using local geometry for topology construction in wireless sensor networks
DESCRIPTION
Using local geometry for Topology Construction in Wireless Sensor Networks. Sameera Poduri Robotic Embedded Systems Lab(RESL) http://robotics.usc.edu/resl University of Southern California Joint work with Prof. Gaurav Sukhatme (RESL, USC), - PowerPoint PPT PresentationTRANSCRIPT
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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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