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Best viewed on-screen in slide-show m Shifted Codes Sachin Agarwal Deutsch Telekom A.G., Laboratories Ernst-Reuter-Platz 7 10587 Berlin Germany Joint work with Andrew Hagedorn and Ari Trachtenberg at Boston Univers 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, sachin.agarwal@telekom.de, January 2008 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, sachin.agarwal@telekom.de, January 2008 2 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbols Received Symbols S. Agarwal, sachin.agarwal@telekom.de, January 2008 3 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbols Received Symbols S. Agarwal, sachin.agarwal@telekom.de, January 2008 4 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbols Received Symbols S. Agarwal, sachin.agarwal@telekom.de, January 2008 5 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbols Received Symbols S. Agarwal, sachin.agarwal@telekom.de, January 2008 6 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbols Received Symbols S. Agarwal, sachin.agarwal@telekom.de, January 2008 7 Partial Information Transmission Channel with Erasures Transmitter Receiver Input symbols Received Symbols S. Agarwal, sachin.agarwal@telekom.de, January 2008 8 Partial Information Multiple Receivers may have different erasures Receiver 1 Given the situation of multiple receivers having partial information, how can all of them be updated to full information efficiently, and over a Transmitter broadcast channel? Receiver 3 Receiver 2 S. Agarwal, sachin.agarwal@telekom.de, January 2008 9 Partial Information Another Example Multiple mobile devices may have out-dated information a. Mobile databases b. Sensor network information aggregation Mobile device 1 c. RSS updates for devices Broadcaster Latest version of information Mobile device 3 Mobile device 2 S. Agarwal, sachin.agarwal@telekom.de, January 2008 10 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, sachin.agarwal@telekom.de, January 2008 11 Information Theoretic Lower Bound Known Result At a minimum, the encoding host would have to send only a lg( k n) little less than the exact C (k n)b contents of the missing input b symbols to the decoding host. Intuition k – Number of input Decoding host is missing k-n symbols input symbols n – Number of symbols Special case of set known a priori at the reconciliation decoding host b – Field size of each symbol S. Agarwal, sachin.agarwal@telekom.de, January 2008 12 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, sachin.agarwal@telekom.de, January 2008 13 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, sachin.agarwal@telekom.de, January 2008 14 Rateless Codes - Encoding Used for content distribution over error-prone channels k input symbols At least k Encoded Symbols A 1 =A+B B 2 =B 3 =A+B+C C 4 =A+C Random choice of edges based on a probability density function S. Agarwal, sachin.agarwal@telekom.de, January 2008 15 Rateless Codes - Decoding Used for content distribution over error-prone channels k input symbols At least k Encoded Symbols 1 =A+B A 2 =B B Solve Gaussian Elimination, Belief Propagation 3 =A+B+C C Irrespective of which encoded symbols are lost in the 4 =A+C communication channel, as long as sufficient encoded symbols are received, the decoding can retrieve all the k input symbols System of Linear Equations S. Agarwal, sachin.agarwal@telekom.de, January 2008 16 Decoding Using Belief Propagation Redundant! k+ Encoded Symbols Decode Decoding host Input Symbols Decoded k Input Symbols S. Agarwal, sachin.agarwal@telekom.de, January 2008 17 Digital Fountain Codes LT Codes 1. Class of rateless erasure codes invented by Asymptotic Properties2 Michael Luby1 Expected number of encoded symbols required for successful decoding 2. Computationally practical (as compared to Random Linear Codes) 3. Fast decoding algorithm based on Belief k O ( k ln 2 k ) propagation instead of Gaussian Elimination Expected decoding computational complexity 4. Form the outer code for Raptor Codes3, which have linear decoding computational complexity O(k ln k ) 5. Designed for the case when no input symbols k: number of input symbols are available at the Decoding host initially 2Assuming a constant probability of failure 1MichaelLuby, “LT codes,” in The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 271–282. 3Amin Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52, no. 6, 2006, pp. 2551–2567. S. Agarwal, sachin.agarwal@telekom.de, January 2008 18 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 0 LT Code (Robust Soliton) -1 log10(Probability) -2 LT code distribution, with parameters k = 1000, c = 0.01, = 0.5. -3 -4 -5 -6 0 200 400 600 800 1000 Degree S. Agarwal, sachin.agarwal@telekom.de, January 2008 19 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, sachin.agarwal@telekom.de, January 2008 20 Inefficiency of LT Codes for our Problem Many redundant encoded symbols k+ Encoded Symbols Decode Decoding host Input Symbols n out of k input symbols are known a priori at the decoding host S. Agarwal, sachin.agarwal@telekom.de, January 2008 21 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 k d n i k (d ) k i d 1 i 0 …which quickly approaches 1 as n → k S. Agarwal, sachin.agarwal@telekom.de, January 2008 22 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 0 LT Code (Robust Soliton) 1) How? -1 log10(Probability) -2 2) By how much? -3 -4 -5 -6 0 200 400 600 800 1000 Degree S. Agarwal, sachin.agarwal@telekom.de, January 2008 23 Shifted Code Construction Definition The shifted robust soliton distribution is given by i k ,n ( j ) 0 k n (i ) for round j 1 n Intuition k n known input symbols at the decoding host reduce the degree of each encoding symbols by an expected fraction 1 1 n k S. Agarwal, sachin.agarwal@telekom.de, January 2008 24 Shifted Code Distribution 0 LT Code (Robust Soliton) -1 Shifted Code log10(Probability) -2 -3 -4 -5 -6 0 200 400 600 800 1000 Degree 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, sachin.agarwal@telekom.de, January 2008 25 Shifted Code – Communication Complexity Lemma IV.2 A decoder that knows n of k input symbols needs 2 k n m (k n) O k n ln 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, sachin.agarwal@telekom.de, January 2008 26 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 k O ln( k n) k n Proof The proof follows from the definitions, since a node with degree d in the μk distribution will correspond to a node with degree roughly d 1 n in the shifted code distribution. k From Luby‘s analysis,the expresson for the average degree of an LT encoded symbol is O(ln k ) S. Agarwal, sachin.agarwal@telekom.de, January 2008 27 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, sachin.agarwal@telekom.de, January 2008 28 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, sachin.agarwal@telekom.de, January 2008 29 Experimental Comparison LT Codes vs. Shifted Codes 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 Y-axis 1200 LT Without Invention Shifted Code With Invention Number of encoded symbols required at the Required encoding symbols at Decoding host 1000 mobile device to obtain the whole data-set X-axis 800 Number of input symbols n available a priori at 600 the mobile device 400 200 0 0 100 200 300 400 500 600 700 800 900 1000 Known input symbols (n) The experiment was repeated 100 times and the error-bars of the standard deviation are also plotted in the graph. S. Agarwal, sachin.agarwal@telekom.de, January 2008 30 Experimental Comparison Constraint Sensors – Deployment on TMotes Total time to Encode Total time to Decode (Measure of computational complexity) (Measure of computational complexity) 3 12 LT (Robust Soliton) LT (Robust Soliton) 2.5 Shifted Code distribution 10 Shifted Code distribution Time To Decode (s) Time to Encode (s) 2 8 1.5 6 1 4 0.5 2 0 0 100 200 300 400 100 200 300 400 Number of Input Symbols Number Input Symbols S. Agarwal, sachin.agarwal@telekom.de, January 2008 31 More Data: Communication Savings k=1000 input symbols, 20 randomized trials 1200 LT Robust Soliton Required encoded symbols for successful \ decoding Shifted Code 1000 800 600 400 200 0 -200 0 200 400 600 800 1000 1200 n, number of known input symbols at decoding host S. Agarwal, sachin.agarwal@telekom.de, January 2008 32 More Data: Communication Savings Normalized k=1000 input symbols, 20 randomized trials Encoded symbols required, normalized with LT-RS LT Robust Soliton 1 Shifted Code 0.8 0.6 0.4 0.2 0 100 200 300 400 500 600 700 800 900 n, number of known input symbols at decoding host S. Agarwal, sachin.agarwal@telekom.de, January 2008 33 More Data: Time Savings, Normalized k=1000 input symbols, 20 randomized trials LT Robust Soliton 1 Shifted Code Time taken to decode, normalized with LT-RS 0.8 0.6 0.4 0.2 0 100 200 300 400 500 600 700 800 900 n, number of known input symbols at decoding host S. Agarwal, sachin.agarwal@telekom.de, January 2008 34 Distribution Shifting When the estimate of n at the Encoding Host is not accurate k 1 i p ( j ) 0 p ( ) k (i ) for round j 1 0 k The Theta distribution shifting decodes input symbols much more quickly than the standard LT codes. S. Agarwal, sachin.agarwal@telekom.de, January 2008 35 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, sachin.agarwal@telekom.de, January 2008 36 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, sachin.agarwal@telekom.de, January 2008 37 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, sachin.agarwal@telekom.de, January 2008 38 Further Reading 1. S. Agarwal, A. Hagedorn and A. Trachtenberg, “Rateless Codes Under Partial Information”, Information Theory and Applications Workshop, UCSD, San Diego, 2008 2. S. Agarwal (Deutsche Telekom A.G.), “Method and System for Constructing and Decoding Rateless Codes with Partial Information”, European Patent Application EP 07 023 243.4 3. Michael Luby, “LT codes,” in The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 271–282. 4. Amin Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52, no. 6, 2006, pp. 2551–2567. S. Agarwal, sachin.agarwal@telekom.de, January 2008 39

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