Geometric Spanners for Routing in Mobile Networks

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 Empty circle certifies the Delaunay edge Node Delaunay Edge

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

6. Select clusterheads Clusterheads select gateways RDG on clusterheads & gateways 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. Clusterheads Disappearing Clusterheads Clients New Appearing Clusterheads Clustering Demo

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 • Compute local Delaunay triangulation. • Information propagation. • Inconsistency resolution. a’s local Delaunay b’s local Delaunay

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

12. Quality Analysis of Routing Graphs • Optimal path length = k, • greedy forwarding path length = O(k2), • perimeter routing in the correct side = O(k2). Greedy forwarding Perimeter 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. Average path length Maximal path length Simulation (Uniform Distribution)

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 Edges in RDG client clusterhead Edges to connect clients to clusterheads gateway