Geometric Spanners for Routing in Mobile

NetworksJie Gao, Leonidas Guibas, John Hershberger, Li Zhang, An

Zhu

MotivationMotivation:

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.

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.

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

NodeDelaunay

Edge

Empty circle certifies the Delaunay edge

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

A Routing Graph Sample

Select clusterheads

Clusterheads select gateways

RDG on clusterheads & gateways

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.

Clustering Demo

Clusterheads

Clients

Disappearing ClusterheadsNew Appearing Clusterheads

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-edgeLong D-edge

Maintaining RDG1. Compute local Delaunay triangulation.2. Information propagation.3. Inconsistency resolution.

a’s local Delaunay b’s local Delaunay

Maintaining GatewaysClusterheads 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

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

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

Simulation (Uniform Distribution)

Average path length

Maximal path length

Simulation (Non-uniform Distribution)

DiscussionScaling 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).

Conclusion

Restricted Delaunay Graph Good spanner. Efficiently maintainable. Performs well experimentally.

Quality analysis of routing paths Under greedy forwarding Under one-sided perimeter routing

Demo

client Edges in RDGEdges to connect clients to clusterheads

clusterheadgateway