Function Computation over Heterogeneous Wireless Sensor Networks Xuanyu Cao, Xinbing Wang, Songwu Lu...

Function Computation over Heterogeneous Function Computation over Heterogeneous Wireless Sensor NetworksWireless Sensor Networks

Xuanyu Cao, Xinbing Wang, Songwu LuDepartment of Electronic Engineering

Shanghai Jiao Tong University, China

Function Computation over Heterogeneous Wireless Sensor Networks

Computation over Heterogeneous Wireless Sensor Networks 2

OutlineOutline IntroductionIntroduction

In-Network ComputationIn-Network Computation Related WorksRelated Works MotivationMotivation

System ModelSystem Model

Main ResultsMain Results

Proof SketchProof Sketch

ConclusionConclusion

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In-Network ComputationIn-Network Computation Scaling law for pure information deliveryScaling law for pure information delivery

Unicast, Multicast, Convergecast.Unicast, Multicast, Convergecast. Homogeneity, Heterogeneity.Homogeneity, Heterogeneity. Static, Mobile.Static, Mobile. Ad hoc, Hybrid.Ad hoc, Hybrid.

Scaling law for function computationScaling law for function computation Symmetric function, Identity function, Divisible Function, Type-Symmetric function, Identity function, Divisible Function, Type-

threshold function, Type-sensitive function, etc.threshold function, Type-sensitive function, etc. Noiseless or Noisy environment.Noiseless or Noisy environment. Broadcast Network and Multihop Network.Broadcast Network and Multihop Network. Energy and LatencyEnergy and Latency

Motivation for function computation Motivation for function computation Sink node is only interested in a function of the data, but not all Sink node is only interested in a function of the data, but not all

the raw data.the raw data.

Computation over Heterogeneous Wireless Sensor Networks

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In-Network Computation In-Network Computation


Performing in-network computation could help save both energy and time in terms of scaling law.

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Related WorksRelated Works

Seminal work [1]Seminal work [1] Multihop and broadcast network.Multihop and broadcast network. Symmetric function, type-sensitive function, type-threshold Symmetric function, type-sensitive function, type-threshold

function.function. Noiseless environment.Noiseless environment. Maximum throughput.Maximum throughput.


[1] A. Giridhar and P. Kumar, “Computing and communicating functions over [1] A. Giridhar and P. Kumar, “Computing and communicating functions over sensor networks,” sensor networks,” IEEE Journal on Selected Areas in CommunicationsIEEE Journal on Selected Areas in Communications, vol. , vol. 23, no. 4, pp. 755-764, 2005.23, no. 4, pp. 755-764, 2005.

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Noisy Networks [2][3]Noisy Networks [2][3] Multihop transmission.Multihop transmission. Symmetric function, Divisible function.Symmetric function, Divisible function. Minimum energy consumptionMinimum energy consumption

[2] L. Ying, R. Srikant and G. E. Dullerud, “Distributed symmetric function [2] L. Ying, R. Srikant and G. E. Dullerud, “Distributed symmetric function computation in noisy wireless sensor networks,” computation in noisy wireless sensor networks,” IEEE Trans. Inf. TheoryIEEE Trans. Inf. Theory, vol. , vol. 53, no. 12, pp. 4826-4833, 2007.53, no. 12, pp. 4826-4833, 2007.[3] C. Li and H. Dai, “Towards efficient designs for in-network computing with [3] C. Li and H. Dai, “Towards efficient designs for in-network computing with noisy wireless channels,” noisy wireless channels,” INFOCOMINFOCOM, pp. 1-8, 2010., pp. 1-8, 2010.


Grid Networks [4] (Grid Networks [4] (most related onemost related one)) Binary input data.Binary input data. Noiseless and noisy networks.Noiseless and noisy networks. Symmetric and identity function.Symmetric and identity function. Energy and time complexity.Energy and time complexity. Matching upper and lower bound.Matching upper and lower bound. Intra-cell and Inter-cell protocols.Intra-cell and Inter-cell protocols.


[4] N. Karamchandani, R. Appuswamy, M. Franceschetti, “Time and [4] N. Karamchandani, R. Appuswamy, M. Franceschetti, “Time and energy complexity of function computation over networks,” energy complexity of function computation over networks,” IEEE Trans. Inf. IEEE Trans. Inf. TheoryTheory, vol. 57, no. 12, pp. 7671-7684, 2011., vol. 57, no. 12, pp. 7671-7684, 2011.

Motivation

Previous works on in-network computation are all Previous works on in-network computation are all forfor homogeneous homogeneous networks. networks.

However, the distribution of sensor nodes can be However, the distribution of sensor nodes can be highly highly heterogeneousheterogeneous in practice [5][6]. in practice [5][6].


[5] G. Alfano, M. Garetto and E. Leonardi, “Capacity scaling of wireless networks [5] G. Alfano, M. Garetto and E. Leonardi, “Capacity scaling of wireless networks with inhomogeneous node density: upper bounds,” IEEE Journal on Selected with inhomogeneous node density: upper bounds,” IEEE Journal on Selected Areas in Communications, vol. 27, no. 7, pp. 1147-1157, 2009.Areas in Communications, vol. 27, no. 7, pp. 1147-1157, 2009.[6] G. Alfano, M. Garetto and E. Leonardi, “Capacity scaling of wireless networks [6] G. Alfano, M. Garetto and E. Leonardi, “Capacity scaling of wireless networks with inhomogeneous node density: lower bounds,” IEEE/ACM Trans. Netw.,vol. with inhomogeneous node density: lower bounds,” IEEE/ACM Trans. Netw.,vol. 18, no. 5, pp. 1624-1636, 2010.18, no. 5, pp. 1624-1636, 2010.

Motivation Two fundamental questions arise:Two fundamental questions arise:

What is the impact of heterogeneity on energy consumption for What is the impact of heterogeneity on energy consumption for function computation?function computation?

How much energy consumption reduction can we get by How much energy consumption reduction can we get by performing in-network computation in heterogeneous networks?performing in-network computation in heterogeneous networks?


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OutlineOutlineIntroductionIntroduction

System ModelSystem Model Network ModelNetwork Model Function ModelFunction Model ObjectiveObjective



ConclusionConclusionComputation over Heterogeneous Wireless Sensor Networks

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Network ModelNetwork Model

The total number of nodes is .The total number of nodes is . The network area is a circle centered at the sink with The network area is a circle centered at the sink with

radius is the network extension exponent.radius is the network extension exponent. Each node independently choose a position in the network Each node independently choose a position in the network

area according to the following probability density function:area according to the following probability density function:

where is the distance from the sink, is the network where is the distance from the sink, is the network area. is specified as follows: area. is specified as follows:

where is the heterogeneity exponent.where is the heterogeneity exponent.

n


( )( )

(|| ||)

sf

s x d x

r r

, 0n

(.)s

( ) min 1,s 2

Network ModelNetwork Model Due to the heterogeneity of the nodes’ distribution, we

assume nodes have different transmission range . The energy consumption of transmitting one bit with range is , where is the path loss exponent.

rr

r 2


Illustration of heterogeneous wireless sensor networksIllustration of heterogeneous wireless sensor networks

Function ModelFunction Model

f

1 2 (1) (2) ( )( , ,..., ) ( , ,..., )n nf y y y f y y y

At one instant, each node gets one binary input data.At one instant, each node gets one binary input data. We consider symmetric function and identity function:We consider symmetric function and identity function:

A function is a symmetric function iff for any permutation A function is a symmetric function iff for any permutation , we have:, we have:

where is arbitrary binary data. Equivalently speaking, where is arbitrary binary data. Equivalently speaking, symmetric function merely depends on the value but not the symmetric function merely depends on the value but not the identity of the input dataidentity of the input data..

The output of identity function is all the raw input data. Hence, The output of identity function is all the raw input data. Hence, computing identity function is equivalent to gather all the raw computing identity function is equivalent to gather all the raw datadata..

iy


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ObjectiveObjective

The objective of this paper is to design energy The objective of this paper is to design energy efficient algorithms which can compute the goal efficient algorithms which can compute the goal function at the sink node with the function at the sink node with the minimum total minimum total energy usageenergy usage..

We prove that the proposed algorithm is energy We prove that the proposed algorithm is energy optimal (except for poly-logarithmic terms) by optimal (except for poly-logarithmic terms) by deriving matching lower bounds.deriving matching lower bounds.


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OutlineOutline

IntroductionIntroduction






Main Result

Energy consumption vs. path loss exponent (\gamma), network extension exponent (\alpha), heterogeneity exponent (\delta).

Identifying three heterogeneous regimes: 1) slightly heterogeneous; 2) significantly heterogeneous; 3) highly heterogeneous

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Symmetric function computationSymmetric function computation

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Identity function computationIdentity function computation

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OutlineOutline

IntroductionIntroduction



Proof SketchProof Sketch TessellationTessellation Transmission schemeTransmission scheme



TessellationTessellation

The key question is how to The key question is how to tessellatetessellate the network in order the network in order to minimize the energy consumption.to minimize the energy consumption.

Transmission Scheme We invoke intra-cell/inter-cell transmission scheme.

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OutlineOutlineIntroductionIntroduction



Proof Sketch Proof Sketch



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Conclusion Conclusion

We have studied the optimal energy consumption of We have studied the optimal energy consumption of function computation in heterogeneous networks.function computation in heterogeneous networks.

For both symmetric function and identity function, weFor both symmetric function and identity function, we design energy efficient algorithm for computation.design energy efficient algorithm for computation. prove the optimality of the proposed algorithm by deriving a prove the optimality of the proposed algorithm by deriving a

matching lower bound. matching lower bound.


Thank you !Thank you !


Function Computation over Heterogeneous Wireless Sensor Networks Xuanyu Cao, Xinbing Wang, Songwu Lu...

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