Shifted Codes Sachin Agarwal Deutsch Telekom A.G., Laboratories Ernst-Reuter-Platz Berlin Germany Joint work with Andrew Hagedorn and Ari Trachtenberg at Boston University Best viewed on-screen in slide-show mode

S. Agarwal, January 1 Outline 1. Motivation & Problem Definition 2. Background a. Rateless Codes b. Digital Fountain Codes 3. Shifted Codes a. Motivation – Inefficiency of LT codes b. Construction of Shifted Codes c. Analysis – Communication and Computation Complexity 4. Experimental Comparison a. LT vs. Shifted Codes b. Constraint Sensors – Deployment on TMotes 5. Discussion and Round-up

S. Agarwal, January 2 Outline 1. Motivation & Problem Definition 2. Background a. Rateless Codes b. Digital Fountain Codes 3. Shifted Codes a. Motivation – Inefficiency of LT codes b. Construction of Shifted Codes c. Analysis – Communication and Computation Complexity 4. Experimental Comparison a. LT vs. Shifted Codes b. Constraint Sensors – Deployment on TMotes 5. Discussion and Round-up

S. Agarwal, January 3 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbolsReceived Symbols

S. Agarwal, January 4 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbolsReceived Symbols

S. Agarwal, January 5 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbolsReceived Symbols

S. Agarwal, January 6 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbolsReceived Symbols

S. Agarwal, January 7 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbolsReceived Symbols

S. Agarwal, January 8 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbolsReceived Symbols

S. Agarwal, January 9 Partial Information Multiple Receivers may have different erasures Transmitter Receiver 1 Receiver 2 Receiver 3 Given the situation of multiple receivers having partial information, how can all of them be updated to full information efficiently, and over a broadcast channel?

S. Agarwal, January 10 Partial Information Another Example Multiple mobile devices may have out-dated information a. Mobile databases b. Sensor network information aggregation c. RSS updates for devices Broadcaster Mobile device 1 Mobile device 2 Mobile device 3 Latest version of information

S. Agarwal, January 11 Problem Definition Given an encoding host with k input symbols and a decoding host with n out of the k input symbols, the goal is to efficiently determine the remaining k-n input symbols at the decoding host. The encoding host has no information of which k-n input symbols are missing at the decoding host. Different decoding hosts may be missing different input symbols Efficiency 1. Communication complexity – Information transmitted from the encoding host to the decoding host should be close in size to the transmission size of the missing k-n input symbols 2. Computational complexity – The algorithm must be computationally tractable

S. Agarwal, January 12 Information Theoretic Lower Bound Known Result At a minimum, the encoding host would have to send only a little less than the exact contents of the missing input symbols to the decoding host. Intuition Decoding host is missing k-n input symbols Special case of set reconciliation k – Number of input symbols n – Number of symbols known a priori at the decoding host b – Field size of each symbol

S. Agarwal, January 13 Outline 1. Motivation & Problem Definition 2. Background a. Rateless Codes b. Digital Fountain Codes 3. Shifted Codes a. Motivation – Inefficiency of LT codes b. Construction of Shifted Codes c. Analysis – Communication and Computation Complexity 4. Experimental Comparison a. LT vs. Shifted Codes b. Constraint Sensors – Deployment on TMotes 5. Discussion and Round-up

S. Agarwal, January 14 Rateless Codes Definition “A class of erasure codes with the property that a potentially limitless sequence of encoding symbols can be generated from a given set of source symbols such that the original source symbols can be recovered from any subset of the encoding symbols of size equal to or only slightly larger than the number of source symbols. ” Wikipedia.org Examples 1. Random Linear Codes 2. LT Codes 3. Raptor Codes 4. Shifted Codes 5. …

S. Agarwal, January 15 Rateless Codes - Encoding Used for content distribution over error-prone channels Random choice of edges based on a probability density function At least k Encoded Symbols k input symbols 1 =A+B 2 =B 3 =A+B+C 4 =A+C A B C

S. Agarwal, January 16 Rateless Codes - Decoding Used for content distribution over error-prone channels At least k Encoded Symbols 1 =A+B 2 =B 3 =A+B+C 4 =A+C k input symbols Solve Gaussian Elimination, Belief Propagation System of Linear Equations Irrespective of which encoded symbols are lost in the communication channel, as long as sufficient encoded symbols are received, the decoding can retrieve all the k input symbols A B C

S. Agarwal, January 17 Decoding Using Belief Propagation Decoded k Input Symbols k+  Encoded Symbols Decoding host Redundant! Decode Input Symbols

S. Agarwal, January 18 Digital Fountain Codes LT Codes 1. Class of rateless erasure codes invented by Michael Luby 1 2. Computationally practical (as compared to Random Linear Codes) 3. Fast decoding algorithm based on Belief propagation instead of Gaussian Elimination 4. Form the outer code for Raptor Codes 3, which have linear decoding computational complexity 5. Designed for the case when no input symbols are available at the Decoding host initially Asymptotic Properties 2 Expected number of encoded symbols required for successful decoding Expected decoding computational complexity k: number of input symbols 2 Assuming a constant probability of failure  1 Michael Luby, “LT codes,” in The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 271– Amin Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52, no. 6, 2006, pp. 2551–2567.

S. Agarwal, January 19 Digital Fountain Codes LT Codes’ Robust Soliton Probability Distribution Robust Soliton Probability Distribution  k, Probability of an encoded symbol with degree d is  k (d) Property of releasing degree 1 symbols at a controlled, near- constant rate throughout the decoding process LT code distribution, with parameters k = 1000, c = 0.01,  = 0.5.

S. Agarwal, January 20 Outline 1. Motivation & Problem Definition 2. Background a. Rateless Codes b. Digital Fountain Codes 3. Shifted Codes a. Motivation – Inefficiency of LT codes b. Construction of Shifted Codes c. Analysis – Communication and Computation Complexity 4. Experimental Comparison a. LT vs. Shifted Codes b. Constraint Sensors – Deployment on TMotes 5. Discussion and Round-up

S. Agarwal, January 21 Inefficiency of LT Codes for our Problem k+  Encoded Symbols Decoding host Decode Input Symbols n out of k input symbols are known a priori at the decoding host Many redundant encoded symbols

S. Agarwal, January 22 Inefficiency of LT Codes for our Problem The number of these redundant encoded symbols grows with the ratio of input symbols known at the decoder (n) to the total input symbols (k) If n input symbols are known a priori, then an additional LT- encoded symbol will provide no new information to the decoding host with probability …which quickly approaches 1 as n → k

S. Agarwal, January 23 Intuitive Fix n known input symbols serve the function of degree 1 encoded symbols, disproportionately skewing the degree distribution for LT encoding We thus propose to shift the Robust Soliton distribution to the right in order to compensate for the additional functionally degree 1 symbols Questions 1) How? 2) By how much?

S. Agarwal, January 24 Shifted Code Construction Definition The shifted robust soliton distribution is given by Intuition n known input symbols at the decoding host reduce the degree of each encoding symbols by an expected fraction

S. Agarwal, January 25 Shifted Code Distribution LT code distribution and proposed Shifted code distribution, with parameters k = 1000, c = 0.01,  = 0.5. The number of known input symbols at the decoding host is set to n = 900 for the Shifted code distribution. The probabilities of the occurrence of encoded symbols of some degrees is 0 with the shifted code distribution.

S. Agarwal, January 26 Shifted Code – Communication Complexity Lemma IV.2 A decoder that knows n of k input symbols needs encoding symbols under the shifted distribution to decode all k input symbols with probability at least 1− . Proof We have k-n input symbols comprising the encoded symbols after the n known input symbols are removed from the decoding graph. The expresson follows from Luby‘s analysis.

S. Agarwal, January 27 Shifted Code – Average Degree of Encoded Symbol Lemma IV.3 The average degree of an encoding node under the  k,n distribution is given by Proof The proof follows from the definitions, since a node with degree d in the μ k distribution will correspond to a node with degree roughly in the shifted code distribution. From Luby‘s analysis,the expresson for the average degree of an LT encoded symbol is

S. Agarwal, January 28 Shifted Codes – Computational Complexity Lemma IV.4* For a fixed , the expected number of edges R removed from the decoding graph upon knowledge of n input symbols at the decoding host is given by R = O (n ln(k − n)) Theorem IV.5 For a fixed probability of decoding failure , the number of operations needed to decode using a shifted code is O (k ln(k − n)) Proof Summing Lemma IV.4 and the computational complexity of (LT) decoding for the unknown k-n input symbols *Proof described in: S. Agarwal, A. Hagedorn and A. Trachtenberg, “Rateless Codes Under Partial Information”, Information Theory and Applications Workshop, UCSD, San Diego, 2008

S. Agarwal, January 29 Outline 1. Motivation & Problem Definition 2. Background a. Rateless Codes b. Digital Fountain Codes 3. Shifted Codes a. Motivation – Inefficiency of LT codes b. Construction of Shifted Codes c. Analysis – Communication and Computation Complexity 4. Experimental Comparison a. LT vs. Shifted Codes b. Constraint Sensors – Deployment on TMotes 5. Discussion and Round-up

S. Agarwal, January 30 Benefit For k = 1000, n = 900, the decoding host needs to download about 700 encoded symbols using conventional LT codes. But using shifted codes, only about 180 encoded symbols are required Experimental Comparison LT Codes vs. Shifted Codes The experiment was repeated 100 times and the error- bars of the standard deviation are also plotted in the graph. LT Shifted Code Y-axis Number of encoded symbols required at the mobile device to obtain the whole data-set X-axis Number of input symbols n available a priori at the mobile device

S. Agarwal, January 31 Experimental Comparison Constraint Sensors – Deployment on TMotes Total time to Encode (Measure of computational complexity) Total time to Decode (Measure of computational complexity)

S. Agarwal, January 32 More Data: Communication Savings

S. Agarwal, January 33 More Data: Communication Savings Normalized

S. Agarwal, January 34 More Data: Time Savings, Normalized

S. Agarwal, January 35 Distribution Shifting When the estimate of n at the Encoding Host is not accurate The Theta distribution shifting decodes input symbols much more quickly than the standard LT codes.

S. Agarwal, January 36 Outline 1. Motivation & Problem Definition 2. Background a. Rateless Codes b. Digital Fountain Codes 3. Shifted Codes a. Motivation – Inefficiency of LT codes b. Construction of Shifted Codes c. Analysis – Communication and Computation Complexity 4. Experimental Comparison a. LT vs. Shifted Codes b. Constraint Sensors – Deployment on TMotes 5. Discussion and Round-up

S. Agarwal, January 37 Many Applications 1. Broadcasting coded updates to synchronize databases 2. Adapting LT codes when partial information has been delivered a. Continuous shifting of the distribution b. Using the partial information in case of unsuccessful decoding (when only some of the input symbols were decoded) 3. Efficient erasure correction when channel characteristics are already known a. For example, input symbols can be first sent as plain-text, and then depending on the estimate of number of lost input symbols, shifted-coded symbols can be transmitted 4. Heterogeneous channel data delivery 5. Application in gossip protocols, particularly in later iterations 6. Sensor networks - data aggregation, routing information, etc. 7. Restoring storage media that are partially erased …

S. Agarwal, January 38 Conclusions & Future-work Conclusions a. Generalization of LT Code when some of the input symbols are already available at the decoding host b. Many applications Future Work a. By adopting Raptor Code concepts (inner code), Shifted codes can be made more efficient b. Analytical expressions for Distribution Shifting c. Application specific shifted codes design d. “Shifting” other rateless codes

S. Agarwal, January 39 Further Reading 1. S. Agarwal, A. Hagedorn and A. Trachtenberg, “Rateless Codes Under Partial Information”, Information Theory and Applications Workshop, UCSD, San Diego, S. Agarwal (Deutsche Telekom A.G.), “Method and System for Constructing and Decoding Rateless Codes with Partial Information”, European Patent Application EP Michael Luby, “LT codes,” in The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 271– Amin Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52, no. 6, 2006, pp. 2551–2567.