Binary Soliton -Like Rateless Coding for the Y-Network

1. 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

2. Outline • Introduction • System model • Soliton-like rateless coding • Simulation results

3. 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

4. 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.

5. 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

6. 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] • 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) [4] M. Champel, K. Huguenin, A. Kermarrec, and N. Le Scouarnec, “LT network codes,” in Proc. ICDCS, 2010.

7. 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)

8. 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.

9. 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.

10. System model • A key component of a fountain code is the packet degree distribution, which characterizes the decoding efficiency and throughput optimality. • RSD

11. 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

12. 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)

13. Soliton-like distribution

14. 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 Ris 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.

15. Soliton-like rateless coding

16. 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.

17. 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:

18. The degree distribution, ,of innovative bufferedS1 packets will be: • Where and are the probabilities that a packet of degree one and two are not forwarded, respectively:

19. 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 :

20. Soliton-like distribution • 4) : is satisfied when 𝜆≥ 0.67 ( By letting ) • => 5) • 6) : Satisfied at each source encodes • => also satisfies • => maintains 6) Since the RSD is used at each source

21. 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.

22. 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 • With K =100, an optimum value of 𝜆 = 0.95 was found for SLRC. [10] D. Sejdinovic, R. Piechocki, and A. Doufexi, “AND-OR tree analysis of distributed LT codes,” in Proc. ITW, 2009, pp. 261–265.

23. Simulation results

24. Simulation results

25. Simulation results

26. 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 Soliton-like • Better at reliable success rates when compared to the DLT and SDLT codes.