Approximately Uniform Random Sampling in Sensor Networks
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Computer Science
Approximately Uniform Random Sampling in Sensor
NetworksBoulat A. Bash, John W. Byers and
Jeffrey Considine
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Computer Science
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
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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
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Computer Science
Outline
Exact uniform random sampling Previous work
Approximately uniform random sampling Naïve biased solution Our almost-unbiased algorithm
Experimental validation Conclusions and future work
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Computer Science
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
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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
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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
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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
Voronoi diagram
n=10
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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
n=10
Ugh…Yes!!!
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Rejection-based Sampling
Problem: Minimum cell area may be small Solution: Under-sample some nodes
Let Athreshold ≥ Amin be globally-known cell area1. Route probe to sensor s closest to random (x,y)2. 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
ε)(1εδ,mink
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Computer Science
James Reserve Sensornet
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James Reserve Sensornet
52 sensors
E[#probes] ε δ1.0 (naïve) 4.3 0.69E[#probes] ε δ1.0 (naïve) 4.3 0.69
1.5 0.48 0.462.2 0.12 0.23
E[#probes] ε δ1.0 (naïve) 4.3 0.69
1.5 0.48 0.462.2 0.12 0.233.1 0.041 0.154.1 0.012 0.0385.0 0.0072 0.019
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Synthetic topology
215 sensors randomly placed on a unit
squareE[#probes] ε δ1.0 (naïve) 3.8 0.57
1.3 0.27 0.412.1 0.051 0.153.1 0.017 0.064.0 0.0079 0.0295.0 0.0042 0.017
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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
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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)