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 that11,...,, XXX NN with weights

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

Since

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

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

),...,|(

),...,,|(

121

121

XXXH

XXXXH

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