Onur G. Guleryuz & Ulas C.Kozat

1. Joint Compression, Detection, and Routing in Capacity Constrained Wireless Sensor Networks Onur G. Guleryuz & Ulas C.Kozat DoCoMo USA Labs, San Jose, CA 95110 {guleryuz,kozat}@docomolabs-usa.com

2. Scenario Phase 1: (NASA, Virgin Galactic,...) Central node Phase 2: (DoCoMo) Low power, low complexity, wireless sensor node Phenomenon (Less exciting applications possible)

3. Overview • This paper is about information theoretic (rate-distortion based) clustering of sensor networks. • How should information emanate from a sensor network? • How should bandwidth/power be allocated to sensor nodes? • If the final application is detection/classification based on the received data, how should the above change? • Based on looking just at the topology of an optimized network, can we tell something about what the network measures?

4. Information Setup : Central node - Node 0 Task: The final collector of all information, interested in: Case 1: Each piece of data Case 2: The average or sum statistic of data (does sensor i think there is an alien around?) (what is the total number of aliens? – we can also handle other linear combinations, several statistics, etc.) : Sensor node – Node i (i=1,...,N) 1. Measure random variable : Set of nodes that have sent information to node i 2. Compress (quantize + entropy code) and communicate: Case 1: ( compressed version of ) Case 2: (Re-compression is allowed)

5. Wireless Network Setup • We assume we know the capacity matrix between nodes. (bits) constrains the point to point bandwidth between i,j. • Communication happens during well defined time intervals. • We assume the capacity matrix remains unchanged over a reasonable duration (>> one time interval). • Bandwidth is constrained: Node i can send at most bits to node j inside a time interval. • Routing is constrained: Every node i (i=1,...,N) can transmit to at most one other node inside a time interval (fan out = 1). (We can also operate under more general frameworks, under some capacity scaling constraints – please see the routing over a depth-two tree example.)

6. Routing Setup Routing is over a tree 0 depth of routing tree 4 5 2 3 1 6 (Nodes 1,3 send their r.v.’s to node 4, which combines the received information with its r.v. (case 1 or case 2), and sends everything to node 0. ...)

7. Problem Statement: Find the Optimal Information Flow Find the jointly optimal compression, detection (case 1, case 2), and routing strategy for the given: i.e., minimize the total distortion at node 0, subject to constraints. : total distortion observed for case 1. : total distortion observed for case 2. We will find optimal solutions for each case and compare them.

8. Example Deployed nodes Optimal Case 1 routing: Optimal Case 2 routing: ? (>,<,=)

9. Mini FAQ Q: Don’t you need to know the distribution of the r.v. before you compress, do rate allocation, etc.? A: No, we use a good upper bound. Practical (achievable) distortion D for encoding any with variance under rate constraint R<=C: <= D <= Using this bound, optimal rate allocation can be done using the “reverse water-filling theorem”. Q: If case 2, shouldn’t the sensor network always send the linear combination since i.e., isn’t the routing problem trivial? A: No. There is a penalty for collecting information within the sensor network due to capacity constraints. The routing problem is combinatorial in the general case.

10. Toy Scenario (given routing) 0 (i=2,..N), Setup: (a) 1 Intra-network bandwidth is sufficient to achieve exponential improvements. … … N i 2 (b) Intra-network bandwidth is the bottleneck. (Skipping many details, reverse water filling, dropping of coefficients, etc.)

11. How many clusters? N(i)? • Which nodes are the cluster heads? Dynamic Programming ~ Optimal Clustering: Harder Scenario Arbitrary routing tree of depth two, with a fan-in constraint. 0 cluster L cluster 1 ... ... ... N(L) nodes N(1) nodes • Intra-cluster bandwidth for cluster i, (or any function of N(i)) • ’s given, .

12. Harder Scenario contd. ( for W=2.5 and W=5.0, N=40) (W=2.5) Range of exponential gains for case 2. (Beyond this range little penalty for case1 optimal routing even if the actual scenario is case 2.)

13. Optimal Clustering: Hardest Scenario Arbitrary Heuristic, steepest descent algorithm ( Central node is at the center)

14. Hardest Scenario (contd.) ( Central node is at the center)

15. Conclusion • Optimal clustering of capacity constrained wireless sensor networks. • Intra-network bandwidth is very important. Without sufficient intra-network bandwidth, no gains for sending statistics instead of the individual data in case 2. • We can solve a dual problem of network lifetime maximization under the constant fidelity. • We can comply with “scaling laws” and find optimal clusters. • Based on looking just at the topology of an optimized network, can we tell something about what the network does? (image from http://www.sruweb.com/~walsh/neuron.jpg)