We study efficient broadcasting for wireless sensor networks, with network
coding. We address this issue for homogeneous sensor networks in the plane. Our
results are based on a simple principle (IREN/IRON), which sets the same rate
on most of the nodes (wireless links) of the network. With this rate selection,
we give a value of the maximum achievable broadcast rate of the source: our
central result is a proof of the value of the min-cut for such networks, viewed
as hypergraphs. Our metric for efficiency is the number of transmissions
necessary to transmit one packet from the source to every destination: we show
that IREN/IRON achieves near optimality for large networks; that is,
asymptotically, nearly every transmission brings new information from the
source to the receiver. As a consequence, network coding asymptotically
outperforms any scheme that does not use network coding.Comment: Dans First International Workshop on Information Theory for Sensor
  Netwoks (WITS 2007) (2007

Adjih, Cédric

Cho, Song Yean

Jacquet, Philippe

English

arXiv

International audienceWe study efficient broadcasting for wireless sensor networks, with network coding. We address this issue for homogeneous sensor networks in the plane. Our results are based on a simple principle (IREN/IRON), which sets the same rate on most of the nodes (wireless links) of the network. With this rate selection, we give a value of the maximum achievable broadcast rate of the source: our central result is a proof of the value of the min-cut for such networks, viewed as hypergraphs. Our metric for efficiency is the number of transmissions necessary to transmit one packet from the source to every destination: we show that IREN/IRON achieves near optimality for large networks; that is, asymptotically, nearly every transmission brings new information from the source to the receiver. As a consequence, network coding asymptotically outperforms any scheme that does not use network coding

HAL-Polytechnique

Near Optimal Broadcast with Network Coding in Large Sensor Networks

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TESTMotivationNetwork Coding FundamentalsOur ApproachNear Optimal Broadcast with Network Codingin Large Sensor NetworksCédric Adjih, Song Yean Cho, Philippe JacquetINRIA/École Polytechnique - Hipercom Team1st Intl. Workshop on Information Theory for Sensor Networks(WITS’07) - Santa Fe - USAC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachMotivationWireless Sensor NetworksI Low cost elements:large networksI Limited battery powerI Wireless communicationC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachEfficient BroadcastProblem studiedI Specific form of communicationI Broadcast: from one source to all nodes in the networkI Energy Efficiency:I minimize the total number of transmissions for broadcastingone message for the sourceC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachNetwork ModelIdealized ModelI Unit-disk graph(boolean)I range: ρI No packet lossI No interferenceI No capacity limitTopologiesI LatticeI Random (uniform)C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachEfficient BroadcastWithout Network CodingI Wireless case: heuristicsexistI Some subset of nodesretransmits messagesI Connected Dominating SetC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachEfficient BroadcastWithout Network CodingI Wireless case: heuristicsexistI Some subset of nodesretransmits messagesI Connected Dominating SetC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachEfficient BroadcastWithout Network CodingI Wireless case: heuristicsexistI Some subset of nodesretransmits messagesI Connected Dominating SetC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachEfficient Broadcast with Network CodingQuestion:I What about Energy Efficient Broadcast with Network Codingin Wireless Networks?C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachEfficient Broadcast with Network CodingResults:I Propose one way to do network coding (“rate selection”)I Show that it is energy efficient: “optimal at the transmissionlevel”– asymptotically for some classes of networksI Offers advantages over routingC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachNetwork Coding PrincipleNetwork Coding: Energy-Efficient BroadcastNetwork Coding Principle 1/2Transmission without codingImage nodes A and B wants tocommunicate, through one relay node RI A sends a packet PA to node RI R forwards the packet PA to node BI B sends a packet PB to node RI R forwards the packet PB to node AC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachNetwork Coding PrincipleNetwork Coding: Energy-Efficient BroadcastNetwork Coding Principle 1/2Transmission without codingImage nodes A and B wants tocommunicate, through one relay node RI A sends a packet PA to node RI R forwards the packet PA to node BI B sends a packet PB to node RI R forwards the packet PB to node AC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachNetwork Coding PrincipleNetwork Coding: Energy-Efficient BroadcastNetwork Coding Principle 2/2With codingI A sends a packet PA to node RI B sends a packet PB to node RI R computes the packet PA xor PBI R sends the coded packet to bothnodes A and BI A and B can decode and get Pb and PaCoding performed inside the networkI Ex. of another form of coding: linear coding, packets arevectors of a Galois Field (ex: GF (28))I linear combination: Psend =∑i αiPiC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachNetwork Coding PrincipleNetwork Coding: Energy-Efficient BroadcastNetwork Coding Principle 2/2With codingI A sends a packet PA to node RI B sends a packet PB to node RI R computes the packet PA xor PBI R sends the coded packet to bothnodes A and BI A and B can decode and get Pb and PaCoding performed inside the networkI Ex. of another form of coding: linear coding, packets arevectors of a Galois Field (ex: GF (28))I linear combination: Psend =∑i αiPiC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachNetwork Coding PrincipleNetwork Coding: Energy-Efficient BroadcastNetwork Coding Principle 2/2With codingI A sends a packet PA to node RI B sends a packet PB to node RI R computes the packet PA xor PBI R sends the coded packet to bothnodes A and BI A and B can decode and get Pb and PaCoding performed inside the networkI Ex. of another form of coding: linear coding, packets arevectors of a Galois Field (ex: GF (28))I linear combination: Psend =∑i αiPiC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachNetwork Coding PrincipleNetwork Coding: Energy-Efficient BroadcastNetwork Coding Principle 2/2With codingI A sends a packet PA to node RI B sends a packet PB to node RI R computes the packet PA xor PBI R sends the coded packet to bothnodes A and BI A and B can decode and get Pb and PaCoding performed inside the networkI Ex. of another form of coding: linear coding, packets arevectors of a Galois Field (ex: GF (28))I linear combination: Psend =∑i αiPiC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachNetwork Coding PrincipleNetwork Coding: Energy-Efficient BroadcastTransmission-Level EfficiencyTransmission-level efficiencyI One transmission reaches several neighbor nodesI Efficiency (at the trans. level): useful for several nodesI Innovative , usefulI Transmission-level optimality , the transmission is useful forevery receiverI Difference with point-to-point linkC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachNetwork Coding PrincipleNetwork Coding: Energy-Efficient BroadcastEnergy-Efficient Broadcast with Network CodingProblem of Energy-Efficient BroadcastI Single source broadcast: perf. depends on avg. rate ofnodes??I Select the rate of each nodeI For these rates: maximum broadcast rateI → number of transmissions per broadcast messageOptimal Solution: Energy-Efficient BroadcastI Finding rate and cost (at same time):I Optimization problemI Linear Program [Wu et al. 2004] [Lun et al. 2004]I Solved in polynomial timeI Distributed algorithms existC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachNetwork Coding PrincipleNetwork Coding: Energy-Efficient BroadcastEnergy-Efficient BroadcastSample solutionI Sensor NetworkI Linear Program:I ≈ N2M variables (heresparse matrix with3710156 coefs)I Solution for the rateI Source rate = 1I Total rate = 9.0625IssuesI Simple rate selection?I How much to expect?C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachNetwork Coding PrincipleNetwork Coding: Energy-Efficient BroadcastEnergy-Efficient BroadcastSample solutionI Sensor NetworkI Linear Program:I ≈ N2M variables (heresparse matrix with3710156 coefs)I Solution for the rateI Source rate = 1I Total rate = 9.0625IssuesI Simple rate selection?I How much to expect?C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingOur Approach: overviewOur ApproachI Start with a simple rate selectionC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingRate SelectionRate SelectionI Starting point:same rate for all nodes = 1Further ReasoningI Assume homogeneous network: each node has ≈ M neighborsI Consider: innovative packetsI If every received transmission is innovative,I Then: node with M neighbors, would receive M usefulpackets per unit timeI Hence: the source needs to send M packets per secondI Problem: nodes on border may have less than M neighborsC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingRate SelectionRate SelectionI Starting point:same rate for all nodes = 1Further ReasoningI Assume homogeneous network: each node has ≈ M neighborsI Consider: innovative packetsI If every received transmission is innovative,I Then: node with M neighbors, would receive M usefulpackets per unit timeI Hence: the source needs to send M packets per secondI Problem: nodes on border may have less than M neighborsC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingProposed Rate SelectionRate Selection: IREN/IRONI Increased Rate forExceptional Nodes (IREN):• rate M for the source, andnodes in the border areaI Identical Rate for OtherNodes (IRON):• rate 1 for other nodes(M: number of neighbors)C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingPerformance of the Rate Selection IREN/IRONKey resultMaximum broadcast rate with IREN/IRONTheoremFor a lattice: the maximum broadcast rate is exactly MTheoremFor a random graphs: maximum broadcast rateMp→ 1, asymptotically,when density M grows (and density M grows fast enough compared tonetwork width L– e.g. M = Lθ with θ > 0)Proof.Discrete geometry, min-cut computation and probabilityC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingOverview of the proof for latticeProof: maximum broadcast rate ≥ MI s source, t any of the dest.I For any U subset : t ∈ U - must showit receives enough packets (cut)I Neighborhood of U (including U):Minkowski sum:(U ⊕ R) , {u + v : u ∈ U, v ∈ R}I Neighbors of U (without U), ∆U:∆U = (U ⊕ R) \ UI Brunn-Minkowski inequality for finitesets: |A⊕ B| ≥ |A|+ |B| − 1I Sum of rates to U ≥ |R| − 1 = MC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingOverview of the proof for latticeProof: maximum broadcast rate ≥ MI s source, t any of the dest.I For any U subset : t ∈ U - must showit receives enough packets (cut)I Neighborhood of U (including U):Minkowski sum:(U ⊕ R) , {u + v : u ∈ U, v ∈ R}I Neighbors of U (without U), ∆U:∆U = (U ⊕ R) \ UI Brunn-Minkowski inequality for finitesets: |A⊕ B| ≥ |A|+ |B| − 1I Sum of rates to U ≥ |R| − 1 = MC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingOverview of the proof for latticeProof: maximum broadcast rate ≥ MI s source, t any of the dest.I For any U subset : t ∈ U - must showit receives enough packets (cut)I Neighborhood of U (including U):Minkowski sum:(U ⊕ R) , {u + v : u ∈ U, v ∈ R}I Neighbors of U (without U), ∆U:∆U = (U ⊕ R) \ UI Brunn-Minkowski inequality for finitesets: |A⊕ B| ≥ |A|+ |B| − 1I Sum of rates to U ≥ |R| − 1 = MC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingOverview of the proof for latticeProof: maximum broadcast rate ≥ MI s source, t any of the dest.I For any U subset : t ∈ U - must showit receives enough packets (cut)I Neighborhood of U (including U):Minkowski sum:(U ⊕ R) , {u + v : u ∈ U, v ∈ R}I Neighbors of U (without U), ∆U:∆U = (U ⊕ R) \ UI Brunn-Minkowski inequality for finitesets: |A⊕ B| ≥ |A|+ |B| − 1I Sum of rates to U ≥ |R| − 1 = MC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingOverview of the proof for latticeProof: maximum broadcast rate ≥ MI s source, t any of the dest.I For any U subset : t ∈ U - must showit receives enough packets (cut)I Neighborhood of U (including U):Minkowski sum:(U ⊕ R) , {u + v : u ∈ U, v ∈ R}I Neighbors of U (without U), ∆U:∆U = (U ⊕ R) \ UI Brunn-Minkowski inequality for finitesets: |A⊕ B| ≥ |A|+ |B| − 1I Sum of rates to U ≥ |R| − 1 = MC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingOverview of the proof for unit disk graphsUnit disk graphsI Seq. of random unit disk graphsI Create a virtual (embedded) latticeI Map the nodes to the virtual latticeI Capacity of an s-t cut, related tocapacity of sL-tL cut on the virtuallattice: Cmin ≥ mminC (L)minI When M →∞ fast enough, the resultsof the lattice are generalized to theseq. of unit disk graphs.C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingOverview of the proof for unit disk graphsUnit disk graphsI Seq. of random unit disk graphsI Create a virtual (embedded) latticeI Map the nodes to the virtual latticeI Capacity of an s-t cut, related tocapacity of sL-tL cut on the virtuallattice: Cmin ≥ mminC (L)minI When M →∞ fast enough, the resultsof the lattice are generalized to theseq. of unit disk graphs.C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingEnergy-efficiencyEnergy-efficiency and optimalityI Assumption behind IREN/IRON:I every transmission is innovative, andreceived rate = M (1 per M neighbors)I Proven: broadcast rate ≈ MI Indeed: ≈ every transmission is innovativeI Optimality: comes from reaching information-theoretic boundfor each transmissionI Transmission-level optimalityI Except: issues for the border nodes and random graphsC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingAsympotic OptimalityBound for Energy-efficiencyI N nodes; and at most Mmax neighbors for any nodeI Minimum transmissions per broadcast: Ebound =NMmaxI Actual transmissions per broadcast: Ecost =transmission ratebroadcast rateTheoremFor lattice graphs, cost per broadcast of IREN/IRON convergestowards cost of the bound, i.e. EcostEbound → 1, when L grows to ∞and range ρ is fixed.TheoremFor random graphs, EcostEboundp→ 1 when L grows to ∞, density growsas M = Lθ, with 0 < θ < 1.C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingAsymptotic OptimalityI Stronger result than optimality of linear programComparison with routingI In wired networks, [Edmonds, 1972],for broadcast routing is sufficient.I In wireless networks, it is different,logic from [Widmer et al. 2005]:I except the source, every retransmitterreceives from a nodeI when retransmit, common neighborsreceiveI Hence asymptotically, at least(2π3 −√32 ) ≈ 0.391 . . . of the areareceives redundant transmissions.C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingAsymptotic OptimalityI Stronger result than optimality of linear programComparison with routingI In wired networks, [Edmonds, 1972],for broadcast routing is sufficient.I In wireless networks, it is different,logic from [Widmer et al. 2005]:I except the source, every retransmitterreceives from a nodeI when retransmit, common neighborsreceiveI Hence asymptotically, at least(2π3 −√32 ) ≈ 0.391 . . . of the areareceives redundant transmissions.C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingAsymptotic OptimalityI Stronger result than optimality of linear programComparison with routingI In wired networks, [Edmonds, 1972],for broadcast routing is sufficient.I In wireless networks, it is different,logic from [Widmer et al. 2005]:I except the source, every retransmitterreceives from a nodeI when retransmit, common neighborsreceiveI Hence asymptotically, at least(2π3 −√32 ) ≈ 0.391 . . . of the areareceives redundant transmissions.C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingSummarySummaryI Presented a rate selection: IREN/IRONI Computed its performance (maximum broadcast rate from thesource)I Shows that it is optimal, asymptoticallyI Offers advantages over routingFuture WorkI Simple rate selection for given graph? Non-homogeneous?I Fully distributed operation?I Realistic wireless models?C.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network CodingTESTMotivationNetwork Coding FundamentalsOur ApproachRate Selection: IREN/IRONPerformance of IREN/IRONComparison with routingThank you for your attentionC.A., Song Yean Cho, Philippe Jacquet — WITS’07 Near Optimal Broadcast with Network Coding

Near Optimal Broadcast with Network Coding in Large Sensor Networks

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