1. Iso-Contour Queries and Gradient Descent with Guaranteed Delivery in Sensor Networks Rik Sarkar, Xianjin Zhu, Jie Gao, Joseph S. B. Micchell, Leonidas J. Guibas IEEE INFOCOM 2008 Speaker: Chen-Yan Wu

2. Outline Introduction Proposed Mechanism Simulation Conclusion

3. Introduction Wireless sensor networks have shown great potential for providing dense monitoring and sensing capabilities with modest cost and management effort. In the application of environmental monitoring, sensors measure readings of the physical space, such as chemical concentration. An iso-contour at an value x is the collection of points with value equal to x. iso-contour 90 80 70 60 50 40 3020

4. Introduction – Scenario The iso-contours encode spatial structures of the signal field, such as boundaries of the ‘hot’ regions. Users with hand-held devices communicate with nearby sensors to obtain directions to places indicated by the sensor data being within a specified range.  Iso-contour query  Value-restricted routing q Dest. 90 80 70 60 50 40 3020

5. Introduction – Challenge Simple gradient descending/ascending routing can typically lead the query message to one iso-contour, unless the query message reaches a local minimum or local maximum, in which case the query gets stuck. q Dest.

6. Overview Preprocessing  Local identification for local maximum and local minimum  Sweep for saddle point  Distributed construction of contour tree + ˙ - local maximumlocal minimum saddle point + + - - - ˙ ˙ a b c e d f g + + - - - ˙ a b c e f ˙ d g

7. Definition  Given a continuous signal field F, and a node id q with value F (q) = x F (q) =70 q 90 80 70 60 50 40 3020 the highest temperature of interior the lowest temperature of interior the highest temperature of exterior the lowest temperature of exterior

8. Preprocessing Local identification for local maximum and local minimum  A node identifies itself as a local maximum if it discovers that all its 1-hop neighbors have value no greater than itself. It then initiates a sweep top down. And vise versa. + - local maximum local minimum + + - - - a b e f g + + i h

9. Preprocessing Sweep  Each sweep is initiated and labeled by a critical node (a maximum, minimum or a saddle node).  In the sweep initiated by a local maximum a, the sweep message carries the tuple (a, F (a)). + - local maximum local minimum + + - - - a b e f g + + i h

10. Preprocessing Sweep – saddle points  If a node gets two sweep messages from different local maximum (minimum), this indicates that two contour components start to merge. Thus a saddle point should be identified. + ˙ - local maximum saddle point local minimum + + - - - a b e f g + + i h ˙ c ˙ d (a, F (a)) (b, F (b))

11. Preprocessing Sweep – saddle point  The sweep messages, the tuple (a, F (a)) and (b, F (b)), encounter at node c. The sweep message will be changed as (c, F (c), M(a,b)).  The sweep messages, the tuple (f, F (f)) and (g, F (g)), encounter at node c. The sweep message will be changed as (d, F (d), S(f,g)). + + + ˙ - - - - ˙ a b c e f ˙ d g local maximum saddle point local minimum (c, F (c), M(a,b)) + + h i

12. Preprocessing Distributed construction of contour tree  a node q with value F (q)= x  After node a, b broadcast.  After node c broadcasts.  After node e, f, g broadcast.  After node d broadcasts. + + + ˙ - - - - ˙ a b c e f ˙ d g local maximum saddle point local minimumq ˙ c - -- e f g + + - - - ˙ ˙ a b c e d f g ++ a b ˙ d

13. Proposed Mechanism Iso-contour queries  Gradient descent routing for iso-contour queries Users at node q want to find value x x=40 + + - - - ˙ ˙ a=80 b=60 c=30 e=20 d=30 f=20 g=20 + + - - - ˙ a b c e f ˙ d g q t s F (s)=30

14. Proposed Mechanism Value restricted routing  Given a source s and destination t, find a path P from s to t such that at every node x on P, y ≤ F (x) ≤ z. z=40 + + - - - ˙ ˙ a=80 b=60 c=30 e=20 d=30 f=20 g=20 y=20 + + - - - ˙ a b c e f ˙ d g q t s F (s)=30

15. Simulation Parameters  1600 nodes  16 by 16 units square region with unit disk graph as the communication model  The average number of neighbors per node is about 21.  The sensors sample from a continuous signal field shown below. Elevation map of West Reno (obtained from usgs.gov) and its sampling

16. The message complexity of contour tree construction

17. The CDF of the node load distribution

18. Conclusion Proposed the distributed construction of a contour tree and its application in iso-contour queries by gradient routing with guaranteed delivery.