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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