1 Worst and Best-Case Coverage in Sensor Networks Seapahn Meguerdichian, Farinaz Koushanfar, Miodrag...
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Worst and Best-Case Coverage
in Sensor Networks
Seapahn Meguerdichian , Farinaz Koushanfar ,
Miodrag Potkonjak , Mani Srivastava
IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL.4, NO. 1,JANUARY-FEBRUARY 2005
IEEE Infocom 2001, Vol. 3, pp. 1380-1387, April 2001.
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Outlines
IntroductionSensing models and assumptionsCoverage formulationsMaximal BreachMaximal SupportExperimentalConclusion
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Coverage
Coverage can be considered as a measure of the quality of service of a sensor network.
Coverage formulations can try to find weak points in a sensor field suggest future deployment or reconfiguration
schemes for improving the overall quality of service.
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Coverage Problem
Given: Field A S sensors, specified by coordinates Initial(I) and final(F) locations of an agent (I , F)
How well can the field be observed ?
Worst Case Coverage:Find a maximal breach path for an agent moving in A.
Best Case Coverage:Find a maximal support path for an agent moving in A.
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Worst Case Coverage
We want to find the closest distance to sensors that an agent traveling on any path in the sensor field must encounter at least once.
We determine the closest distance to sensors even if the agent tries to optimally avoid the sensors.
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Best Case Coverage
We want to find the farthest distance to sensors that an agent traveling on any path in the sensor field must have from sensors, even if it tries to stay as close to sensors as possible.
At some points, the agent must move away from sensors in order to be able to traverse the field.
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Key Highlight
Transform the difficult to represent coverage problems to discrete-domain optimization using computational geometry(計算幾何 )
and graph theory constructs:
Voronoi DiagramDelaunay Triangulation
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Sensing Model
KpsdpsS
),(),(
We express the general sensing model S at an arbitrary point p for a sensor s as:
where d(s,p) is the Euclidean distance between the sensor s and the point p, and positive constants and K are sensor technology dependent parameters
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Assumption
Sensing effectiveness diminishes as distance increases
Homogeneous sensor nodes Sensor node locations are known Non-directional sensing technology Centralized computation model
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Coverage Formulation
How well can the field be observed ?
Worst Case Coverage: Maximal Breach Path
Best Case Coverage: Maximal Support Path
The “paths” are generally not unique. They quantify the best and worst case observability (coverage) in the sensor field.
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Maximal Breach Path
Given: Field A instrumented with sensors S; areas I and F.
Breach: the minimum Euclidean distance from P to any sensor in S.
Problem: Identify PB, the Maximal Breach Path in
S, starting in I and ending in F.
PB is defined as a path with the property that for any
point p on the path PB, the distance from p to the
closest sensor is maximized.
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Enabling Step: Voronoi Diagram
By construction, each line-segment maximizes distance from the nearest point (sensor).
Consequence: Path of Maximal Breach of Surveillance in the sensor field lies on the Voronoi diagram lines.
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Graph-Theoretic Formulation
Given: Voronoi diagram D with vertex set V and line segment set L and sensors S
Construct graph G(N,E): • Each vertex viV corresponds
to a node ni N
• Each line segment li L
corresponds to an edge ei E
• Each edge eiE, Weight(ei) = Distance of li from closest sensor sk S
Formulation: Is there a path from I to F which uses no edge of weight less than K?
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Finding Maximal Breach Path
Algorithm
1. Generate Voronoi Diagram2. Apply Graph-Theoretic Abstraction3. Search for PB
Check existence of path I --> F using BFS Search for path with maximal, minimum edge weights This is a Maximal Breach Path, PB, and it is not
unique.
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Critical Regions
I FPB
PSsupport_weight breach_weight
30 sensors are deployed at random.
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Bounded Voronoi Diagram
I F
PB
Sensor field with Voronoi Diagram and a Maximal Breach Path.
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Maximal Support Path
Given: Field A instrumented with sensors S; areas I and F.
Support : the maximum Euclidean distance fromthe path P to the closest sensor in S..Problem: Identify Ps, the Maximal Support Path in
S, starting in I and ending in F.
Only requirement: the distance from the farthest point on Ps to the closest sensor is minimized.
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Maximal Support Path
Given: Delaunay Triangulation
of the sensor nodes
Construct graph G(N,E): The graph is dual to the Voronoi
graph previously described
Formulation: what is the path from which the agent can best be observed while moving from I to F? (The path is embedded in the Delaunay graph of the sensors)
Solution: Similar to the max breach algorithm, use BFS and Binary Search to find the shortest path on the Delaunay graph.
I F
PS
Sensor field with Delaunay triangulation and a Maximal Support Path (Ps)
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Maximal Breach Path Example (50 nodes)
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Maximal Breach Path Example (200 nodes)
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Maximal Breach Path – Sensor Deployment
0%
10%
20%
30%
40%
50%
60%
5 10 15 25 30 65 100Number of Sensors
Bre
ach
Im
pro
vem
en
t Add 4
Add 3
Add 2
Add 1
Even after deploying 100 sensors, breach coverage can be improved by about 10 percent by deploying just one more sensor.
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Maximal Support Path – Sensor Deployment
0%
10%
20%
30%
40%
50%
60%
70%
80%
5 10 15 25 30 65 100Number of Sensors
Su
pp
ort
Im
pro
vem
en
t
Add 4
Add 3
Add 2
Add 1
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Asymptotic Behavior
On average, after deploying about 100 sensors, additional random sensors do not improve coverage very significantly.
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Conclusions
Best and Worst case coverage formulations Efficient optimal algorithms using computational
geometry and graph theory Maximal Breach Path (worst-case coverage) Maximal Support Path (best-case coverage)
Applications in: Deployment Asymptotic analysis