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 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

Wireless Sensor Networks Energy constraint (limited battery) Communication cost >> computation cost

Internet Cost metrics: bandwidth, delay etc.

Problem:

What is the Minimum total cost (e.g. communication) to collect correlated data at single sink?

Model Formalization Source Graph: GX

Undirected graph G(V, E) Source nodes {1, 2, …, N }, sink t e=(i, j) E — comm. link, weight we

Discrete Sources: X={ X1, X2, …, XN }

Arbitrary distribution p( X1=x1, X2=x2, …, XN=xN ) Generate i.i.d. samples, arbitrary sample rate

Task: collect source data with negligible loss at t

8

Sink-t1

2

34

5

6

7

9

10

11

12

Model Formalization: continued Linear costs

g( Re, we ) = Re · we , e E

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

Goal: Minimize total Cost

Minimal Communication Cost -Uncapacitated and data correlation ignored

Sink-t1

2

34

5

6

8

9

11

12

7

10

Link-Path FormulationECMP Shortest-Path Routing: Uncapacitated Minimum Cost

indicesd = 1, 2, ...,D demandsp = 1, 2, ..., Pd paths for demand d e = 1, 2, ...,E links

constantshd volume of demand dδedp = 1 if link e belongs to path p realizing demand d

variablesWe metric of link e, w = (w1, w2, ...,wE)Xdp(w) (non-negative) flow induced by link metric system w for demand d on path p

minimizeF = Σe WeΣd Σpδedp Xdp(w)

constraintsΣp Xdp(w) = hd, d= 1, 2, ...,D

Data correlation – Tradeoffs: path length vs. data rate

Routing vs. Coding (Compression) Shorter path or fewer bits?

Example: Two sources X1 X2

Three relaying nodes 1, 2, 3 R - data rate in bits/sample Joint compression reduces redundancy

X1

X2

1

t

3

2

X1

X2

R1

R2

t

X1

X2

R1

R2

R3<R1+R2

t

Data correlation - Previous Work 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 Routing - Spanning Tree (ST) Finding optimal ST NP-hard

X2

X3

H(X1)H(X2)

Sink-t1

2

34

5

6

7

8

9

10

11

12

X1

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)

Sink-t1

2

34

5

6

8

9

11

12

Correlation Factor

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

As long as ρ= 1- r/R > 1/2, the SPT is no longer optimal

All edge weights are 1

Minimal Communication Cost – local data correlation ： Add Heuristic Algorithm 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.

Step 1: For j Mi\Si(k), do∈ Fij(k+1) = Fi(k) – RiWij’+RiWij + Σe (Ri – ρij) We

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.

Step 3: Update Si(k+1) = {j} Set Fi(k+1) = Fij(k+1) and k := k + 1 and go to Step 1.

Step 4: No more improvement possible; stop.

Add Heuristic: example

First Step: Shortest path routing

Sink-t1

2

34

5

6

9

10

12

7

118

After Heuristic:

When ρij >1/2, j will be the next hop of i.

Local data correlation: analysis Information from neighbors needed Optimal? Approximation algorithm Other factors took into account: energy,

capacity…

Thanks!