1. Global Connectivity from Local Geometric Constraints for Sensor Networks with Various Wireless Footprints Authors: Raissa D’Souza, David Galvin, Cristopher Moor, Dana Randall Venue: IPSN’2006 Presentator: Yunhuai LIU

2. Outline Introduction Knowledge before this paper Global connectivity of G θ Sparseness of G θ When will greedy routing works

3. Adaptive Power Topology Control Save energy by reduced transmission power Connectivity must be preserved

4. Localized Algorithms of θ-graph (G θ ) Use local information to guarantee global connectivity Assume location information or direction information Θ-graph, or G θ Neighbor nodes divide the circle of a node to many sectors with the largest angle < θ θ-constraint θ Key issue: what is the critical value of θ that can guarantee the global connectivity?

5. What We Have Known Before θ<5π/6 By Wattenhofer in Infocom’01 and PODC’01 Under unit disk model The first proposal of APTC θ<π By the same author of this paper in Infocom’03 Again, disk model No boundary effect is considered

6. Three Issue in This Paper Boundary effect and various wireless footprint The sparseness of G θ Geography-based routing on APTC topologies

7. Boundary Effect Boundary nodes and interior nodes Special care of boundary nodes θ B --- θ of interior nodes θ I --- θ of interior nodes Wireless footprint – in contrast with disk model

8. Three Conclusions Guaranteed global connectivity when: 1. Boundary node set is inner connected and θ I < π, with any wireless footprint 2. θ B < 3π/2, θ I < π, and wireless footprint is “week- monotonicity” 3. θ I < π and the average of footprints is approximately to be uniform disks (no θ B constraint)

9. Week-monotonicity It is less restrictive than unit disk model Weak-monotonicity if {I,j} is an edge and k is a node where ∟jik = α and d(i, k) ≤ cos(α)∙d(i, j), then ~ik is also an edge. Disk model (monotonicity)Weak-monotonicity

10. Sparseness of G θ How sparse is G θ : θ= π The expected out-degree = 5 Variance=4 θ= 2π/3 The expected out-degree = 8.875 12.2344

11. Geography-based Routing Footprint eccentricity α Defined as the smallest constant with the property that for every u and v, if u and v are connected, then u is connect to every w such that d(u,w)<d(u,v)/ α

12. When Will Greedy Routing FAIL? When α<2, we can always find a θ so that by satisfying θ-constraint, the network is globally connected When α>2, we can arrange the nodes to let greedy geography-based routing fail

13. Question and Answer Thanks for you patient