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.
1
Outlines
Introduction
Sensing models and assumptions
Coverage formulations
Maximal Breach
Maximal Support
Experimental
Conclusion
2
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.
3
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.
4
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.
6
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 Diagram
Delaunay Triangulation
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Sensing Model
We express the general sensing model S at an
arbitrary point p for a sensor s as:
S ( s, p ) 

d (s, p)
K
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
8
Assumption
Sensing effectiveness diminishes as
distance increases




Homogeneous sensor nodes
Sensor node locations are known
Non-directional sensing technology
Centralized computation model
9
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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13
14
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 viV corresponds
to a node ni N
• Each line segment li L
corresponds to an edge ei E
• Each edge eiE, 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 Diagram
2. Apply Graph-Theoretic Abstraction
3. 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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17
Critical Regions
I
PB
support_weight
breach_weight
30 sensors are deployed at random.
F
PS
18
Bounded Voronoi Diagram
I
F
PB
Sensor field with Voronoi Diagram and a Maximal Breach Path.
19
Maximal Support Path
Given: Field A instrumented with sensors S; areas I
and F.
Support : the maximum Euclidean distance from
the 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
I
F
PS
Sensor field with Delaunay triangulation and a
Maximal Support Path (Ps)
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
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graph.
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
Add 4
Breach Improvement
60%
Add 3
50%
Add 2
40%
Add 1
30%
20%
10%
0%
5
10
15
25
30
Num be r of Se ns ors
Even after deploying 100 sensors, breach coverage can be
improved by about 10 percent by deploying just one more
sensor.
65
100
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Maximal Support Path – Sensor Deployment
Support Improvement
80%
70%
Add 4
60%
Add 3
50%
Add 2
40%
Add 1
30%
20%
10%
0%
5
10
15
25
30
Num ber of Sensors
65
100
25
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
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