Function Computation over Heterogeneous
Wireless Sensor Networks
Xuanyu Cao, Xinbing Wang, Songwu Lu
Department of Electronic Engineering
Shanghai Jiao Tong University, China
Function Computation over Heterogeneous Wireless Sensor Networks
Outline
 Introduction
 In-Network Computation
 Related Works
 Motivation
 System Model
 Main Results
 Proof Sketch
 Conclusion
Computation over Heterogeneous Wireless Sensor Networks
2
In-Network Computation
 Scaling law for pure information delivery




Unicast, Multicast, Convergecast.
Homogeneity, Heterogeneity.
Static, Mobile.
Ad hoc, Hybrid.
 Scaling law for function computation
 Symmetric function, Identity function, Divisible Function, Typethreshold function, Type-sensitive function, etc.
 Noiseless or Noisy environment.
 Broadcast Network and Multihop Network.
 Energy and Latency
 Motivation for function computation
 Sink node is only interested in a function of the data, but not all the
raw data.
Computation over Heterogeneous Wireless Sensor Networks
3
In-Network Computation
Performing in-network computation could help save
both energy and time in terms of scaling law.
Computation over Heterogeneous Wireless Sensor Networks
4
Related Works
 Seminal work [1]
 Multihop and broadcast network.
 Symmetric function, type-sensitive function, type-threshold
function.
 Noiseless environment.
 Maximum throughput.
[1] A. Giridhar and P. Kumar, “Computing and communicating functions over
sensor networks,” IEEE Journal on Selected Areas in Communications, vol. 23,
no. 4, pp. 755-764, 2005.
Computation over Heterogeneous Wireless Sensor Networks
5
Related Works
 Noisy Networks [2][3]
 Multihop transmission.
 Symmetric function, Divisible function.
 Minimum energy consumption
[2] L. Ying, R. Srikant and G. E. Dullerud, “Distributed symmetric function
computation in noisy wireless sensor networks,” IEEE Trans. Inf. Theory, vol.
53, no. 12, pp. 4826-4833, 2007.
[3] C. Li and H. Dai, “Towards efficient designs for in-network computing with
noisy wireless channels,” INFOCOM, pp. 1-8, 2010.
Computation over Heterogeneous Wireless Sensor Networks
6
Related Works
 Grid Networks [4] (most related one)






Binary input data.
Noiseless and noisy networks.
Symmetric and identity function.
Energy and time complexity.
Matching upper and lower bound.
Intra-cell and Inter-cell protocols.
[4] N. Karamchandani, R. Appuswamy, M. Franceschetti, “Time and energy
complexity of function computation over networks,” IEEE Trans. Inf. Theory,
vol. 57, no. 12, pp. 7671-7684, 2011.
Computation over Heterogeneous Wireless Sensor Networks
Motivation
Previous works on in-network computation are all
for homogeneous networks.
However, the distribution of sensor nodes can be
highly heterogeneous in practice [5][6].
[5] G. Alfano, M. Garetto and E. Leonardi, “Capacity scaling of wireless networks
with inhomogeneous node density: upper bounds,” IEEE Journal on Selected
Areas in Communications, vol. 27, no. 7, pp. 1147-1157, 2009.
[6] G. Alfano, M. Garetto and E. Leonardi, “Capacity scaling of wireless networks
with inhomogeneous node density: lower bounds,” IEEE/ACM Trans. Netw.,vol.
18, no. 5, pp. 1624-1636, 2010.
Computation over Heterogeneous Wireless Sensor Networks
Motivation
 Two fundamental questions arise:
 What is the impact of heterogeneity on energy consumption for
function computation?
 How much energy consumption reduction can we get by
performing in-network computation in heterogeneous networks?
Computation over Heterogeneous Wireless Sensor Networks
Outline
Introduction
System Model
 Network Model
 Function Model
 Objective
Main Results
Proof Sketch
Conclusion
Computation over Heterogeneous Wireless Sensor Networks
10
Network Model
 The total number of nodes is n.
 The network area is a circle centered at the sink with

radius n ,   0 is the network extension exponent.
 Each node independently choose a position in the network
area according to the following probability density function:
f ( ) 

s(  )
r
r
s (|| x ||) d x
where  is the distance from the sink,  is the network
area. s (.) is specified as follows:
s (  )  min 1,   
where   2 is the heterogeneity exponent.
Computation over Heterogeneous Wireless Sensor Networks
11
Network Model
 Due to the heterogeneity of the nodes’ distribution, we
assume nodes have different transmission range r . The
energy consumption of transmitting one bit with range r

is r , where   2 is the path loss exponent.
Illustration of heterogeneous wireless sensor networks
Computation over Heterogeneous Wireless Sensor Networks
Function Model
 At one instant, each node gets one binary input data.
 We consider symmetric function and identity function:
 A function f is a symmetric function iff for any permutation
we have:

f ( y1, y2 ,..., yn )  f ( y (1) , y (2) ,..., y (n) )
where y i is arbitrary binary data. Equivalently speaking,
symmetric function merely depends on the value but not the
identity of the input data.
 The output of identity function is all the raw input data. Hence,
computing identity function is equivalent to gather all the raw
data.
Computation over Heterogeneous Wireless Sensor Networks
,
Objective
 The objective of this paper is to design energy
efficient algorithms which can compute the goal
function at the sink node with the minimum total
energy usage.
 We prove that the proposed algorithm is energy
optimal (except for poly-logarithmic terms) by
deriving matching lower bounds.
Computation over Heterogeneous Wireless Sensor Networks
14
Outline
Introduction
System Model
Main Results
Proof Sketch
Conclusion
Computation over Heterogeneous Wireless Sensor Networks
15
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
Symmetric function computation
17
Identity function computation
18
Outline
Introduction
System Model
Main Results
Proof Sketch
 Tessellation
 Transmission scheme
Conclusion
Computation over Heterogeneous Wireless Sensor Networks
19
Tessellation
 The key question is how to tessellate the network in order
to minimize the energy consumption.
Transmission Scheme
 We invoke intra-cell/inter-cell transmission scheme.
Outline
Introduction
System Model
Main Results
Proof Sketch
Conclusion
Computation over Heterogeneous Wireless Sensor Networks
22
Conclusion
 We have studied the optimal energy consumption of
function computation in heterogeneous networks.
 For both symmetric function and identity function, we
 design energy efficient algorithm for computation.
 prove the optimality of the proposed algorithm by deriving a
matching lower bound.
Computation over Heterogeneous Wireless Sensor Networks
23
Thank you !
Computation over Heterogeneous Wireless Sensor Networks