1. On the interdependence of routing and data compression in multi-hop sensor networks Anna Scaglione, Sergio D. Servetto

2. Background Broadcast communication Multi-hop sensor network Objective: each node obtains an estimation of the entire field Constraint: distortion < prescribed constant

3. Main Idea Jointly compress data generated by different nodes as this information travels over multiple hops. In order to eliminate correlations in the representation of the sample field.

4. Problem setup N nodes placed on the closed set [0,1]x[0,1]. Each node i observes a sample S i. How can we describe S i ? Rate distortion function R s (D) Correlation between samples.

5. Rate/Distortion Function R S (D) Distortion function defined: d(s,s’) Hamming distortion: 1{s  s’} Square error distortion: (s-s’) 2 Average distortion: E[d(S,S’)]  D Physical meaning Given distribution S and a constant D, what’s the minimum bits we need to represent S so that the mean distortion is less than or equal to D?

6. Correlation Correlation between samples increases as the distance between then in the grid decreases.

7. Problem setup (reminder) N nodes placed on the closed set [0,1]x[0,1]. Each node i observes a sample S i How can we describe S i Rate distortion function R s (D) e Correlation between samples.

8. Why independent Encoders Fail? Assumption: each node encodes its own data independently. Consider a general case when each S i is uniform in [0,1]. Mean-square distortion function.

9. Why independent Encoders Fail? Each node uses a scalar quantizer with B bits of resolution (i.e., step size 2 -B ). Previous result: average distortion is (1/12)x2 -B Total average distortion D= (N/12)x2 -B B=(1/2)log 2 (N/12D) Total information bits: O(N logN)

10. Why independent Encoders Fail? Regardless of the routing strategy, the total information is O(N logN). Amount of bandwidth is O(L N 1/2 ) for a cut of the network, where L is the amount of bandwidth for each node.

11. Routing and Data Compression What have we learnt? Data compression is needed to remove correlation between samples. What are the choices? Distributed Source Coding (not focused) Routing and Source Coding (that’s it!)

12. Routing and Source Coding Idea: Re-encode data at intermediate nodes to remove correlation. Easy to implement than distributed source coding. Use scalar quantizer locally and then forward the data in files compressed using universal source coding algorithms (e.g. Lempel-Ziv).

13. Transmission Time Vs. Compression Ratio Example 1: Amount of traffic: 3H(X 1,X 2,X 3,X 4 ) bits Transmission time: 8 rounds Example 2: Amount of traffic: 2H(X 1,X 2,X 3,X 4 )+H(X 1,X 2 )+H(X 3,X 4 ) bits Transmission time: 4 rounds

14. Questions posted Under what conditions on the statistics of the source can a network transport all the data generated by the sources? What are the tradeoffs between bandwidth requirement and transmission delay?

15. Transport Capacity We know an upper bound: O(L N 1/2 ). Construct a particular flow, calculate the amount of bandwidth needed. Suppose q(S i ) is the quantized version of S i such that E[d(Si,q(S i ))]  D. We need H q (S i ) bits for a source.

16. Transport Capacity Partition the nodes into four groups. Total traffic to go through the cut: 3H(G 1, G 2, G 3, G 4 ) (plus O(N 1/2 ) transmissions to spread data within cuts) Trivial modification for random grid.

17. General Constraints Facts: O(H q (S 1,…,S N )) bits must go across the 4- way cut. Capacity of the 4-way cut is O(LN 1/2 ). From rate/distortion theory, H q (S 1,…,S N )  R S1,…,SN (D), since q is a quantizer with mean distortion D. Condition: O(R S1,…,SN (D))  O(L N 1/2 )

18. The remaining of the story Existence of routing algorithm and source codes that requires no more than O(R S1,…,SN (D)) bits in O( N 1/2 ) transmissions. Proof that O(R S1,…,SN (D))  O(log(N/D) When D/N is kept constant, the field generates a bounded amount of information. If the total distortion is kept constant, the growth of R is only logarithm in N, well below O(L N 1/2 ).