Digital Fountain with Tornado Codes and LT Codes

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

2. 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. 40-45, Sep 2001. • Michael Luby,Michael Mitzenmacher, M. Shokrollahi,Daniel Spielman, “Efficient Erasure Correcting Codes,” IEEE Transactions on Information Theory, vol. 47, no. 2, Feb 2001. • 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. 1528-1540, Oct 2002. • Michael Luby, “LT Codes.” • http://www.digitalfountain.com

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

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

5. 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

6. 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

7. Digital Fountain Download on demand? Resume download? Packet loss? Server load? Network load? Scalability?

8. Concept x1 x2 xk y1 y2 yn Receive any k encoding packets to reconstruct the source packets

9. Reed-Solomon Codes • k source packets  n encoding packets • n = 2A - 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 = 16. • 80000 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

10. k1/2 mk Bipartite Graph Encoding (Tornado Codes) • k source packets  n encoding packets • n = i = 0 to mik. • fixed n. • Coding time  Number of edges k k 2k 3k 0<  <1 x1 y1 = x1 x2  x3 x2  x3 Sparse: Avg. # of variables per equation is small Poisson distribution Soliton distribution

11. 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. x1x2x3x4x5…xk d yi d xi1 xi2 yi = xid

12. Bipartite Graph Encoding • Decoding process x3 x3 x3x2 x3x2 x3x2x1 x3x2x1 x3x2x1x4

13. 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. • y1 = x3, y2 = x2 x3, y3 = x3 x1, y4 = x1 x2  x4 • y1 = x3 y’2 = y2  x3 = x2, y’3 = y3  x3 = x1. • y’2 = x2, y’3 = x1  y’’4 = y4  x1  x2 = x4. • y’’4 = x4.

14. Tornado Codes And LT Codes Tornado LT Pre-determine infinite n/k structure pre-construct dynamically construct decoder must know the graph constructed at encoder degree and set of each encoding symbol

15. Tornado Codes And LT Codes • Decoding inefficiency • Use of sparse codes • Reception of duplicate packets

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

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