Geometric Spanners for Routing in Mobile Networks

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Geometric Spanners for Routing in Mobile Networks

• Jie Gao, Leonidas Guibas,
• John Hershberger, Li Zhang, An Zhu

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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.

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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.

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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
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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
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A Routing Graph Sample
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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.

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Clustering Demo
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Restricted Delaunay Graph

• A RDG
• Is planar (no crossing edges).
• Contains all short Delaunay edges (lt1).
• 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
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Maintaining RDG

1. Compute local Delaunay triangulation.
2. Information propagation.
3. Inconsistency resolution.

as local Delaunay
bs local Delaunay
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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.

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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
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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
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Simulation (Uniform Distribution)
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Simulation (Non-uniform Distribution)
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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) O(n).

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Conclusion

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

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