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Presentation at IEEE AWSITC, June 4, 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

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

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 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.

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Presentation at IEEE AWSITC, June 4, Illustration: Traditional JosJasper

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

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

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

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 4, Bits are not cars! (Ralf Koetter) = =

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Presentation at IEEE AWSITC, June 4, Wireless Example 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

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Presentation at IEEE AWSITC, June 4, Wireless Circular Network Traditional MethodNetwork Coding 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

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Presentation at IEEE AWSITC, June 4, Random Network Coding … … … … … 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

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Presentation at IEEE AWSITC, June 4, 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).

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Presentation at IEEE AWSITC, June 4, 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.

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 4, Network Coding in 2010 Also other (theoretical) results on network coding have been derived since the start in 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

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 4, Energy Benefit: Wireless Example Revisited 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

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

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

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Presentation at IEEE AWSITC, June 4, Network Used in Proof

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

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 4, 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)

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Presentation at IEEE AWSITC, June 4, Result for Triangle Network

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Presentation at IEEE AWSITC, June 4, 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

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

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

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

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 4, 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.

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 4, 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.

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 4, 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

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Presentation at IEEE AWSITC, June 4, 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

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

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