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: G X  Undirected graph G(V, E) Source nodes {1, 2, …, N }, sink t e=(i, j)  E — comm. link, weight w e  Discrete Sources: X={ X 1, X 2, …, X N } Arbitrary distribution p( X 1 =x 1, X 2 =x 2, …, X N =x N ) Generate i.i.d. samples, arbitrary sample rate Task: collect source data with negligible loss at t 8 Sink- t

Model Formalization: continued Linear costs  g( R e, w e ) = R e · w e,  e  E R e - data rate on edge e, in bits/sample w e - weight depends on application For communication cost of wireless links w e  l , 2    4, l – Euclidean distance Goal: Minimize total Cost

Minimal Communication Cost - Uncapacitated and data correlation ignored Sink- t 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 volume of demand d δedp = 1 if link e belongs to path p realizing demand d variables We 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 minimize F = Σe WeΣd Σpδedp Xdp(w) constraints Σp 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 X 1 X 2  Three relaying nodes 1, 2, 3  R - data rate in bits/sample  Joint compression reduces redundancy X1X1 X2X2 1 t 3 2 X1X1 X2X2 R1R1 R2R2 t X1X1 X2X2 R1R1 R2R2 R 3 <R 1 +R 2 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 X2X2 X3X3 H(X 1 ) H(X 2 ) Sink- t X1X1 H(X 1,X 2, X 3 )

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- t

Correlation Factor For each node in the Graph G (V,E), find correlation factors with its neighbors. Correlation factor ρ uv, representing the correlation between node u and v. ρ uv = 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 – ρ ij) 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- t 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!