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MAP: Medial Axis Based Geometric Routing in Sensor Networks

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### MAP: Medial Axis Based Geometric Routing in Sensor Networks

MobiCom’05

Jehoshua Bruck, Jie Gao, Anxiao(Andrew) Jiang

Ku Dara

Contents

- Introduction
- Medial axis
- Medial Axis based naming and routing Protocol (MAP)
- In continuous region in the Euclidean plane
- In discrete sensor field
- Simulation
- Summary

MAP

Introduction(1/2)

- Design of Routing algorithm
- Routing is elementary in all communication networks
- is tightly coupled with auxiliary infrastructure that abstracts the network connectivity
- Stable link & powerful nodes (Internet) : routing table
- Fragile links & constantly changing topologies &nodes with less resouceful h/w (ad-hoc mobile wireless networks) : flooding

Too energy-expensive for sensor networks

Flooding for route discovery

Light infrastructure of sensor networks for efficient and localized routing

Medial Axis

MAP

Introduction(2/2)

- Medial Axis
- Set of points with at least two closest neighbors on the boundaries of the shape
- ‘Skeleton’ of a region
- Capture both geometric and topological features by using the connectivity information
- MAP
- Medial axis based naming and routing protocol as a routing infrastructure
- Depend only connectivity graph
- Consist of 2 protocols
- Madial Axis Construction Protocol(MACP)
- Medial Axis based Routing Protocol(MARP)

MAP

Medial Axis - Definition

- Given a bounded region R, boundary
- A is the collection of points with two or more closest points in
- A cord is a line segment on the medial axis and its closest points on
- A point on the medial axis with 3 or more closest points on is called a medial vertex
- Canonical cell : the medial axis, 2 chords,

MAP

Naming w.r.t. medial axis

- Point p is named by the chord x(p)y(p) it stays on (x(p), y(p), d(p))
- x(p) is a point on the medial axis
- y(p) is the closest point of x(p) on ∂R
- d(p) is height. i.e. relative distance from x(p): |px(p)|/|x(p)y(p)|

Theorem: Every point is given a unique name

MAP

Routing between canonical cells(1/2)

- The naming system naturally builds a Cartesian coordinate system
- x-longitude curve --The chord with medial point x
- h-latitude curve --The collection of points with the height h (0 ≤ h ≤ 1)
- The canonical cells are glued together by the medial axis.
- With the knowledge of the medial axis – route from cells to cells by checking only local neighbor information

MAP

C2

C1

C1

Routing between canonical cells(1/2)- Two canonical cells adjacent to the same medial vertex may not share a chord
- Build rotary systems around medial vertices
- Polar coordinate system: (|ap|/r, ), r is the maximum radius of a ball centered at a medial vertex a

MAP

Routing scheme(1/2)

- Routing is done in 2 steps
- Check the medial axis graph, find a route connecting the corresponding points on the medial axis as guidance
- Realize the route by local gradient descending, in either the Cartesian coordinate system inside a canonical piece, or a polar coordinate system around a medial vertex

MAP

Routing scheme(2/2)

- Routing is done in 2 steps
- Check the medial axis graph, find a route connecting the corresponding points on the medial axis as guidance
- Realize the route by local gradient descending, in either the Cartesian coordinate system inside a canonical piece, or a polar coordinate system around a medial vertex

MAP

MAP in discrete networks – naming(1/5)

- Detect boundaries of the sensor field
- Find sample nodes on boundaries
- By manual identification, or automatic detection [Fekete’04, funke’05]

MAP

MAP in discrete networks – naming(2/5)

- Detect boundaries ( the curve construction problem)
- Use local flooding to connect nearby boundary nodes
- Include nodes on the shortest path between them as boundary nodes

MAP

MAP in discrete networks – naming(3/5)

- Construct the media axis graph
- Detect medial nodes (the sensors with 2 or more closest boundary nodes) by restricted flooding
- Flooding message: Sensor’s ID, boundary, hop count

MAP

MAP in discrete networks – naming(4/5)

- Construct the medial axis graph
- Connect medial nodes into a graph and clean it up
- Remove very short branches

Broadcast this simple graph to all sensors

MAP

MAP in discrete networks – naming(5/5)

- Assign names to sensors for discrete networks
- Replace chords by approximate shortest path trees
- “Medial axis with dangling trees”
- Shortest path forest rooted at the medial axis
- Nodes are assigned names w.r.t. where it lies in the tree

All the computation is

simple and local

MAP

Medial axis based routing

- Medial Axis based Routing Protocol
- Find the shortest path in the medial axis graph A
- Route in parallel to the shortest path
- Route along the shortest path trees rooted at that medial point to reach the destination q
- Guaranteed delivery
- If there is no better choice, route toward the medial axis
- Maintain balanced load
- Try to route in parallel with the medial axis as much as possible to avoid overloading nodes near the medial axis

MAP

Simulation (1/2)

- Outdoor sensor field: Campus(650mX620m) 5735 nodes

The simple medial axis graph: 18nodes, 27edges

MAP

Simulation (2/2)

Routing path comparison

Load balance comparison

MAP

destination

source

GPSR

Normalized standard deviation of traffic load on sensors

MAP

Summary

- MAP
- Topology-enabled naming and routing schemes that based purely on link connectivity information
- Advantage
- Takes only connectivity graph as input
- Infrastructure is lightweight
- Routing is efficient and local

MAP

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