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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
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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
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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
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Slepian–Wolf coding
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Slepian–Wolf coding
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Slepian–Wolf coding
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Slepian–Wolf coding
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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.
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Preposition
• Proposition 1: Separation of source coding and transmission structure optimization.
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Single-Sink Data Gathering
• Optimal Transmission Structure: – Shortest Path Tree
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Single-Sink Data Gathering
Optimization problem
Rate Allocation
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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
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Rate Allocation
R1: the largest
R1: the smallest
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Example
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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)
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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.
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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
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The Minimum Steiner Tree
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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.
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Proposed Heuristic Approximation Algorithms
Assumption : Nodes that are outside k-hop neighborhood count very little, in terms of rate, in the local entropy conditioning,
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Numerical Simulations
• Source model: multivariate Gaussian random field.
• Correlation model: an exponential model that decays exponentially with the distance between the nodes.
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Numerical Simulations
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Numerical Simulations
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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