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 Keeping tracks of events entering or leaving the region Communication purposes to the outside Communication purposes to the outside Routing along shortest paths puts increased loads on boundary nodes Routing along shortest paths puts increased loads on boundary nodes Exhausting energy supply Exhausting energy supply Moderately sized holes caused by failed nodes or obstacles may tend to grow larger and larger 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 Computing exact coordinates requires use of special hardware like GPS or scanning devices Limits size and structure of network Limits size and structure of network Continuous range modulation Continuous range modulation Inaccuracies from local measurements accumulate to become significant errors Inaccuracies from local measurements accumulate to become significant errors Can be cumbersome if desiring high accuracy 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 Positioning of nodes is a result of a random distribution Reasonable node density Reasonable node density Each node can communicate with at least 100 other nodes Each node can communicate with at least 100 other nodes Network is overall connected Network is overall connected

5. How to detect boundaries? Communication range of boundary nodes intersects a smaller than average portion of the region 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 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 Using a simple local rule to let nodes decide whether they are close to a boundary Node density μ Node density μ Threshold α Threshold α Check if number of neighbors falls below αμ 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 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 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 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 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* 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 Root first queries tree for Δ, and then for historgram… then determines μ_est Network flood passing on value of αμ_est Network flood passing on value of αμ_est Nodes then decide if they belong to a boundary before passing on the flood 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) 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 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) 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 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 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 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 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 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 Example network with inner and outer boundaries Boundaries are detected by choosing a correct α Boundaries are detected by choosing a correct α

11. Conclusions Topology of a large, dense sensor network is possible without location hardware 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 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) (May lead to routing and energy management)