Networked Slepian–Wolf: Theory, Algorithms, and Scaling Laws R˘azvan Cristescu, Member, IEEE, Baltasar Beferull-Lozano, Member, IEEE, Martin Vetterli, Fellow, IEEE IEEE Transactions on Information Theory, Dec., 2005

Outline Introduction – Slepian–Wolf Coding Problem Formulation – Single Sink Case – Multiple Sink Case Single Sink Data Gathering Multiple Sink Data Gathering – Heuristic Approximation Algorithms Numerical Simulations Conclusion

Introduction Independent encoding/decoding Low coding gain Optimal transmission structure: Shortest path tree 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. Distributed source coding: Slepian–Wolf coding – Allow nodes to use joint coding of correlated data without explicit communication Assume a prior knowledge of global network structure and correlation structure is availlable Exploiting data correlation without explicit communication (coding at each node Independent ly) – Node can exploit data correlation among nodes without explicit communication. Optimal transmission structure: Shortest path tree

Slepian–Wolf coding

Problem Single Sink Case Multiple Sink Case Assume the Slepian–Wolf coding is used. Then, (1)Find a rate allocation that minimizes the total network cost. (2) Find an optimal transmission structure.

Preposition Proposition 1: Separation of source coding and transmission structure optimization.

Single-Sink Data Gathering Optimal Transmission Structure: – Shortest Path Tree

Single-Sink Data Gathering Optimization problem Rate Allocation

Proof Consider that with weights Since Thus, assigningYields optimal

Rate Allocation R 1 : the largest R 1 : the smallest

Example

Multiple Sink Case For Node X 3, the optimal transmission structure is the minimum-weight tree rooted at X3 and span the sinks S 1 and S 2. the minimum Steiner tree (NP-complete)

Steiner Tree Euclidean Steiner tree problem – Given N points in the plane, it is required to connect them by lines of minimum total length in such a way that any two points may be interconnected by line segments either directly or via other points and line segments.line segments

Steiner Tree Steiner tree in graphs – Given a weighted graph G(V, E, w) and a subset of its vertices S  V, find a tree of minimal weight which includes all vertices in S Terminal Steiner points

The Minimum Steiner Tree

Existing Approximation If the weights of the graph are the Euclidean distances, – the Euclidean Steiner tree problem – The existing approximation PTAS [3], with approximation ratio (1+  ),  > 0.

Proposed Heuristic Approximation Algorithms Assumption : Nodes that are outside k-hop neighborhood count very little, in terms of rate, in the local entropy conditioning,

Numerical Simulations Source model: multivariate Gaussian random field. Correlation model: an exponential model that decays exponentially with the distance between the nodes.

Numerical Simulations

Conclusions This paper addressed the problem of joint rate allocation and transmission structure optimization for sensor networks. It was shown that – in single-sink case the optimal transmission structure is the shortest path tree. – in the multiple-sink case the optimization of transmission structure is NP-complete. Steiner tree problem