1. Geometric Spanners for Routing in Mobile Networks Jie Gao, Leonidas Guibas, John Hershberger, Li Zhang, An Zhu

2. Motivation Motivation: Efficient routing is difficult in ad hoc mobile networks. Geographic forwarding, e.g., the Greedy Perimeter Stateless Routing (GPSR) protocol, can be used with a location service. GPSR is based on the Relative Neighborhood Graph (RNG) or the Gabriel Graph (GG) for connectivity. Our approach: Restricted Delaunay Graph (RDG). Combined with a mobile clustering algorithm. Good spanner in both Euclidean & Topological distance. Efficient maintenance in a distributed setting.

3. Prior Work Many routing protocols: Table-driven Source-initiated on-demand Greedy Perimeter Stateless Routing (GPSR) by Karp and Kung, Bose and Morin. Clustering in routing: Lowest-ID Cluster Algorithm by Ephremides et al.

4. From Computational Geometry.. Graph Spanner G’ G Shortest path in G’ const optimal path in G Stretch factor Delaunay Triangulation. Voronoi diagram. Empty-circle rule. Good spanner. Voronoi cell Node Delaunay Edge Empty circle certifies the Delaunay edge

5. Construction of the Routing Graph Assume the visible range = disk with radius 1. 1. Clusterheads. 2. Gateways. 3. Restricted Delaunay Graph on clusterheads and gateways. Routing graph = RDG + edges from clients to clusterheads. u v

6. A Routing Graph Sample

7. Mobile Clustering Algorithm 1-level clustering algorithm: Lowest-ID Cluster algorithm. Hierarchical algorithm: proceed the 1-level clustering algorithm in a number of rounds. Constant Density Property for hierarchical clustering: # of clusterheads and gateways in any unit disk is a constant in expectation.

8. Clustering Demo Clusterheads Clients Disappearing Clusterheads New Appearing Clusterheads

9. Restricted Delaunay Graph A RDG Is planar (no crossing edges). Contains all short Delaunay edges (<=1). RDG is a spanner Euclidean Stretch factor: 5.08. Topological spanner. Routing graph is a spanner, too. Both Euclidean & topological distance. Short D-edge Long D-edge

10. Maintaining RDG 1. Compute local Delaunay triangulation. 2. Information propagation. 3. Inconsistency resolution. a ’s local Delaunay b ’s local Delaunay

11. Maintaining Gateways Clusterheads maintain a maximal matching. Update cost = constant time per node. (1)The original maximal matching between clients of two clusterheads. (2)A pair of nodes become invisible. (3)A node leaves the cluster. (4)A new node joins the cluster. Edges in matching Edges in bipartite graph, not in matching

12. Quality Analysis of Routing Graphs Optimal path length = k, greedy forwarding path length = O(k 2 ), perimeter routing in the correct side = O(k 2 ). Greedy forwardingPerimeter routing

13. Simulation (Uniform Distribution) 300 random points. Inside a square of size 24. Visible range: radius-2 disk. 1-level clustering algorithm. RNG v.s. RDG under GPSR protocol. Static case only. RNG RDG

14. Simulation (Uniform Distribution) Average path lengthMaximal path length

15. Simulation (Non-uniform Distribution)

16. Discussion Scaling vs. spanner property Cannot be achieved at the same time. Efficiency of clustering No routing table. Update cost: constant per node. Changes happen only when topology changes. Forwarding cost RDG: constant. RNG (or GG): Ω(n).

17. Conclusion Restricted Delaunay Graph Good spanner. Efficiently maintainable. Performs well experimentally. Quality analysis of routing paths Under greedy forwarding Under one-sided perimeter routing

18. Demo client Edges in RDG Edges to connect clients to clusterheads clusterhead gateway