicity Codes Swastik Kopparty (Rutgers) (based on [K-Saraf-Yekhanin ’11], [K ‘12], [K ‘14])](/10/2717695/big_thumb.jpg "Multiplicity Codes Swastik Kopparty (Rutgers) (based on [K-Saraf-Yekhanin ’11], [K ‘12], [K ‘14])>")
