Using local geometry for Topology Construction in Wireless Sensor Networks

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Using Using local geometrylocal geometry for Topology Construction in for Topology Construction in

Wireless Sensor Networks Wireless Sensor Networks

Sameera PoduriRobotic Embedded Systems Lab(RESL)

http://robotics.usc.edu/reslUniversity of Southern California

Joint work with Prof. Gaurav Sukhatme (RESL, USC), Sundeep Pattem & Prof. Bhaskar Krishnamachari

(ANRG, USC)

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MotivationMotivation

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Different Coverage & Connectivity requirements

local control, global requirements

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ProblemProblem

Given a set of nodes, construct an efficient topology

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Local conditions that influence global network properties

Control instruments- Power control- Sleep scheduling- Position control

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ApproachApproach

• What are the desirable properties? (global/local?)• What topologies have these properties?• Can they be constructed with local rules?• How can we design deployment algorithms to implement these

rules?

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Talk OutlineTalk Outline

• Network properties• Proximity graphs• Local rules for construction• Neighbor-Every-Theta graphs

– Connectivity Properties– Coverage optimization

• Deployment Algorithms• Results• Related Work• Summary & Future directions

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ModelModel

• Communication – binary disk– Different communication ranges

• Coverage – binary disk– Nodes can sense the angle and distance of neighbors

• Very large network• No localization/GPS

Construction Rules

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Network PropertiesNetwork Properties

• Connectivity • Coverage • Sparseness• Degree• Spanner Ratio

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ConnectivityConnectivity

- 0/1 : Path between any two given nodes

- “degree” of connectivity (k-connectivity)

- Path Connectivity = minimum (vertex disjoint) paths between any

two given nodes

- Vertex Connectivity = minimum vertices to disconnect the network

- Edge Connectivity = minimum edges to disconnect the network

Network Properties - 1

Menger’s ThmMenger’s Thm

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Coverage Coverage – Net area “sensed”

DegreeDegree– # neighbors

SparsenessSparseness– #edges = O(#nodes)

Network Properties - 2

| | . | , |P c u v≤

SpannerSpanner – efficiency of paths– , c = spanner ratio

–

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Proximity GraphsProximity Graphs

• Encode spatial arrangement of nodes.• Can model network communication graph

• Popular graphs– Minimum Spanning Tree (MST)– Relative Neighborhood Graph (RNG)– Gabriel Graph (GG)– Delaunay Graph (DG)– Yao Graph (YG)

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PropertiesProperties

– All are connected and sparse

– RNG: low power consumption, low degree and good connectivity

– GG & DG: optimal power spanners

– GPSR derives it’s scalability from the RNG and GG (routing decisions based on local state only)

– YG: low spanner

Proximity Graphs

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RNG: No node closer to both X and Y

GG: No node in the circle of minimum radius passing through X and Y

DG: No node in the circumcircle of X, Y, Z

DefinitionsDefinitions

YG(θ): No node closer than Y in θ sector X

Y

θ

Proximity Graphs







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Hierarchical RelationshipHierarchical Relationship

Proximity Graphs

Average degree,

Connectivity

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ModelModel

• Communication – binary disk– Different communication ranges

• Coverage – binary disk– Nodes can sense the angle and distance of neighbors

• Very large network• No localization/GPS

Construction Rules

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GOAL: Communication graph = Proximity graph

Construction RulesConstruction Rules

Comm. GraphRNG

Problem: Comm Graph is Disk graph

(Only edges < Rc)

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Relative Neighborhood GraphRelative Neighborhood Graph

Theorem1: If each node has at least one neighbor in every 2/3 sector around it, the communication graph is a super-graph of RNG.

Construction Rules

XYY

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RNG…RNG…

2/3 result -

• Sufficient but not necessary

• Best you can do with no global knowledge

• “tight” bound

Construction Rules

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Gabriel GraphGabriel GraphTheorem 2: If each node has at least one neighbor in every θ = arccos(r/R)

sector around it, the communication graph is a super-graph of GG.

Construction Rules

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Delaunay GraphDelaunay Graph• Corollary : If each node has at least one neighbor in every

θ = arccos(r/R) sector around it, the communication graph is a super-graph of DG.

Construction Rules

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Neighbor-Every-ThetaNeighbor-Every-Theta Condition Condition

NET Graph: A graph in which every node satisfies NET condition

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Connectivity of Connectivity of NETNET graph graph

Theorem3: An infinite NET graph is at least 2/ connected for <

Every polygon has at least 3 exterior angles >

NET Graphs

#Edges cut 3 / 2/

#nodes > 2

#nodes = 2

#nodes = 1

#Edges cut 2 2/ - 1 k

#Edges cut 2/

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Connectivity of Connectivity of NET NET graph..graph..

For = , NET graph is guaranteed to be 1-connected

Result by D’Souza et al. *,

If each node has at least one neighbor in every sector around it, then the graph is guaranteed to be connected.

* R. M. D'Souza, D. Galvin, C. Moore, D. Randall. A local topology control algorithm guaranteeing global connectivity and greedy routing. (Working paper)

NET Graphs

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NET NET graphsgraphs

• Each node has at least one neighbor in every sector

Single parameter family of graphs Connectivity ≥ 2/

= 2/3 RNG

NET Graphs

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Coverage OptimizationCoverage Optimization

• Suppose that a node needs k neighbors to satisfy the sector conditions for the proximity graphs

• To maximize coverage from the node’s local perspective:

- All neighbors must lie on the perimeter of the communication range

- They should be placed symmetrically around the node

NET Graphs

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Theorem 3 For , the area coverage is maximized when

the nodes are placed at the edges of disjoint

sectors of .

s cR R=

2

k

πk k

( , )cC X R

NET Graphs

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Tiling GraphsTiling Graphs• When k = 3, 4, 6, the locally optimal symmetric placement can be

replicated globally

• This results in Tiling graphs

NET Graphs

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Tiling Graph propertiesTiling Graph properties

• Globally optimal in terms of coverage

• A number of other global properties:

• While the RNG and GG have spanning ratios of and in general, the spatial arrangement of nodes in the tilings result in constant spanning ratios.

( )O n ( )O n

NET Graphs

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SignificanceSignificance

Traditional approaches -

1. Sleep Scheduling -• network is deployed with high density

• Nodes decide locally whether to stay awake

2. Power Control - • Static & mobile ad-hoc networks

• Smallest transmission power

Deployment 1. Incremental deployment

• Static nodes by a mobile agent

2. Distributed deployment• Self-deployment of mobile nodes

NET Graphs

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Incremental DeploymentIncremental Deployment

• Deploy nodes one at a time

• Pick new position based on geometry of existing nodes, cost of travel, etc

• Can be implemented for mobile nodes too

• Works best when the topology is known a priori

Deployment Algo

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Incremental Deployment - topologies Incremental Deployment - topologies

No Error Gaussian error 3o and 15% range

Non

- til

ing

angl

e(2

/5)

Tili

ng a

ngle

(

/3)

Deployment Algo

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Distributed DeploymentDistributed Deployment

• Nodes make decisions independently• Potential Field Approach

Algorithm• Start state

– all constraints satisfied– all edges are preserved

• Spread out and trim unnecessary edges

Deployment Algo

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Distributed DeploymentDistributed Deployment

2

reprep

KF

d=−

attF =otherwise

If edge is not required

( )/rep attx F F x mυ′′ ′= + − (m=1)

2( )

attrep

r

KF

d Rη=−

−0

Deployment Algo

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SimulationSimulation

• Fast

• No negotiations

• Conservative

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Deployment Algo

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Distributed Deployment - topologies Distributed Deployment - topologies

Incremental No Error Distributed

Non

- til

ing

angl

e(2

/5)

Tili

ng a

ngle

(

/3)

Deployment Algo

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CoverageCoverage

Deployment Results

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ConnectivityConnectivity

Deployment Results

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DegreeDegree

Deployment Results

4

14

12

10

8

6

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Constraint SatisfactionConstraint Satisfaction

Deployment Results

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Comparison with RNG Comparison with RNG

Deployment Results

Comm. graph DifferenceRNG

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Related Work Related Work

• Topology Control: – X. Li’05, Santi’03 (surveys)

• Power Control: – Wattenhofer’05, Brendin’05, Jennings’02, Borbash’02

• Sleep scheduling: – Zhang’05, Wang’03

• Deployment of static network by mobile agent:– Batalin’04, Corke’04

• Deployment of mobile network:– Howard’02, Cortes’04, Poduri’03

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SummarySummary

• NET graphs – based on purely local geometric conditions

– single parameter

– range of coverage-connectivity trade-offs

• Applications – Power control, Sleep scheduling (dense networks)

– Controlled deployment

• Assumptions:– Disk model for communication (but ranges could be different)

– Directional information about neighbors

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ExtensionsExtensions

Relax assumptions:• Irregular communication range• Vary Rs/Rc• Formalize notion of boundary

Deployment Algorithm:• Improve Sparseness• Negotiations? - Coloring• Rendezvous problem

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Documents

Using local geometry for Topology Construction in Wireless Sensor Networks