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Energy-Efficient Communications via Network Coding

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. Outline. Introduction on Network Coding

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Energy-Efficient Communications via Network Coding

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  1. 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. Outline • Introduction on Network Coding • Energy Benefit for Multiple Unicast in Wireless Networks • Multi-Rate Network Coding for Minimum-Cost Multicasting • Conclusions

  3. 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. 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. Illustration: Traditional Jos Jasper

  6. Illustration: Network Coding Jos Jasper

  7. Illustration: Combining Messages Since + = Jos can decode - = and Jasper can decode - =

  8. Illustration: Result after Decoding Jos Jasper

  9. Network Coding Example S1 S2 S1 S2 m1 m2 m1 m2 m3=m1+m2 • Possible benefits: • throughput gain • energy efficiency • robustness • adaptability • security • … m1 m1 m2 m1 m3 m2 m1 m1 m3 m3 R1 R2 R1 R2 m2? m2=m3-m1 m1=m3-m2 without network coding with network coding

  10. “Bits are not cars!”(Ralf Koetter) 00001100 + 10101010 = 10100110 + =

  11. Wireless Example Traditional Method Network Coding 1 2 3 1 2 3 m1 m1 m3 m3 m1 m1 m1+m3 m1+m3 m3 m3 4 transmissions 3 transmissions Information exchange between nodes 1 and 3 using node 2

  12. Wireless Circular Network Traditional Method Network Coding 1 1 m1,m8,m2 m1,m8,m2 8 2 8 2 m8,m7,m1 m2,m1,m3 m2 m8,m7,m1 m2,m1,m3 m2+m4 7 3 7 3 m7,m6,m8 m3,m2,m4 m7,m6,m8 m3,m2,m4 m6,m5,m7 m4,m3,m5 m6,m5,m7 m4,m3,m5 m2 m2+m4 6 4 6 4 m5,m4,m6 m5,m4,m6 5 5 N(N-2)=8×6=48 transmissions N(N-1)/2=8×7/2=28 transmissions

  13. Random Network Coding 2 4 1 5 3 … … m3+m4+m5 … m4 … m1+m3 m1+m2+m3 m2+m3 … y1= m3+m4+m5 … … m3+m4+m5 y2= m3+m4 m4 … y3=m1 … … m2 y4=m1 +m3+m4+m5 … y5=m1+m2 m3+m4 R y6=m1 +m3 +m5

  14. Encoding Assume n original packets m1, m2, …, mn generated by one or several sources; Each packet consists of K symbols from GF(2s): mi=(mi,1,mi,2,…,mi,K); At a certain node, encoding vector g=(g1,g2,…,gn), with each giєGF(2s); Information vector x=g1m1+g2m2+…+gnmn=(x1,x2,…,xK), where xk=g1m1,k+g2m2,k+…+gnmn,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. Decoding Solving a linear system of equations with n unknowns (the original messages m1, m2, …, mn); With random network coding, the probability of linearly dependent combinations becomes small if the field size 2s is sufficiently large; Therefore, only (few more than) n information vectors need to be received in order to retrieve the original packets.

  16. Max-Flow Min-Cut Source 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! R1 R2 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.

  17. 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. 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. 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. Energy Benefit: Wireless Example Revisited Traditional Routing Network Coding 1 2 3 1 2 3 m1 m1 m3 m3 m1 m1 m1+m3 m1+m3 m3 m3 4 transmissions 3 transmissions Energy benefit of network coding in comparison to traditional routing is 4/3

  21. Generalization of the Example 1 2 3 … N-1 N Multiple Unicast: 1→N & N→1 Energy Benefit: 2(N-1)/N → 2

  22. Research Challenge Find the maximum energy benefit that network coding can offer Line network example: ≥ 2 Effros et al.: ≥ 2.4 Our contribution: ≥ 3

  23. Network Used in Proof

  24. Three Sets of Unicast Connections Senders Receivers Receivers Senders Senders Receivers

  25. Number of Transmissions Routing: 3K(K-1)/2 ≈ 1.5K2 Network Coding: 3(K+1)K/2- (K-2)(K-3) ≈ 0.5K2 Hence, energy benefit of 1.5/0.5=3 for large K

  26. 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. Result for “Triangle Network”

  28. 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. Example S Butterfly Network: • One source • Four relay nodes • Two receivers • Nine unit capacity edges of cost 1 R1 R2

  30. Throughput versus Cost • Throughput 2 • Cost/symbol 4.5 x y x x x y x x+y y x x x+y x+y • Throughput 1 • Cost/symbol 4

  31. Goal To construct a network code that enables the source to control the throughput, achieving the minimum possible cost at all throughputs.

  32. 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. 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. 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. 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. Example Revisited Operating Point 1 • Throughput 2 • Cost/symbol 4.5 x x+y x x+y Operating Point 2 • Throughput 1 (y=0) • Cost/symbol 4 y x+y x y y

  37. 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. 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. 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. 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. Wireless Example Revisited Once More Traditional Routing Network Coding PL Network Coding 1 2 3 1 2 3 1 2 3 m1 m1 m1 m3 m1 m3 m1+m3 m1+m3 m3 m1+m3 m1+m3 m3 4 transmissions 3 transmissions 2 transmissions Exploiting Broadcast Exploiting Broadcast & MA

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