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Approximately Uniform Random Sampling in Sensor Networks

This talk discusses the challenges of sampling in sensor networks and presents an approximate unbiased algorithm for uniform random sampling. It also explores ways to reduce messaging costs and extend the algorithm to non-geographic predicates.

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Approximately Uniform Random Sampling in Sensor Networks

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  1. Approximately Uniform Random Sampling in Sensor Networks Boulat A. Bash, John W. Byers and Jeffrey Considine

  2. Introduction • What is this talk about? • Selecting (sampling) a random node in a sensornet • Why is sampling hard in sensor networks? • Unreliable and resource-constrained nodes • Hostile environments • High inter-node communication costs • How do we measure costs? • Total number of fixed-size messages sent per query

  3. Motivation • Sampling makes data aggregation simpler • Approximations to AVG, MEDIAN, MODE • A lot of work on aggregation queries in sensornets • TAG, Cougar, FM Sketches … • Sampling is crucial to randomized algorithms • e.g. randomized routing

  4. Outline • Exact uniform random sampling • Previous work • Approximately uniform random sampling • Naïve biased solution • Our almost-unbiased algorithm • Experimental validation • Conclusions and future work

  5. Sampling Problem • Exact uniform random sampling • Each sensor s is returned from network of n reachable sensors with probability 1/n • Existing solution (Nath and Gibbons, 2003) • Each sensor s generates (rs, IDs) where rs is a random number • Network returns ID of the sensor with minimal rs • Cost: Ө(n) transmissions

  6. Relaxed Sampling Problem • (ε, δ)-sampling • Each sensor s is returned with probability no greater than (1+ε)/n, and at least (1-δ)n sensors are output with probability at least 1/n

  7. Naïve Solution • Spatial Sampling • Return the sensor closest to a random (x,y) • Possible with geographic routing (GPSR 2001) • Nodes know own coordinates (GPS, virtual coords, pre-loading) • Fully distributed; state limited to neighbors’ locations • Cost: Ө(D) transmissions, D is network diameter Yes!!! n=10

  8. Pitfall in Spatial Sampling • Bias towards large Voronoi cells • Definition: Set of points closer to sensor s than any other sensor (Descartes, 1644) • Areas known to vary widely n=10 Voronoi diagram

  9. Removing Bias • Rejection method • In each cell, mark area of smallest Voronoi cell • Only accept probes that land in marked regions • In practice, use Bernoulli trial for acceptance with P[acc] = Amin/As (von Neumann, 1951) • Find own cell area As using neighbor locations (from GPSR) • Need Aavg/Amin probes per sample on average Ugh… Yes!!! n=10

  10. Rejection-based Sampling • Problem: Minimum cell area may be small • Solution: Under-sample some nodes • Let Athreshold≥ Amin be globally-known cell area • Route probe to sensor s closest to random (x,y) • If As < Athreshold, then sensor s accepts Else, sensor accepts with Pr[acc] = Athreshold/As • Athreshold set by user • For (ε, δ)-sampling, set to the area of the cell that is the k-quantile, where • Cost: Ө(cD) transmissions, where c is the expected number of probes

  11. James Reserve Sensornet

  12. James Reserve Sensornet • 52 sensors

  13. Synthetic topology • 215 sensors randomly placed on a unit square

  14. Improving Algorithm • Put some nodes with small cells to sleep • No sampling possible from sleeping nodes • Similar to power-saving schemes (Ye et al. 2002) • Virtual Coordinates (Rao et al. 2003) • Hard lower bound on inter-sensor distances • Pointers • Large cells donate their “unused” area to nearby small cells • When a large cell rejects, it can probabilistically forward the probe to one of its small neighbors

  15. Conclusions • New nearly-uniform random sampling algorithm • Cost proportional to sending a point-to-point message • Tunable (and generally small) sampling bias • Future work • Extend to non-geographic predicates • Reduce messaging costs for high number of probes • Move beyond request/reply paradigm • Apply to DHTs like Chord (King and Saia, 2004)

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