Date post: | 31-Dec-2015 |
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Given a wireless sensor network (WSN), where each node measures a value, how many sensors satisfy a given query predicate?
Example: monitoring a building Count sensors that detect a critical temperature If a threshold is reached turn on air conditioning
ID Location Time °C
s1 Room 101 11:40 50
s2 Room 102 11:40 35
s3 Room 103 11:40 38
s4 Room 203 11:40 40
... ... ... ...3
How many sensors measure 38°C?
Exactly one. But ... what if the sensors produce uncertain data?
Given a WSN and a query Q, each sensor has a probability of satisfying Q.
Probability Distribution Probability that 1 sensor satisfies Q? Probability that 2 sensors satisfy Q? ...
ID Location Time Prob. °C Prob. Humidity Tuple-Probability
s1 Room 101 11:40 0.7 50 0.6 10% 0.2
s2 Room 102 11:40 0.7 35 0.6 20% 0.8
s3 Room 103 11:40 0.7 38 0.6 15% 0.7
s4 Room 203 11:40 0.7 40 0.6 40% 0.4
... ... ... ... ... ... ... ... 4
A probabilistic count query on a set of uncertain sensors S consists of a query Q, a count k and returns a probability value for every k.
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Exactly k sensors in a set of sensors S satisfy Q only if sensor sx in S satisfies Q and k-1 other sensors in S satisfy Q or
sensor sx in S doesn‘t satisfy Q and k other sensors in S satisfy Q
Each probability value is an independent Bernoulli random variable with
Possible World Wk,j Probability P(Wk,j)
W4,1 = {s1,s2,s3, s4} 0.2*0.8*0.7*0.4=0.0448
W3,1 = {s1,s2,s3} 0.2*0.8*0.7*(1-0.4)=0.0672
W3,2 = {s1,s2, s4} 0.2*0.8*(1-0.7)*0.4=0.0192
... ...
W0,1 = {} (1-0.2)*(1-0.8)*(1-0.7)*(1-0.4)=0.0288
We can solve the problem of computing the probabilistic counts efficiently by using the Poisson Binomial Recurrence
and
Exactly k: , at most k: and at
least k:
k
j
SjPSkP0
),(),(
1
0
),(1),(k
j
SjPSkP
),( SkP
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Zero Probability
Adding a zero probability doesn‘t affect the count histogram Zero probabilities can be ignored
One Probability
Adding a one probability shifts all probabilistic counts to the right
Introduction of counter for one probabilities
)\,(),(: xsSjPSjPj
)\,1(),(: xsSjPSjPj
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Wireless Sensor Network Consideration of underlying chracteristics (Topology, Routing,...)
Ultimate goal: reduce communication cost
Time dimension Continuous data stream Values and probabilities change over time t = 0, t = t +1, ...
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All probabilities are sent up to sink in every round Centralized computation of count histogram in sink node
s2
s3 s4
s0
s1
{0.4}{0.7}
{0.8}{0.2} {0.8, 0.7, 0.4}
{0.2, 0.8, 0.7, 0.4}ID Location Time Probability
s1 Room 101 11:40 0.2
s2 Room 102 11:40 0.8
s3 Room 103 11:40 0.7
s4 Room 203 11:40 0.4
)1)(\,()\,1(),(xx sxsx PsSjPPsSjPSjP
{0.0288, 0.2088, 0.4408, 0.2768, 0.0448}
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Initial computation as in centralized algorithm In subsequent rounds only nodes that changed send update Sink node computes new probabilistic counts incrementally
Phase 1 Remove the effect that the previous probability Temporary probabilistic counts
Phase 2 Incorporate new probability Result
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Case 1:Nothing has to be done to incorporate the effect of
Case 2:Increment Counter
Case 3:
),(),(ˆ SjPSjP tt
),(),(ˆ 1 SjPSjP tt
)1)(\,(ˆ)\,1(ˆ),(1,1,
1
txtx sxsxt PsSjPPsSjPSjP
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Initialization as in centralized algorithm Only updates are sent in subsequent rounds
s2
s3 s4
s0
s1
{0.7} {}, C=1
{0.2} {} {0.7} {}, C=1
{0.7, 0.2} {}, C=1
ID Location Time Probability
s1 Room 101 11:40 0.2
s2 Room 102 11:40 0.8
s3 Room 103 11:40 0.7
s4 Room 203 11:40 0.4
ID Location Time Probability
s1 Room 101 11:50 0.0
s2 Room 102 11:50 0.8
s3 Room 103 11:50 1.0
s4 Room 203 11:50 0.4
{0.12, 0.56, 0.32, 0.0, 0.0}, C=1{0.0288, 0.2088, 0.4408, 0.2768, 0.0448 }, C=0
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{0.0, 0.12,0.56, 0.32, 0.0}
All probabilities are sent up to sink in every round Distributed computation of count histogram in every
intermediate node (stopping at k) Component-wise multiplication
s2
s3 s4
s0
s1
{0.6, 0.4}{0.3, 0.7}
{0.2, 0.8}{0.8, 0.2}
{{0.2, 0.8}, {0.3, 0.7}, {0.6, 0.4}}
{{0.8, 0.2}, {0.036, 0.252}}
ID Location Time Probability
s1 Room 101 11:40 0.2
s2 Room 102 11:40 0.8
s3 Room 103 11:40 0.7
s4 Room 203 11:40 0.4
{0.036, 0.252, ...}
{0.0288, 0.2088, ...}
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Initialization as in in-network algorithm Only updates are sent in subsequent rounds
{0.036, 0.252}s2: {0.2,0.8}s3: {0.3,0.7}s4: {0.6,0.4}
s2
s3 s4
s0
s1
{0.3, 0.7}
s1:{0.8, 0.2}
{0.012, 0.56}s2: {0.2,0.8}s3: {}, C=1s4: {0.6,0.4}
{0.0288, 0.2088}s1: {0.8, 0.2}s2: {0.036, 0.252}
{0.0288, 0.2088}s1: {}s2: {0.12, 56}
{0.12, 0.56}s1: {}s2: {0.12, 0.56}
{0.036, 0.252}s2: {0.2,0.8}s3: {}, C=1s4: {0.6,0.4}
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ID Location Time Probability
s1 Room 101 11:40 0.2
s2 Room 102 11:40 0.8
s3 Room 103 11:40 0.7
s4 Room 203 11:40 0.4
ID Location Time Probability
s1 Room 101 11:50 0.0
s2 Room 102 11:50 0.8
s3 Room 103 11:50 1.0
s4 Room 203 11:50 0.4
s1:{0.8, 0.2} {0.2}{}
{0.3, 0.7} {0.7}, {}, C=1
Positions of sensors randomly chosen Hop-wise shortest-path tree 10 simulation runs with 100 timestamps each Varying parameters
Parameter Values
n (Number of Nodes) 100, 500, 1000, 2500
γ (Ratio of Uncertain Sensors) 25%, 50%, 100%
δ (Probability of Change) 25%, 50%, 75%, 100%
k (Count) 1, 5, 10, 25
m (Message Size in bytes) 64, 128, 256
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The incremental in-network algorithm offers the best overall performance among all four investigated approaches In particular when the message size is small, there is a small
probability of updates, large networks and high uncertainty and small values of k
Future work: Other types of aggregate queries, e.g., count Explore correlations between sensor readings Explore different network topologies
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Sensors with can safely be ignored in computation For Sensors we use a counter variable Ct
s2
s3 s4
s0
s1
{0.4}{1.0}Ct=1
{0.8}{0.0} {0.8, 0.4} Ct=1
{0.8, 0.4} Ct=1ID Location Time Probability
s1 Room 101 11:50 0.0
s2 Room 102 11:50 0.8
s3 Room 103 11:50 1.0
s4 Room 203 11:50 0.4
{0.12, 0.56, 0.32} Ct=1
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