1. Boundary Recognition in Sensor Networks by Topology Methods Yue Wang, Jie Gao Dept. of Computer Science Stony Brook University Stony Brook, NY Joseph S.B. Mitchell Dept. of Applied Math. And Statistics Stony Brook University Stony Brook, NY MobiCom ‘ 06

2. Outline Introduction Topological Boundary Recognition Simulations Conclusions

3. Introduction WSNs are tightly coupled with the geometric environment in which they are deployed WSN applications such as environment monitoring and data collection require sufficient coverage over the region interest Global topology of a WSN plays an important role such as point-to-point routing and data gathering mechanism

4. Introduction This paper focus on the discovering the global geometry of the sensor field including both inner and outer boundaries Understanding the global geometry and topology of the sensor field is important for networking operation The goal of this paper is to find the boundary nodes by using only connectivity information

5. Contributions Developing a practical distributed algorithm for boundary detection using only the communication graph Do not need assume any location, angular or distance information The propose algorithm doesn ’ t require follow the unit disc graph model

6. Contributions The boundary detection algorithm is motivated by an observation that holes in a sensor field create irregularities in hop count distance

7. Topology Boundary Recognition Assumption Sensors are random scattered in a geometric region Nearby nodes communicate with each other directly

8. Topology Boundary Recognition Goal Discovering the nodes on the boundary of the senor field Using only local connectivity information Proposing a distributed algorithm that identifies boundary cycles for the sensor field

9. Topology Boundary Recognition Basic idea Exploit special structure of the shortest path tree to detect the existence of holes Inner holes of the sensor field “ disrupt ” the natural flow of the shortest path tree Shortest paths diverge prior to a hole and then meet after the hole

10. Topology Boundary Recognition Flood the network from an arbitrary node, r. Each node records the minimum hop count to r. Generally prefer to select r as a node on the outer boundary of the sensor field

11. Topology Boundary Recognition Determine the nodes that form the cut, where the shortest paths of distinct homotopy type meet after passing around holes. The nodes of a branch of the cut have their least common ancestor (LCA) relatively far away and their paths to the LCA well separated

12. Topology Boundary Recognition

13. Determine a shortest cycle, R, enclosing the composite hole; R serves as a coarse inner boundary

14. Topology Boundary Recognition Flood the network from the cycle R. Each node in the network records its minimum hop count to R.

15. Topology Boundary Recognition Detect “ extremal nodes ” whose hop counts to R are locally maximal

16. Topology Boundary Recognition Refine the coarse inner boundary R to provide tight inner and outer boundaries. These boundaries are in fact cycles of shortest paths connecting adjacent extremal nodes.

17. Topology Boundary Recognition Undelete the nodes of the removed cut branches and restore the real inner boundary locally, and it can compute the medial axis of the sensor field

18. Topology Boundary Recognition Build a shortest path tree Find cuts in the shortest path tree Detect a coarse inner boundary Find extremal nodes Find the outer boundary and refine the coarse inner boundary Restore the inner boundary The medial axis of the sensor field

19. Build a shortest path tree Flood the network from an arbitrary root node r Each sensor node p sets a timer with a random remaining time When the timer of p reaches 0 then flood and build a shortest tree T(p) The tree with a timer of minimum value starts first and suppresses the other trees

20. Build a shortest path tree When the frontier of T(p) encounters a node q, there are two cases If q does not belong to any other tree, then q is included in the tree T(p) and q will broadcast the packet If q already included in another tree T(p ‘ ), then the start time of T(p) and T(p ’ ) are compared

21. Find cuts in the shortest path tree Definition of a cut pair (p,q) The (hop) distance between p or q and y = LCA(p,q) is above a threshold δ 1 The maximum (hop) distance between a node on the path in T from p to y and the path in T from q to y is above a threshold δ 2

22. Find cuts in the shortest path tree

23. Detect a coarse inner boundary The cut pair p and q will find the shortest paths between them that do not go through any cut node Using any shortest path algorithm to find this path We can use the two shortest paths from q and q to LCA(p,q)

24. Detect a coarse inner boundary

25. Find extremal nodes An extremal node is a node whose minimum hop count to nodes in R is locally maximal

26. Find the outer boundary and refine the coarse inner boundary

27. Restore the inner boundary The final step is to recover the inner holes in the sensor field and find their boundaries For each cut, we find a cut pair (p,q) such that the inclusion of edge pq in the refined inner boundary R

28. The medial axis of the sensor field The medial axis is defined as the set of nodes with at least two closest boundary nodes Medial axis can be used to generate virtual coordinates for efficient greedy routing

29. Simulations Effect of node distribution and density Random distribution of sensors Low density, sparse graphs

30. Random distribution of sensors AVG Degree : 7 AVG Degree : 10 AVG Degree : 13AVG Degree : 16

31. Random distribution of sensors Based on original neighbors Using 2-hop neighbors as fake 1-hop neighbors Using 3-hop neighbors as fake 1-hop neighbors

32. Low density, sparse graphs 3443 nodes AVG degree 35 2628 nodes AVG degree 25 1742 nodes AVG degree 16 842 nodes AVG degree 7

33. Conclusions This paper proposes an algorithm to discover both inner and outer boundary nodes The proposed mechanism only needs the connected graph to apply to the algorithm The simulations show that the proposed algorithm is efficient to discover the boundaries

34. Thank You!!