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The Minimal Communication Cost of Gathering Correlated Data over Sensor Networks

The Minimal Communication Cost of Gathering Correlated Data over Sensor Networks. EL 736 Final Project Bo Zhang. Motivation: Correlated Data Gathering.

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The Minimal Communication Cost of Gathering Correlated Data over Sensor Networks

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  1. The Minimal Communication Cost of Gathering Correlated Data over Sensor Networks EL 736 Final Project Bo Zhang

  2. 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

  3. 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.

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

  5. 7 5 9 2 1 Sink-t 12 4 3 6 10 8 11 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

  6. 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 costof wireless links we l , 2    4 , l – Euclidean distance • Goal: Minimize total Cost

  7. 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 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 7 5 9 2 1 Sink-t 12 4 3 10 6 11 8

  8. X2 X1 2 3 1 t X2 X2 X1 R2 X1 R1 R2 R3<R1+R2 R1 t t 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

  9. 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 X1 H(X2) H(X1) X3 H(X1,X2, X3) 7 5 9 2 1 Sink-t 12 4 3 6 10 11 8

  10. Data correlation - Previous Work (Cont’d) • Slepian-Wolf Coding (SWC): • Optimal SWC scheme • routes? Shortest path routing • rates? LP formulation (Cristecu et al, INFOCOM04) 5 9 2 1 Sink-t 12 4 3 6 11 8

  11. 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

  12. 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

  13. 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.

  14. Add Heuristic: example First Step: Shortest path routing 7 5 9 2 1 After Heuristic: When ρij >1/2, j will be the next hop of i. Sink-t 12 4 3 6 10 11 8

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

  16. Thanks!

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