1. Network Correlated Data Gathering With Explicit Communication: NP- Completeness and Algorithms R˘azvan Cristescu, Member, IEEE, Baltasar Beferull-Lozano, Member, IEEE, Martin Vetterli, Fellow, IEEE, Roger Wattenhofer IEEE Transactions on Networking, Feb. 2006

2. Outline Introduction to Compression in Sensor Networks Problem Formulation NP-Completeness Approximation Algorithms Numerical Simulations Conclusion

3. Introduction Independent encoding/decoding Low coding gain Optimal transmission structure: Shortest path tree Distributed source coding: Slepian–Wolf coding – Allow nodes to use joint coding of correlated data without explicit communication Lossless Assume global network structure and correlation structure Without explicit communication (Independent encoding) – Node can exploit data correlation among nodes without explicit communication. Optimal transmission structure: Shortest path tree

4. Slepian–Wolf coding

8. Introduction Encoding with explicit communication – Nodes can exploit the data correlation only when the data of other nodes is locally at them). – Without knowing the correlation among nodes a priori. The objective of this paper Find an optimal transmission structure ? (Minimum Cost Data Gathering Tree Problem)

9. Problem Formulation (Minimum Cost Data Gathering Tree Problem) Let G(V, E) be a weighted graph, where each edge e i  E has a weight w i. Minimum Cost Data Gathering Tree Problem – Given a weighted graph G, find a spanning tree T of G that minimizes

10. Assumptions Assume the coding rates of internal nodes are i i No side information with side information R r + R+2r r r R constant

11. Assumptions i X i is only correlated with the nearest node X j r + R+2r r r R

12. Examples

13. Problem Formulation

14. Case 1:  =0 – Independent data – Shortest path tree Case 2:  =1 – Maximal correlated data – K-TSP problem (multiple traveling salesman) NP-hard

15. NP-Completeness

16. Heuristic Approximation Algorithms 1.Shortest path tree – If data is near independent, this approach is good. 2.Greedy algorithm – Start from an initial subtree containing only the sink. – Add successively, to the existing subtree, the node whose addition results in the minimum cost increment. 3.Simulated Annealing – A provably optimal but computationally heavy optimization method

17. Simulated Annealing

18. Heuristic Approximation Algorithms 4.Balanced SPT/TSP Tree 5.Leaves Deletion Approximation 6.Shallow Light Tree (SLT) [2][5] -- A spanning tree that approximates both the MST and TSP for a given node.

19. Balanced SPT/TSP Tree

21. Optimal Radius

22. Leaves Deletion Algorithm Step 1: construct the global SPT. Step 2: make the leaf nodes change their parent node to some other leaf node in their neighborhood if this change reduces the total cost.

23. Leaves Deletion Algorithm

24. Shallow Light Tree (SLT) Given a graph G(V, E) and a positive number  The SLT has two properties:

25. Numerical Simulations Leaves Deletion(LD) vs. SPT N=200  = 0.9

26. Numerical Simulations N=100  = 0.5

27. Numerical Simulations N=200  = 0.2 SPT LD SPT/TSP

28. Numerical Simulations

29. N=100  = 0.8

30. Numerical Simulations C SLT / C SPT/TSP

31. Conclusions This paper formulates the network correlated data gathering tree problem with coding by explicit communication. This paper proved that the minimum cost data gathering tree Problem is NP-hard, even for scenarios with several simplifying assumptions. Several approximation algorithms are proposed and shown to have significant gains over the shortest path tree.