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New Degree Distribution to improve LT-Code in Network Coding for

Broadcasting in Ad-hoc Wireless Networks

Nour KADI, Khaldoun Al AGHA

21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications

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Introduction Previous work The drawback of LT codes Switched Code

◦ Analysis of Switched Distribution

Simulation Conclusions

Outline

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A wireless ad hoc network is a decentralized type of wireless network.◦ wireless Sensor networks (WSN)

Each node participates in routing by forwarding data for other nodes, and so the determination of which nodes forward data is made dynamically based on the network connectivity. 

Introduction(1/3) - ad-hoc network

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Overcoming problems associated with resource scarcity and unreliable channels has been a challenge for data broadcasting protocols in ad-hoc networks.

The network coding combined with network broadcasting can reduce the bandwidth and power consumption and increase the throughput.

Rateless codes achieve a reliable broadcast transmission.

Introduction(2/3)

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Network Coding

b1 b2

b1

b1

b1

b2

b2

b2

b2

b1

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The number of transmissions(1/2)

(for channel coding)

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The number of transmissions(2/2)

(For network coding)

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Linear Network Coding [4] LT Code

Encoding & Decoding

complexity

Low complexity

Intermediate nodes

Decode & re-encode received information

Just forward the encoded

packets

Result

Save the network resource and increase the throughput

Consume lot of network resource

Previous work[4]D. S. Lun, M. M´edard, R. Koetter, and M. Effros, “On coding for reliable communication over packet networks,” CoRR, vol. abs/cs/0510070, 2005.

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To achieve reliable broadcasting and efficient bandwidth utilization ：◦ Using LT-code guarantees the simple complexity

of the proposed coding scheme.

◦ Using network coding increases the throughput and saves the network resources.

We introduce a coding scheme that efficiently broadcasts the source packets.

Introduction(3/3)

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Combines LT-code with network coding, by enabling intermediate nodes to perform coding.

Switched Code(1/2)

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The RSD is designed to decode plenty of symbols only when it receives sufficiently large number of encoded packets.

Hence, using LT- code in its original form increases significantly the end-to-end packet delay.

The drawback of the LT codes

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We present a degree distribution that increases the symbol recovery probability at any time during the decoding process, while keeping the overhead as small as possible.

Switched Code(2/2)

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The sender switches from the first distribution to the other according to the number of encoded packets which have been sent.

Switched distribution

[6] S. Agarwal, A. Hagedorn, and A. Trachtenberg, “Adaptive rateless coding under partial information,” in Information Theory and Applications Workshop, 2008, 2008, pp. 5–11.

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Degree distribution of LT codes

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Analysis of Switched Distribution(1/8)

1 5

Codeword with coding candidate {1, 3, 4}, and degree = 3

2 3 4

Definition 1. (codeword and degree):

Definition 2. Binary Exponential Distribution():

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Definition 3. (The Decoding Probability): Let be the probability to decode a codeword of degree d when r − 1 of the source symbols has been recovered.

Analysis of Switched Distribution(2/8)

𝐷(𝑟|𝑑 )=¿

(𝑘−𝑟+11 )( 𝑟− 1𝑑−1)

(𝑘𝑑)¿

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Definition 4. (The Symbol Recovery Probability): Let be the probability to recover the source symbol. So , where p(d) is the probability to have a codeword of degree d.

Definition 5. Let be The expected number of recovered symbols after sending y codewords. And let the overhead

= Y − k where = k.

Analysis of Switched Distribution(3/8)

In addition to minimize the overhead, our interest is to maximize ,

This could be obtained by maximizing , r.

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Lemma 2. To recover the first symbol at the destination, it is more useful that the source chooses the degree of the codeword according to Binary Exponential Distribution than using Robust Soliton Distribution (RSD).

Analysis of Switched Distribution(4/8)

Because

(i) ()

(ii) , where , and ( H()+ ), H(n) is the Harmonic number of n. (RSD)

We can prove that .

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Lemma 3. To recover the last symbol, it is more useful to use Soliton Distribution.

Analysis of Switched Distribution(5/8)

In this case r=k, from proposition 1,

(BED)

(Ideal Soliton distribution)

And we can prove that .

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Lemma 4. RSD outperforms only after recovering 70% of the source packets at the destination.

Analysis of Switched Distribution(6/8)

Using a dichotomic technique, we find that when r − 1 is inferior to 0.70*k then is superior to and this is reversed when r − 1 becomes superior to 0.70*k.

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Definition 6. Let’s consider an incremental decoder S. If the decoder S receives r codewords, then it decodes them in an ascending order according to their degree d. First step, it decodes the packets with d = 1. Then, using the packets which are recovered from the first step, it decodes the packets with d = 2 . For a step i, it decodes the packets with d = i by using the packets which are recovered from steps 1,2,. . . ,i-1. If a packet of degree d couldn’t be decoded at step i = d, then it will be ignored in the next steps.

Analysis of Switched Distribution(7/8)

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Proposition 2. The expected number of recovered symbols after sending k codewords according to is at least 0.70*k.

Prove this proposition for the incremental decoder. Assume that S receives Y codewords, then

Step 1 ： =

Step 2 ： =

Analysis of Switched Distribution(8/8)

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Then, the total expected number of recovered symbols is

Using a dichotomic technique, we find that when Y=k then =0.70*k.

And this result is valid for a non-restricted decoder.

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To adapt LT code to the case where some input symbols are already known at the destination.◦ Generate high degree with higher probability

Shifted Robust Soliton Distribution SRSD[6]

k : total number of input symbolsl : the number of input symbols already known at the destination.

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Using Opnet simulator

The simulation continues until k source packets generated by the source node are delivered to all other nodes.

At each time slot, an intermediate node can transmit one packet, which is received by its neighbors with a probability (1-).

Using distributed TDMA scheduling

Running the simulation 30 times

The parameters of RSD

◦ C=0.2,

Simulation(1/2)

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Comparison with 2 other LT-code based schemes.◦ Original LT code

◦ Forward & Re-code[4] (for line network) Adapt LT-code with RSD to network coding

Comparison with a random linear network coding schemes.

◦ Probabilistic NC[8]

Random linear network coding with a probabilistic approach

Simulation(2/2)

[4] P. Pakzad, C. Fragouli, and A. Shokrollahi, “Coding schemes for line networks,” CoRR, vol. abs/cs/0508124, 2005.

[8] C. Fragouli, J. Widmer, and J.-Y. L. Boudec, “Efficient broadcasting using network coding,” IEEE/ACM Trans. Netw, vol. 16, no. 2, pp. 450–463, 2008.

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Compare with LT-code based schemes(1/2)

loss rate

30 static node in line network

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Comparison with LT-code based schemes(2/2)

30 static node in line network , k=10

The deliver ratio is the number of delivered packets at the destination over the total number of the source packets.

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Comparison with a random linear network coding schemes

50 nodes placed randomly on a area.

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Considered the broadcast traffic in ad-hoc wireless networks, We have presented a novel rateless code which outperforms LT code as it allows intermediate nodes to process the received packets.

The proposed distribution increases the decoding probability of any received symbol.

The simulation shows that our scheme reduces the number of transmissions and increases the packet delivery ratio.

Conclusions