1. 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

2. Computation over Heterogeneous Wireless Sensor Networks 2 Outline  Introduction  In-Network Computation  Related Works  Motivation  System Model  Main Results  Proof Sketch  Conclusion

3. 3 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, Type- threshold 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

4. 4 In-Network Computation Computation over Heterogeneous Wireless Sensor Networks Performing in-network computation could help save both energy and time in terms of scaling law.

5. 5 Related Works  Seminal work [1]  Multihop and broadcast network.  Symmetric function, type-sensitive function, type-threshold function.  Noiseless environment.  Maximum throughput. Computation over Heterogeneous Wireless Sensor Networks [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.

6. 6 Related Works Computation over Heterogeneous Wireless Sensor Networks Seminal work [1] Multihoptransmission. Symmetricfunction, type- sensitivefunction, type- thresholdfunction. Noiselessenvironment. Maximalthroughput.  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.

7. 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. Computation over Heterogeneous Wireless Sensor Networks [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.

8. 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]. Computation over Heterogeneous Wireless Sensor Networks [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.

9. 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

10. 10 Outline Introduction System Model  Network Model  Function Model  Objective Main Results Proof Sketch Conclusion Computation over Heterogeneous Wireless Sensor Networks

11. 11 Network Model  The total number of nodes is.  The network area is a circle centered at the sink with radius is the network extension exponent.  Each node independently choose a position in the network area according to the following probability density function: where is the distance from the sink, is the network area. is specified as follows: where is the distance from the sink, is the network area. is specified as follows: where is the heterogeneity exponent. where is the heterogeneity exponent. Computation over Heterogeneous Wireless Sensor Networks

12. Network 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. Computation over Heterogeneous Wireless Sensor Networks Illustration of heterogeneous wireless sensor networks

13. Function Model  At one instant, each node gets one binary input data.  We consider symmetric function and identity function:  A function is a symmetric function iff for any permutation, we have: where is arbitrary binary data. Equivalently speaking, symmetric function merely depends on the value but not the identity of the input data. where 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

14. 14 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

15. 15 Outline Introduction System Model Main Results Proof Sketch Conclusion Computation over Heterogeneous Wireless Sensor Networks

16. 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

17. 17 Symmetric function computation

18. 18 Identity function computation

19. 19 Outline Introduction System Model Main Results Proof Sketch  Tessellation  Transmission scheme Conclusion Computation over Heterogeneous Wireless Sensor Networks

20. Tessellation  The key question is how to tessellate the network in order to minimize the energy consumption.

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

22. 22 Outline Introduction System Model Main Results Proof Sketch Conclusion Computation over Heterogeneous Wireless Sensor Networks

23. 23 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

24. Thank you ! Computation over Heterogeneous Wireless Sensor Networks