Digital Fountain with Tornado Codes and LT Codes K. C. Yang.
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Transcript of Digital Fountain with Tornado Codes and LT Codes K. C. Yang.
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Digital Fountain with Tornado Codes and LT Codes
K. C. Yang
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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
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Outline
• Current Delivery Solutions• Digital Fountain• Reed-Solomon Codes• Bipartite Graph Encoding
• Tornado Codes• Luby Transform Codes
• Experimental Results
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Current Delivery Solutions
• Point-to-Point vs. Broadcast
P2P Broadcast
Download on demand?
Packet loss?
Server load?
Network load?
Scalability?
Pause-resume download?
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Digital Fountain
• The receiver can reconstruct the original data after receiving a sufficient number of packets – regardless of order or sequence.
serverencoded packet
file
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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
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Digital Fountain
Digital fountain
Yes
Yes
Good
Low
Low
High
Download on demand?
Resume download?
Packet loss?
Server load?
Network load?
Scalability?
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Concept
x1 x2 xk
y1 y2 yn
knknn
kk
kk
n xaxaxa
xaxaxa
xaxaxa
y
y
y
2211
2222121
1212111
2
1
Receive any k encoding packets to reconstruct the source packets
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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 = 6k = 5
n = 12k = 5
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Bipartite Graph Encoding (Tornado Codes)
• k source packets n encoding packets• n = i = 0 to m ik.• fixed n.
• Coding time Number of edgesk
k 2k 3k mkk1
/2
x1
x2
x3
y1 = x1 x2 x3
0< <1
Poisson distribution
Soliton distribution
Sparse:Avg. # of variables per equation is small
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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.
xi1 xi2 xid
d
x1x2x3x4x5…xk
yi =
yid
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Bipartite Graph Encoding
x3
x3 x2
x3 x2 x1
x3 x2 x1 x4
x3
x3 x2
x3 x2 x1
• Decoding process
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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.
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Tornado Codes And LT Codes
Tornado LT
n/k Pre-determine infinite
structure pre-construct dynamically construct
decoder must know
the graph constructed at encoder
degree and set of each encoding symbol
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Tornado Codes And LT Codes
LT Tornado Reed-Solomon
Decoding inefficiency
Asymptotically 1 1 + 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
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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
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Experimental Results
• Decoding inefficiency (c)
• Distinctness inefficiency (d)
• Reception inefficiency ( = cd)
packets data source ofnumber
tionreconstruc before received packetsdistinct ofnumber c
received packetsdistinct ofnumber
received packets ofnumber totald
packets data source ofnumber
tionreconstruc before received packets ofnumber