The Minimal Communication
Cost of Gathering Correlated
Data over Sensor Networks
EL 736 Final Project
Bo Zhang
Motivation: Correlated Data Gathering
Correlated data gathering core
component of many applications, real
life information processes
 Large scale sensor applications
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Scientific data collection: Habitat Monitoring
High redundancy data: temperature, humidity,
vibration, rain, etc.
Surveillance videos
Resource Constraint
Data collection at one or more sinks
 Network: Limited Resources
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Wireless Sensor Networks
 Energy constraint (limited battery)
 Communication cost >> computation cost
 Internet
 Cost metrics: bandwidth, delay etc.
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Problem:
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What is the Minimum total cost (e.g.
communication) to collect correlated data at
single sink?
Model Formalization
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Source Graph: GX
 Undirected graph G(V, E)
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Source nodes {1, 2, …, N }, sink t
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e=(i, j) E — comm. link, weight we
 Discrete Sources: X={ X1, X2, …, XN }
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Arbitrary distribution p( X1=x1, X2=x2, …, XN=xN )
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Generate i.i.d. samples, arbitrary sample rate
Task: collect source data with negligible loss at t
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Sink-t
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Model Formalization: continued
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Linear costs
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g( Re, we ) = Re · we ,  e  E
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Re - data rate on edge e, in bits/sample
we - weight depends on application
For communication cost of wireless links
we  l , 2    4 , l – Euclidean distance
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Goal: Minimize total Cost
Minimal Communication Cost Uncapacitated and data correlation ignored
Link-Path Formulation
ECMP Shortest-Path Routing: Uncapacitated Minimum Cost
indices
d = 1, 2, ...,D
demands
p = 1, 2, ..., Pd paths for demand d
e = 1, 2, ...,E links
constants
hd
δedp = 1
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variables
We
Xdp(w)
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metric of link e, w = (w1, w2, ...,wE)
(non-negative) flow induced by link metric system
w for demand d on path p
minimize
F = Σe WeΣd Σpδedp Xdp(w)
Sink-t
10
volume of demand d
if link e belongs to path p realizing demand d
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constraints
Σp Xdp(w) = hd, d= 1, 2, ...,D
Data correlation –
Tradeoffs: path length vs. data rate
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X2
Routing vs. Coding (Compression)
X1
Shorter path or fewer bits?
Example:
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2
Two sources X1 X2
Three relaying nodes 1, 2, 3
R - data rate in bits/sample
Joint compression reduces redundancy
X2
X1
t
X2
X1
R1
R2
R1
t
3
1
R2
R3<R1+R2
t
Data correlation - Previous Work
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Explicit Entropy Encoding (EEC)
 Joint encoding possible only with side info
 H(X1,X2,X3)= H(X1)+ H(X2|X1)+ H(X3|X1,X2)
 Coding depends on routing structure
X1
 Routing - Spanning Tree (ST)
 Finding optimal ST NP-hard
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Sink-t
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X2
H(X2)
H(X1)
X3
H(X1,X2, X3)
Data correlation - Previous Work (Cont’d)
Slepian-Wolf Coding (SWC):
 Optimal SWC scheme
 routes? Shortest path routing
 rates? LP formulation
(Cristecu et al, INFOCOM04)
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Sink-t
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Correlation Factor
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For each node in the Graph G (V,E), find
correlation factors with its neighbors.
Correlation factor ρuv , representing the correlation
between node u and v.
ρuv = 1 – r / R
R - data rate before jointly compression
r - data rate after jointly compression
Correlation Factor (Cont’d)
Shortest Path Tree (SPT):
Total Cost: 4R+r
 Jointly Compression:
Total Cost: 3R+3r
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All edge weights are 1
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As long as ρ= 1- r/R > 1/2, the SPT is no longer
optimal
Minimal Communication Cost – local data
correlation ：Add Heuristic Algorithm
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Step 0: Initially collecting data at sink t via shortest path. Compute Cost Fi(0) = Σe Ri We, where We is
the weight of link e realizing demand Ri. Set Si(0) = {j’}, where j is the next-hop of node i. i, j = 1,
2… N, i ≠ j . Set iteration count to k = 0. Let Mi denote the neighbors of node i.
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Step 1: For j ∈ Mi\Si(k), do
Fij(k+1) = Fi(k) – RiWij’+RiWij + Σe (Ri – ρij) We
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Step 2: Determine a new j such that
Fij(k+1) = min {Fij(k+1)} < Fi(k).
If there is no such j, go to step 4.
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Step 3: Update
Si(k+1) = {j}
Set Fi(k+1) = Fij(k+1) and k := k + 1 and go to Step 1.
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Step 4: No more improvement possible; stop.
Add Heuristic: example
First Step: Shortest path routing
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After Heuristic:
Sink-t
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When ρij >1/2, j will be
the next hop of i.
Local data correlation: analysis
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Information from neighbors needed
Optimal?
Approximation algorithm
Other factors took into account: energy,
capacity…
Thanks!