Continuous Probabilistic Count Queries in Wireless Sensor Networks

1. ContinuousProbabilistic Count Queriesin Wireless Sensor Networks Anna Follmann, Mario A. Nascimento, Andreas Züfle, Matthias Renz, Peer Kröger, Hans-Peter Kriegel

2. Outline • Motivation • Background • Continuous Count Queries on WSNs • Performance Evaluation • Conclusion

3. Count Query • 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 • How many sensors measure 38°C? Exactly one. But ... what if the sensors produce uncertain data?

4. Probabilistic Count Query • 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? • ...

5. Probabilistic Count and Count Histogram • 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.

6. Data Model and Possible Worlds Semantics • Eachprobabilityvalueis an independent Bernoulli random variable with • 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 kother sensors in S satisfy Q

7. PoissonBinomialRecurrence • 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:

8. Computation of BPR

9. AllowingCertainty • 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

10. ContinuousProbabilistic Count Query in Wireless Sensor Network • 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, ...

11. A CentralizedAlgorithm • All probabilities are sent up to sink in every round • Centralized computation of count histogram in sink node {0.0288, 0.2088, 0.4408, 0.2768, 0.0448} {0.2, 0.8, 0.7, 0.4} {0.8} {0.8, 0.7, 0.4} {0.2} s2 s0 s3 s1 s4 {0.7} {0.4}

12. A CentralizedIncrementalAlgorithm • 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

13. RemovingtheEffect of a previousProbability • Case 1: • Nothing has to be done to remove the effect of • Case 2: • Decrement Counter

14. RemovingtheEffect of a previousProbability • Case 3:

15. IncorporatingtheEffect of a newProbability • Case 1: Nothing has to be done to incorporate the effect of • Case 2: Increment Counter • Case 3:

16. A CentralizedIncrementalAlgorithm • Initialization as in centralized algorithm • Only updates are sent in subsequent rounds {0.0288, 0.2088, 0.4408, 0.2768, 0.0448 }, C=0 {0.0, 0.12,0.56, 0.32, 0.0} {0.12, 0.56, 0.32, 0.0, 0.0}, C=1 {0.7, 0.2} {}, C=1 {0.2} {} {0.7} {}, C=1 s2 s0 s3 s1 s4 {0.7} {}, C=1

17. An In-NetworkAlgorithm • 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 {0.0288, 0.2088, ...} {{0.8, 0.2}, {0.036, 0.252}} {{0.2, 0.8}, {0.3, 0.7}, {0.6, 0.4}} {0.036, 0.252, ...} {0.2, 0.8} s2 s0 {0.8, 0.2} s3 s1 s4 {0.3, 0.7} {0.6, 0.4}

18. An IncrementalIn-NetworkAlgorithm • Initialization as in in-network algorithm • Only updates are sent in subsequent rounds {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: {0.3,0.7} s4: {0.6,0.4} {0.012, 0.56} s2: {0.2,0.8} s3: {}, C=1 s4: {0.6,0.4} {0.036, 0.252} s2: {0.2,0.8} s3: {}, C=1 s4: {0.6,0.4} s0 s2 s1:{0.8, 0.2} {0.2}{} s1:{0.8, 0.2} s3 s1 s4 {0.3, 0.7} {0.3, 0.7} {0.7}, {}, C=1

19. Performance Evaluation – Set up • Positions of sensors randomly chosen • Hop-wise shortest-path tree • 10 simulation runs with 100 timestamps each • Varying parameters

20. Number of Nodes

21. Ratio of Uncertain Sensors

22. Probability of Change

23. Count k

24. MessageSize

25. Conclusions • 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

26. Thanks …

27. A CentralizedAlgorithm ...

28. A CentralizedAlgorithm – AllowingCertainty • Sensors withcansafelybeignored in computation • For Sensors weuse a counter variable Ct {0.12, 0.56, 0.32} Ct=1 {0.8, 0.4} Ct=1 {0.8} {0.8, 0.4} Ct=1 {0.0} s2 s0 {1.0} Ct=1 {0.4} s3 s1 s4

29. A CentralizedAlgorithm – AllowingCertainty C=1

30. A CentralizedIncrementalAlgorithm

31. An In-NetworkAlgorithm