A Decentralised Coordination Algorithm
for Maximising Sensor Coverage in
Large Sensor Networks
Ruben Stranders, Alex Rogers and Nicholas R. Jennings
School of Electronics and Computer Science
University of Southampton, UK
1
This work is about constructing large
sensor networks
Frequency
assignment
problem
Maintain good
sensor quality
Efficient
(polynomial time)
algorithms
2
These networks consist of many
resource constrained sensing devices
Sensor
1. Deployment
3
These networks consist of many
resource constrained sensing devices
Radio Link
2. Construct
communication network 4
Sensing quality is modelled by a
submodular set function
1
1
3
3
2
Q({1, 3}) – Q({1}) ≥ Q({1, 2, 3}) – Q({1, 2})
Models the diminishing returns of adding a sensor
5
Sensing quality is modelled by a
submodular set function
1
1
3
3
2
Examples (Guestrin 2005):
• Mutual Information
• Area Coverage
• Entropy
6
Frequency allocation is one of the key
challenges
Equivalent to (multi-agent) graph colouring
Communication graph
7
Frequency allocation is one of the key
challenges
Communication graph
8
Frequency allocation is one of the key
challenges
Garbled Reception
Colouring the communication graph is not sufficient
9
Frequency allocation is one of the key
challenges
We need to consider the conflict graph
(Square of the communication graph)
10
Frequency allocation is one of the key
challenges
We need to consider the conflict graph
(Square of the communication graph)
11
The frequency allocation is one of the
key challenges
Multi-agent graph colouring occurs often in sensor networks
e.g. Coordination of sense/sleep cycles
12
Frequency allocation is a difficult
challenge for two reasons
1. Might need many
frequencies
Reduced bandwidth
2. NP-hard problem
or
Poor approximations
13
Requires lots of resources
Our approach deactivates sensors to
simplify the problem
14
Specifically, our approach is to make
the communication graph triangle-free
Arbitrary Graph
Triangle-free Graph
(K3-minor free)
Might need many
colours
Colourable with three
colours
Colouring is NP-hard
Colouring can be found
15
in linear time
Specifically, our approach is to make
the communication graph triangle-free
Arbitrary Graph
Triangle-free Graph
(K3-minor free)
Might need many
colours
Colourable with three
colours
Colouring is NP-hard
Colouring can be found
16
in linear time
Specifically, our approach is to make
the communication graph triangle-free
Triangle-free Graph
(K3-minor free)
Colourable with three
colours
Colouring can be found
17
in linear time
Specifically, our approach is to make
the communication graph triangle-free
Triangle-free Graph
(K3-minor free)
Communication
Graph
Square of
Triangle-free Graph
Conflict Graph
Colourable with three
colours
Colourable with six
colours
Colouring can be found
in linear time
Colouring is easy
18
However, by deactivating sensors, we
lose sensing quality
Sensor
coverage
area 19
However, by deactivating sensors, we
lose sensing quality
Sensing quality is given by submodular function
20
Maximising quality while simplifying
frequency allocation is still NP-hard
Maximise sensing quality
subject to graph being
triangle-free
Maximising submodular
function subject to
p-independence constraint
21
Therefore, we developed two efficient
approximate algorithms
Arbitrary Graph
Triangle-free Graph
22
The centralised algorithm iteratively
selects sensors that improve quality
Each iteration, activate
the sensor that:
• Maximises
quality increase
without
• Creating a triangle
23
The centralised algorithm iteratively
selects sensors that improve quality
24
The centralised algorithm iteratively
selects sensors that improve quality
Step 1
25
The centralised algorithm iteratively
selects sensors that improve quality
Step 2
26
The algorithm terminates when no
remaining sensor can be activated
Can’t add:
creates
triangle!
Can’t select any more sensors.
27
The algorithm terminates when no
remaining sensor can be activated
Can’t select any more sensors.
Done
28
The centralised algorithm achieves at
least 1/7th of the optimal quality
Greedy
Optimal
This follows from submodularity and p-independence29
The centralised algorithm achieves at
least 1/7th of the optimal quality
p-independence
system
Need to remove at
most p sensors
after adding an
arbitrary sensor to
retain trianglefreeness
30
The centralised algorithm achieves at
least 1/7th of the optimal quality
p-independence
system
Need to remove at
most p sensors
after adding an
arbitrary sensor to
retain trianglefreeness
31
The centralised algorithm achieves at
least 1/7th of the optimal quality
p-independence
system
Need to remove at
most p sensors
after adding an
arbitrary sensor to
retain trianglefreeness
p=6
32
The centralised algorithm achieves at
least 1/7th of the optimal quality
Greedily maximising
submodular function
subject to
p-independence
constraint
QG ≥ 1/(1+p) Q*
(Nemhauser, 1978)
QG ≥ 1/7 Q*
33
Using similar techniques, we created a
decentralised algorithm
34
Using similar techniques, we created a
decentralised algorithm
Central Idea
In every triangle
deactivate the
sensor that blocks
the two with
highest quality
1
2
3
4
35
Using similar techniques, we created a
decentralised algorithm
Sensors activate
themselves
asynchronously
1
2
3
4
36
Sensors check if activating themselves
block sensors with higher quality
Sensor checks if
it is part of a
triangle
1
2
3
4
37
Sensors check if activating themselves
block sensors with higher quality
Is the sensor
part of a
triangle?
Yes: we have to
deactivate at
least one of
these
No: the sensor
can remain
active
1
2
3
4
38
Sensors check if activating themselves
block sensors with higher quality
Sensor checks if
its contribution
is smaller than
that of the other
two
1
2
3
4
Q({1, 2}) ≤ Q({2, 3})
and
Q({1, 3}) ≤ Q({2, 3})
39
Sensors check if activating themselves
block sensors with higher quality
Sensor checks if
its contribution
is smaller than
that of the other
two
Q({1, 2}) ≤ Q({2, 3})
and
Q({1, 3}) ≤ Q({2, 3})
1
2
3
4
✓
✓
40
Sensors check if activating themselves
block sensors with higher quality
Sensor checks if
its contribution
is smaller than
that of the other
two
1
2
3
4
If so, it
deactivates itself
41
Sensors check if activating themselves
block sensors with higher quality
1
2
3
4
42
Sensors check if activating themselves
block sensors with higher quality
1
2
3
4
✘
Q({2, 3}) ≤ Q({3, 4})
and
✓
Q({2, 4}) ≤ Q({3, 4})
43
Sensors check if activating themselves
block sensors with higher quality
1
2
3
4
44
The algorithm terminates when the
sensor is no longer part of a triangle
Done45
Both algorithms efficiently compute a
triangle-free network
Original
49
Both algorithms efficiently compute a
triangle-free network
Centralised
50
Both algorithms efficiently compute a
triangle-free network
Decentralised
51
To evaluate the algorithms, we
simulated sensor deployments
0
Unit square
environment
1
0
R
300 sensors
1
0
52
0
Both algorithms provide >70% sensing
quality of the original deployment
Sensing Quality
1
Loss from
restricting
solution
( < 20% )
0.9
0.8
Optimal
Centralised
Decentralised
0.7
Loss from
suboptimal
solution
( < 10% )
0.6
0.1
0.2
0.3
Sensing Radius
0.4
0.5
53
We also considered a dynamic
environment, where sensors can fail
0
When a sensor fails:
1
0
R
Centralised:
rerun algorithm
with remaining
0
sensors
Decentralised:
rerun algorithm
if a neighbour
fails
0
1
55
Both algorithms achieve a coverage
over time close to the optimal
Coverage x Time
1500
Upper bound on
achievable performance
1000
One at a time
Centralised
Decentralised
All active
500
0
0.10
0.20
0.30
Sensing Radius
0.40
0.50
56
In conclusion, our algorithms create
sensor networks with high quality
Simplify the
frequency
assignment
problem
Provide good
sensor quality
Polynomial time
algorithms for
constructing and
colouring
57
57