Digital Fountain with Tornado Codes and LT Codes K. C. Yang

References Gavin B. Horn, Per Knudsgaard, Soren B. Lassen, Michael Luby, Jens E. Rasmussen, “A Scalable and Reliable Paradigm for Media on Demand,” IEEE Computer, vol. 34, no. 9, pp , Sep Michael Luby,Michael Mitzenmacher, M. Shokrollahi,Daniel Spielman, “Efficient Erasure Correcting Codes,” IEEE Transactions on Information Theory, vol. 47, no. 2, Feb John W. Byers, Michael Luby, Michael Mitzenmacher, “A Digital Fountain Approach to Asynchronous Reliable Multicast,” IEEE Journal on Selected Areas in Communications, vol. 20, no. 8, pp , Oct Michael Luby, “LT Codes.”

Outline Current Delivery Solutions Digital Fountain Reed-Solomon Codes Bipartite Graph Encoding Tornado Codes Luby Transform Codes Experimental Results

Current Delivery Solutions Point-to-Point vs. Broadcast P2PBroadcast Download on demand? Packet loss? Server load? Network load? Scalability? Pause-resume download?

Introduction MoD Play the requested video or music without interruptions after a given start-up delay User-centered MoD Allocate the bandwidth at the server depending on when and how many client requests. e.g. Batching. Data-centered MoD Allocate a constant bandwidth over the period of time. e.g. Periodical broadcasting. time

Digital Fountain The receiver can reconstruct the original data after receiving a sufficient number of packets – regardless of order or sequence. server encoded packet file

Digital Fountain Take a file of k packets. Encode it into ck encoded packets. Given any set of k encoded packets, the original file can be recovered. Don’t care which packets the client receives. source

Digital Fountain Digital fountain Yes Good Low High Download on demand? Resume download? Packet loss? Server load? Network load? Scalability?

Concept x1x1 x2x2 xkxk y1y1 y2y2 ynyn Receive any k encoding packets to reconstruct the source packets

Reed-Solomon Codes k source packets  n encoding packets n = 2 A - 1, where A is the length of a symbol. k(n-k)A/2 exclusive-ORs of source packets. e.g. k = 10000, n = 20000, A = exclusive-ORs of source packets per source packet. Finite stretch factor (n/k). Receive many useless duplicate transmissions when packet loss and parallel download. n = 6 k = 5 n = 12 k = 5

Bipartite Graph Encoding (Tornado Codes) k source packets  n encoding packets n =  i = 0 to m  i k. fixed n. Coding time  Number of edges k  k k  2k 2k  3k 3k  mk mk  k 1/2 x1x1 x2x2 x3x3  y 1 = x 1  x 2  x 3 0<  <1 Poisson distribution Soliton distribution Sparse: Avg. # of variables per equation is small

Bipartite Graph Encoding (LT Codes) Each packet is independently generated. Encoding process (Infinite iterations) Randomly choose the degree d of encoding symbol by a degree distribution. Uniformly choose d input symbols. Exclusive-or these d symbols. Decoder needs to know the degree and set of neighbors of each encoding symbol. Sparse codes, too. xi1xi1 xi2xi2 x id d x 1 x 2 x 3 x 4 x 5 …x k y i = yiyi d

Bipartite Graph Encoding x 3 x 3 x 2 x 3 x 2 x 1 x 3 x 2 x 1 x 4 x 3 x 3 x 2 x 3 x 2 x 1 Decoding process

Bipartite Graph Encoding Decoding process (Iteration) 1. Find any equation with exactly one variable, recover the value. 2. Combine the recovered variable in all equations with exclusive-ORs. y 1 = x 3, y 2 = x 2  x 3, y 3 = x 3  x 1, y 4 = x 1  x 2  x 4 y 1 = x 3  y’ 2 = y 2  x 3 = x 2, y’ 3 = y 3  x 3 = x 1. y’ 2 = x 2, y’ 3 = x 1  y’’ 4 = y 4  x 1  x 2 = x 4. y’’ 4 = x 4.

Tornado Codes And LT Codes TornadoLT n/kn/k Pre-determineinfinite structure pre-constructdynamically construct decoder must know the graph constructed at encoder degree and set of each encoding symbol

Tornado Codes And LT Codes LTTornadoReed-Solomon Decoding inefficiency Asymptotically  1 Encoding time O(lnk) O(nln(1/  )) O(k(n-k)A) Decoding time O(klnk) O(nln(1/  )) O(k(n-k)A) Decoding inefficiency Use of sparse codes Reception of duplicate packets

Experimental Results 4132 source packets  8264 encoding packets 512 B packet size with 500 B of data and 12 B of information Berkeley Carnegie Mellon Cornell

Experimental Results Decoding inefficiency (  c ) Distinctness inefficiency (  d ) Reception inefficiency (  =  c  d )