Binary Soliton-Like Rateless Coding for the Y
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Transcript Binary Soliton-Like Rateless Coding for the Y
Binary Soliton-Like Rateless Coding
for the Y-Network
Andrew Liau, ShahramYousefi, Senior Member, IEEE,
and Il-Min Kim Senior Member, IEEE
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 12, DECEMBER 2011
Outline
Introduction
System model
Soliton-like rateless coding
Simulation results
Introduction
In today’s telecommunication applications, content can originate
from multiple sources and may travel through many transport nodes
to reach one or more receivers.
Currently, intermediate nodes in a communications network
perform
Buffer-and-forward (BF)
Not an optimal strategy in the sense of overall network throughput
Network coding (NC)
Each transport node linearly combines packets received
Provides the maximum throughput for all users simultaneously
The complexity increases on the decoder side
LT code and Raptor code
Provide practical capacity-achieving solutions by way of carefully-
designed encoding degree distributions
The complexities for these rateless codes are very low (logarithmic
to linear scale).
For multicast scenarios for the binary erasure channel (BEC)
When the encoder uses the Robust Soliton Distribution (RSD)
Capacity over the BEC is achieved universally
The erasure rate of the channel does not need to be known a priori
The original LT codes provide optimality for single-source, single-
hop, and single-sink networks.
Motivation
We want a scheme that has good information diffusion
Using a channel code providing Maximum Distance Separable (MDS)
-type (every coded bit is the same) properties
Loss resilience
NC linearly combines packets at intermediate nodes
Fountain codes linearly combine packets at the sources and provides
low decoding complexity
Advantages of marrying NC and fountain codes
The low complexity decoder
The ability to increase the effective length of the fountain code
Previous works
[8] R. Gummadi and R. S. Sreenivas, “Relaying a fountain code across
multiple nodes,” in Proc. ITW, 2008, pp. 49–153.
[8] describes a system where encoding is superimposed at each
transport node resulting in multi-layer fountain coding.
The performance of the code is equivalent to a single hop as the RSD is
preserved.
Multi-layer fountain decoding might be impractical to use due to its high
complexity.
LT Network Codes [4]
[4] M. Champel, K. Huguenin, A. Kermarrec, and N. Le Scouarnec, “LT
network codes,” in Proc. ICDCS, 2010.
Generalizing the setting to any network with a single source and sink.
Using complex data structures, transport nodes selectively combine
packets to form the RSD at each hop.
NP-hard problem at each transport node
Other shortcomings
Not resilient to nodes churn rates
Not scalable (complexity and dependencies on the network configurations)
System model
Soliton-like degree distribution
Allowing each source to use the RSD regardless of the number of total
sources.
We consider a two-user, two-hop , single-sink network.(Y-network)
System model
At each source (S1 and S2): The information is encoded by an LT
code.
At the relay (𝑅): Either BF or NC is performed.
At the sink (𝐷): After successful decoding, the sink transmits a
single acknowledgment (ACK) bit indicating the termination of
the session.
System model
Each
performs LT coding [5]
Over the sets
To produce the packets
R:
If NC is applied, re-encode
and
to generate
If BF is applied , forwards packets from S1 in even time slots and
packets from S2 in odd time slots.
System model
A key component of a fountain code is the packet degree distribution,
which characterizes the decoding efficiency and throughput
optimality.
RSD
Soliton-like rateless coding
RSD :
The literature scale poorly with network size
Sensitive to node churn rates
=>SLRC
With the RSD at each source , we need a intelligent NC at R to
preserve important properties of the RSD.
=> NC at R
Some attributes of the best distribution
𝑝(⋅) is an aggregate degree distribution seen from D.
The probability of degree-two packet is the maximum of the
distribution
=>
=>
=>
(fountain code ,in single-source, single-hop)
(in more practical scenarios)
For BP decoding to start, degree-one packets are required
=>
=>
(too many of them cause inefficient decoding)
p(1)<<p(2) (Otherwise , distributions result in significantly larger
minimum overhead)
Soliton-like distribution
Soliton-like rateless coding : At R
We protect degree-one and two packets by forwarding them with
probability λ ,where λ will be optimized.
If the packets are not forwarded by R, then they are buffered for
future use.
The memory of R is restricted to K for each source.
R is restricted to form a new packet by combining a single packet
from S1 with a single packet from S2.
Although a Soliton-like distribution is generated at R, redundancy
must also be addressed.
Soliton-like rateless coding
Soliton-like rateless coding
Definition 3 (Soliton-like rateless coding (SLRC)):
The SLRC protocol requires LT coding at each source
Combining at R according to Algorithm 1 where
and
are
innovative.
This means that Algorithm 1 reuses a packet
or more than
once only if there are no unused packets in the corresponding
buffers.
Soliton-like rateless coding
Theorem 1: The aggregate distribution produced by the SLRC with
𝜆 ≥ 0.67 is Soliton-like.
Proof :
We can determine the degree distribution, 𝜇(𝑘), seen at 𝐷 from
the set of packets forwarded from either source:
𝑞(𝑘) is the probability of a packet of degree k being forwarded:
The degree distribution,
,of innovative buffered S1 packets will
be:
Where
and
are the probabilities that a packet of degree
one and two are not forwarded, respectively:
When a packet is not forwarded, the relay distribution due to only
linear combining is :
The aggregate distribution in
is a mixture of forwarded and
linearly combined packets :
The probability, 𝜃, that a packet is from either distribution is defined
as :
Soliton-like distribution
Since the RSD is
used at each source
4) : is satisfied when 𝜆≥ 0.67 ( By letting
=> 5)
6) : Satisfied at each source encodes
=>
=>
also satisfies
maintains 6)
)
Soliton-like distribution
Corollary 1: The aggregate distribution produced by the SLRC
protocol in the presence of a single source in a session is the RSD.
Proof: Suppose that S2 has left the network. In this case, R can assume
that only degree-zero packets have been received from S2.
which results in the aggregate distribution being equal to the
RSD.
Simulation results
[9] S. Puducheri, J. Kliewer, and T. E. Fuja, “The design and performance of
distributed LT codes,” IEEE Trans. Inf. Theory, vol. 53, no. 10, pp.
3740–3754, Oct. 2007.
The DLT [9]code is based on the RSD :
With values of 𝑐, 𝛿, and a message length of 2K
The proposed SLRC :
With values of 𝑐, 𝛿, and a message length of K
The SDLT [10]:
With values of 𝑐, 𝛿, and a message length of K
A coding distribution, Λ(𝑥), at R
BF
[10] D. Sejdinovic, R. Piechocki, and A. Doufexi, “AND-OR tree analysis
of distributed LT codes,” in Proc. ITW, 2009, pp. 261–265.
With K =100, an optimum value of 𝜆 = 0.95 was found for
SLRC.
Simulation results
Simulation results
Simulation results
Conclusion
We propose a scheme that exploits the benefits of network coding
and fountain coding
SLRC
Not affected by node churn rates in that if a source node left, no
changes to the protocol are needed.
By preserving key properties of the RSD as packets travel through
the network, we show that the aggregate distribution is Solitonlike
Better at reliable success rates when compared to the DLT and
SDLT codes.