1. Presentation at IEEE AWSITC, June 4, 20101 Energy-Efficient Communications via Network Coding Jos Weber Delft University of Technology The Netherlands Visiting Professor at Presentation at IEEE AWSITC, June 4, 2010 Based on joint work with Jasper Goseling

2. Presentation at IEEE AWSITC, June 4, 20102 Outline Introduction on Network Coding Energy Benefit for Multiple Unicast in Wireless Networks Multi-Rate Network Coding for Minimum-Cost Multicasting Conclusions

3. Presentation at IEEE AWSITC, June 4, 20103 Part 1 Introduction on Network Coding Energy Benefit for Multiple Unicast in Wireless Networks Multi-Rate Network Coding for Minimum-Cost Multicasting Conclusions and Future Work

4. Presentation at IEEE AWSITC, June 4, 20104 Network Coding Paradigm Traditional routing solutions for communication networks keep independent data streams separate. Network coding solutions allow nodes in the network to combine independent data streams.

5. Presentation at IEEE AWSITC, June 4, 20105 Illustration: Traditional JosJasper

6. Presentation at IEEE AWSITC, June 4, 20106 Illustration: Network Coding JosJasper

7. Presentation at IEEE AWSITC, June 4, 20107 Illustration: Combining Messages Since Jos can decode + = and Jasper can decode - - = =

8. Presentation at IEEE AWSITC, June 4, 20108 Illustration: Result after Decoding JosJasper

9. Presentation at IEEE AWSITC, June 4, 20109 Network Coding Example S1S1 R2R2 S2S2 R1R1 without network coding m1m1 m2m2 m1m1 m1m1 m1m1 m1m1 S1S1 S2S2 R1R1 R2R2 with network coding m1m1 m1m1 m2m2 m2m2 m3m3 m3m3 m3m3 m 2 =m 3 -m 1 m 3 =m 1 +m 2 m 1 =m 3 -m 2 Possible benefits: throughput gain energy efficiency robustness adaptability security … m2?m2? m2m2

10. Presentation at IEEE AWSITC, June 4, 201010 Bits are not cars! (Ralf Koetter) 00001100 + 10101010 = 10100110 + =

11. Presentation at IEEE AWSITC, June 4, 201011 Wireless Example 123213 Traditional MethodNetwork Coding m1m1 m3m3 m1m1 m1m1 m3m3 m3m3 m1m1 m 1 +m 3 m3m3 4 transmissions3 transmissions Information exchange between nodes 1 and 3 using node 2

12. Presentation at IEEE AWSITC, June 4, 201012 Wireless Circular Network Traditional MethodNetwork Coding 1 46 82 3 5 7 46 82 5 73 1 m 1,m 8,m 2 m 8,m 7,m 1 m 2,m 1,m 3 m 7,m 6,m 8 m 3,m 2,m 4 m 6,m 5,m 7 N(N-2)=8×6=48 transmissions m 4,m 3,m 5 m 5,m 4,m 6 m2m2 m2m2 m 2 +m 4 N(N-1)/2=8×7/2=28 transmissions m 6,m 5,m 7 m 7,m 6,m 8 m 8,m 7,m 1 m 1,m 8,m 2 m 2,m 1,m 3 m 3,m 2,m 4 m 4,m 3,m 5 m 5,m 4,m 6

13. Presentation at IEEE AWSITC, June 4, 201013 Random Network Coding 2 3 4 5 … … … … … R 1 … … … … … … m 1 +m 3 m 1 +m 2 +m 3 m 2 +m 3 m4m4 m 3 +m 4 +m 5 m4m4 y 1 = m 3 +m 4 +m 5 m2m2 m 3 +m 4 y 2 = m 3 +m 4 y 3 =m 1 y 4 =m 1 +m 3 +m 4 +m 5 y 5 =m 1 +m 2 y 6 =m 1 +m 3 +m 5

14. Presentation at IEEE AWSITC, June 4, 201014 Encoding Assume n original packets m 1, m 2, …, m n generated by one or several sources; Each packet consists of K symbols from GF(2 s ): m i =(m i,1,m i,2,…,m i,K ); At a certain node, encoding vector g=(g 1,g 2,…,g n ), with each g i єGF(2 s ); Information vector x=g 1 m 1 +g 2 m 2 +…+g n m n =(x 1,x 2,…,x K ), where x k =g 1 m 1,k +g 2 m 2,k +…+g n m n,k ; Encoding can be performed recursively (to already encoded packets); Encoding vector can be deterministic or random (in which case it is transmitted together with the information vector).

15. Presentation at IEEE AWSITC, June 4, 201015 Decoding Solving a linear system of equations with n unknowns (the original messages m 1, m 2, …, m n ); With random network coding, the probability of linearly dependent combinations becomes small if the field size 2 s is sufficiently large; Therefore, only (few more than) n information vectors need to be received in order to retrieve the original packets.

16. Presentation at IEEE AWSITC, June 4, 201016 Max-Flow Min-Cut Assume each link has unit capacity. Min-cut is two for both receiver nodes. Max-flow is two for each receiver node. Not achievable simultaneously by traditional routing! Achievable simultaneously by network coding! This works for all multicast networks: The upper bound on the obtainable data rate imposed by the smallest maximum flow from the source to some receiver can be achieved simultaneously for all receivers using coding. Source R2R2 R1R1

17. Presentation at IEEE AWSITC, June 4, 201017 Network Coding in 2010 Also other (theoretical) results on network coding have been derived since the start in 2000. Possible benefits with respect to throughput, energy efficiency, robustness, adaptability, security, … Potential for practical applications is under investigation, first results are available. N.B. Work of North-West University, Potchefstroom

18. Presentation at IEEE AWSITC, June 4, 201018 Part 2 Introduction on Network Coding Energy Benefit for Multiple Unicast in Wireless Networks Multi-Rate Network Coding for Minimum-Cost Multicasting Conclusions and Future Work

19. Presentation at IEEE AWSITC, June 4, 201019 Energy Benefit Energy benefit of network coding for a wireless multiple unicast configuration: minimum energy consumption of any routing solution minimum energy consumption of any network coding solution

20. Presentation at IEEE AWSITC, June 4, 201020 Energy Benefit: Wireless Example Revisited 123213 Traditional RoutingNetwork Coding m1m1 m3m3 m1m1 m1m1 m3m3 m3m3 m1m1 m 1 +m 3 m3m3 4 transmissions3 transmissions Energy benefit of network coding in comparison to traditional routing is 4/3

21. Presentation at IEEE AWSITC, June 4, 201021 Generalization of the Example 123N-1…N Energy Benefit: 2(N-1)/N 2 Multiple Unicast: 1 N & N 1

22. Presentation at IEEE AWSITC, June 4, 201022 Research Challenge Line network example: 2 Effros et al.: 2.4 Our contribution: 3 Find the maximum energy benefit that network coding can offer

23. Presentation at IEEE AWSITC, June 4, 201023 Network Used in Proof

24. Presentation at IEEE AWSITC, June 4, 201024 Three Sets of Unicast Connections SendersReceivers Senders ReceiversSenders Receivers

25. Presentation at IEEE AWSITC, June 4, 201025 Number of Transmissions Routing: 3K(K-1)/2 1.5K 2 Network Coding: 3(K+1)K/2- (K-2)(K-3) 0.5K 2 Hence, energy benefit of 1.5/0.5=3 for large K

26. Presentation at IEEE AWSITC, June 4, 201026 Rx Energy Energy benefit when taking also Rx energy into account: Line network: 2E(Tx) + 2E(Rx) E(Tx) + 2E(Rx) Triangle network: 3E(Tx) + 3E(Rx) E(Tx) + 6E(Rx)

27. Presentation at IEEE AWSITC, June 4, 201027 Result for Triangle Network

28. Presentation at IEEE AWSITC, June 4, 201028 Part 3 Introduction on Network Coding Energy Benefit for Multiple Unicast in Wireless Networks Multi-Rate Network Coding for Minimum-Cost Multicasting Conclusions and Future Work

29. Presentation at IEEE AWSITC, June 4, 201029 Example S R2R2 R1R1 Butterfly Network: One source Four relay nodes Two receivers Nine unit capacity edges of cost 1

30. Presentation at IEEE AWSITC, June 4, 201030 Throughput versus Cost xy x xy y x+y Throughput 2 Cost/symbol 4.5 Throughput 1 Cost/symbol 4 xx xx

31. Presentation at IEEE AWSITC, June 4, 201031 Goal To construct a network code that enables the source to control the throughput, achieving the minimum possible cost at all throughputs.

32. Presentation at IEEE AWSITC, June 4, 201032 Model and Definitions Acyclic directed graph Capacity and cost on edges Multicast traffic Single network use Throughput: number of symbols transmitted Cost (per symbol) = (Σ costs of all edges used)/throughput Operating point: throughput-cost pair

33. Presentation at IEEE AWSITC, June 4, 201033 Network Coding at Minimum Cost For a given throughput, find minimum-cost subgraph satisfying min-cut conditions: [Lun et al., IEEE IT, 2006] Construct a code on the subgraph: [Jaggi et al., IEEE IT, 2005] [Ho et al., IEEE IT, 2006] Multi-rate network coding: one subgraph for each operating point! Challenge: Find a code that works on all subgraphs

34. Presentation at IEEE AWSITC, June 4, 201034 Related Work Variable-Rate Linear Network Coding, [Fong & Yeung, IEEE ITW, 2006]: Variable throughput Single subgraph Changing set of receivers, i.e., those nodes in the network that have the min-cut satisfied Network Coding for Link Failures, [Koetter & Medard, IEEE/ACM TN, 2003], [Jaggi et al., IEEE IT, 2005]: Single throughput Different subgraphs

35. Presentation at IEEE AWSITC, June 4, 201035 Outline of Code Construction The source selects the throughput and encodes the data using one set of coding vectors. Take size of global coding vectors equal to maximum supported throughput. At lower throughputs, fix unused symbols at zero. The chosen throughput is communicated to other nodes in the network, e.g., by including it in the header of a packet. Intermediate nodes know the subgraphs used at each operating point and perform the same linear coding operation at all throughputs, i.e., there is only one set of local coding vectors. Receivers know which symbols are used at each operating point and can decode accordingly.

36. Presentation at IEEE AWSITC, June 4, 201036 Example Revisited xx+y x x y y y Operating Point 1 Throughput 2 Cost/symbol 4.5 Operating Point 2 Throughput 1 (y=0) Cost/symbol 4

37. Presentation at IEEE AWSITC, June 4, 201037 Main Result Theorem: For any network, a multi-rate code can be constructed achieving the minimum possible cost at all throughputs. Proof (sketch): Consider transfer matrices for each receiver for each operating point; Require all transfer matrices to have full rank; Consider product of all determinants; Follow [Koetter & Medard, IEEE/ACM TN, 2003] algebraic framework.

38. Presentation at IEEE AWSITC, June 4, 201038 Part 4 Introduction on Network Coding Energy Benefit for Multiple Unicast in Wireless Networks Multi-Rate Network Coding for Minimum-Cost Multicasting Conclusions and Future Work

39. Presentation at IEEE AWSITC, June 4, 201039 Conclusions Network coding is a promising technique with possible benefits with respect to throughput, energy efficiency, robustness, adaptability, security, … A better lower bound on the maximum possible energy benefit for multiple unicast on wireless networks has been derived A multi-rate network code for minimum-cost multicasting has been proposed

40. Presentation at IEEE AWSITC, June 4, 201040 Other/Future Research Studying combined channel and network coding Further exploring the possible energy benefit of network coding Taking into consideration stochastic packet arrivals Physical-layer network coding

41. Presentation at IEEE AWSITC, June 4, 201041 Wireless Example Revisited Once More 123213 Traditional RoutingNetwork Coding m1m1 m3m3 m1m1 m1m1 m3m3 m3m3 m1m1 m 1 +m 3 4 transmissions3 transmissions PL Network Coding 312 2 transmissions m3m3 m 1 +m 3 Exploiting Broadcast Exploiting Broadcast & MA