Neighborhood-Based Topology Recognition in Sensor Networks

1. Neighborhood-Based Topology Recognition in Sensor Networks S.P. Fekete, A. Kröller, D. Pfisterer, S. Fischer, and C. Buschmann Corby Ziesman

2. Finding Boundaries Useful for: • Keeping tracks of events entering or leaving the region • Communication purposes to the outside • Routing along shortest paths puts increased loads on boundary nodes • Exhausting energy supply • Moderately sized holes caused by failed nodes or obstacles may tend to grow larger and larger

3. Exact Coordinates? • Computing exact coordinates requires use of special hardware like GPS or scanning devices • Limits size and structure of network • Continuous range modulation • Inaccuracies from local measurements accumulate to become significant errors • Can be cumbersome if desiring high accuracy But we can calculate the boundaries without needing the exact coordinates….

4. How to detect boundaries? Assumptions for this approach: • Positioning of nodes is a result of a random distribution • Reasonable node density • Each node can communicate with at least 100 other nodes • Network is overall connected

5. How to detect boundaries? • Communication range of boundary nodes intersects a smaller than average portion of the region • May be natural fluctuations in density, so probabilistic tools are employed • Using a simple local rule to let nodes decide whether they are close to a boundary • Node density μ • Threshold α • Check if number of neighbors falls below αμ • It’s important to get good estimates for the average density μ of fully connected nodes, and determining a good threshold α

6. Determining μ Compute node degree histogram Δ = max neighborhood size μ = average density

7. Determining α • If α is too small, no node will be part of the boundary • As α increases, connected boundary pieces grow until different pieces of the same boundary merge together correctly • If α is too large, false boundaries in low density areas emerge, until eventual the entire network is a single boundary Plateau indicates correct boundaries α can be calculated then through sampling values of α and keeping track of # boundaries

8. Communicating • Using a min spanning tree in a graph with n nodes in a distributed fashion, using only local communication* • Root first queries tree for Δ, and then for historgram… then determines μ_est • Network flood passing on value of αμ_est • Nodes then decide if they belong to a boundary before passing on the flood • Forms connected boundaries by constructing tree (two nodes connected if hop distance is at most 2) • Root assigns resulting tree a unique ID (it’s node ID), which is broadcast • Nodes then determine hop count to closest boundary (nodes equidistant to different boundaries are called Voronoi nodes) • Root sends out message token, recipients decide who to send token to based on smallest common neigborhood • Neighbors not selected to pass token along to exclude themselves from further token passing • After awhile, the root ID is prioritized when searching for the token’s next hop, closing the loop, forming the boundary * R. G. Gallager, P. A. Humblet, and P. M. Spira. A distributed algorithm for minimum-weight spanning trees. ACM Transactions on Programming Languages and Systems, 1983.

9. Inner and Outer Boundaries • If the geometry of the boundaries are not too convoluted, it can be assumed that the outer boundary will be the longest, and consist of the largest number of nodes • Future work may involve taking into account possibility of convoluted inner boundaries by keeping track of curvature along the boundary

10. Example network with inner and outer boundaries • Boundaries are detected by choosing a correct α

11. Conclusions • Topology of a large, dense sensor network is possible without location hardware • Future work may involve taking into account higher-order information of the neighborhood structure to overcome the limitation of requiring high node density • (May lead to routing and energy management)