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

Slepian–Wolf coding

Slepian–Wolf coding

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

),...,|( 11 XXXHR NNN

Consider that 11,...,, XXX NN with weights

),(...),(),( 11 SXdSXdSXd STPNSTPNSTP

Since

Thus, assigning ),...,|( 11 XXXHR NNN Yields optimal

),...,|,( 1211 XXXXHRR NNNNN

),...,|(),...,,|(

121

121

XXXHXXXXH

NN

NNN

Rate Allocation

R1: the largest

R1: the smallest

Example

Multiple Sink Case

• For Node X3, the optimal transmission structure is the

minimum-weight tree rooted at X3 and span the sinks S1 and S2.

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.

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.

5

52

6

2

2

3

4

13

2

23 4

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

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