1

A Decentralised Coordination Algorithm for Maximising Sensor Coverage in

Large Sensor NetworksRuben Stranders, Alex Rogers and Nicholas R. Jennings

School of Electronics and Computer ScienceUniversity of Southampton, UK

2

This work is about constructing large sensor networks

Frequency assignment problem

Maintain good sensor quality

Efficient (polynomial time) algorithms

3

These networks consist of many resource constrained sensing devices

Sensor

1. Deployment

4

These networks consist of many resource constrained sensing devices

2. Construct communication network

Radio Link

5

Sensing quality is modelled by a submodular set function

Q({1, 3}) – Q({1}) ≥ Q({1, 2, 3}) – Q({1, 2})Models the diminishing returns of adding a sensor

1 1

33

2

6

Sensing quality is modelled by a submodular set function

Examples (Guestrin 2005):• Mutual Information• Area Coverage• Entropy

1 1

33

2

7

Frequency allocation is one of the key challenges

Equivalent to (multi-agent) graph colouring

Communication graph

8

Frequency allocation is one of the key challenges

Communication graph

9

Frequency allocation is one of the key challenges

Garbled Reception

Colouring the communication graph is not sufficient

10

Frequency allocation is one of the key challenges

We need to consider the conflict graph(Square of the communication graph)

11

Frequency allocation is one of the key challenges

We need to consider the conflict graph(Square of the communication graph)

12

The frequency allocation is one of the key challenges

Multi-agent graph colouring occurs often in sensor networkse.g. Coordination of sense/sleep cycles

13

Frequency allocation is a difficult challenge for two reasons

1. Might need many frequencies

Reduced bandwidth

2. NP-hard problem Poor approximationsRequires lots of resources

or

14

Our approach deactivates sensors to simplify the problem

15

Specifically, our approach is to make the communication graph triangle-free

Colourable with threecolours

Colouring can be foundin linear time

Might need many colours

Colouring is NP-hard

Arbitrary Graph Triangle-free Graph(K3-minor free)

16

Specifically, our approach is to make the communication graph triangle-free

Colourable with threecolours

Colouring can be foundin linear time

Might need many colours

Colouring is NP-hard

Arbitrary Graph Triangle-free Graph(K3-minor free)

17

Specifically, our approach is to make the communication graph triangle-free

Triangle-free Graph(K3-minor free)

Colourable with threecolours

Colouring can be foundin linear time

Specifically, our approach is to make the communication graph triangle-free

Colourable with threecolours

Colouring can be foundin linear time

Triangle-free Graph(K3-minor free)

18

Colourable with sixcolours

Colouring is easy

Square of Triangle-free Graph

Communication Graph

Conflict Graph

19

However, by deactivating sensors, we lose sensing quality

Sensor coveragearea

20

However, by deactivating sensors, we lose sensing quality

Sensing quality is given by submodular function

21

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

22

Therefore, we developed two efficient approximate algorithms

Arbitrary Graph Triangle-free Graph

23

The centralised algorithm iteratively selects sensors that improve quality

• Creating a triangle

Each iteration, activate the sensor that:

without

• Maximises quality increase

24

The centralised algorithm iteratively selects sensors that improve quality

25

The centralised algorithm iteratively selects sensors that improve quality

Step 1

26

The centralised algorithm iteratively selects sensors that improve quality

Step 2

27

The algorithm terminates when no remaining sensor can be activated

Can’t add:creates triangle!

Can’t select any more sensors.

28

The algorithm terminates when no remaining sensor can be activated

DoneCan’t select any more sensors.

29

The centralised algorithm achieves at least 1/7th of the optimal quality

This follows from submodularity and p-independence

Greedy Optimal

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 triangle-freeness

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 triangle-freeness

32

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 triangle-freeness

p = 6

33

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*

QG ≥ 1/7 Q*

(Nemhauser, 1978)

34

Using similar techniques, we created a decentralised algorithm

35

Using similar techniques, we created a decentralised algorithm

In every triangle deactivate the sensor that blocks the two with highest quality

1 2

3 4

Central Idea

36

Using similar techniques, we created a decentralised algorithm

Sensors activate themselves asynchronously 1 2

3 4

37

Sensor checks if it is part of a triangle

Sensors check if activating themselves block sensors with higher quality

1 2

3 4

38

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

1 2

3 4No: the sensor can remain active

39

Sensor checks if its contribution is smaller than that of the other two

Q({1, 2}) ≤ Q({2, 3})

Q({1, 3}) ≤ Q({2, 3})and

Sensors check if activating themselves block sensors with higher quality

1 2

3 4

40

and✓✓

Sensors check if activating themselves block sensors with higher quality

1 2

3 4

Q({1, 2}) ≤ Q({2, 3})

Q({1, 3}) ≤ Q({2, 3})

Sensor checks if its contribution is smaller than that of the other two

41

If so, it deactivates itself

Sensors check if activating themselves block sensors with higher quality

1 2

3 4

Sensor checks if its contribution is smaller than that of the other two

42

Sensors check if activating themselves block sensors with higher quality

1 2

3 4

43

and✘✓

Sensors check if activating themselves block sensors with higher quality

1 2

3 4

Q({2, 3}) ≤ Q({3, 4})

Q({2, 4}) ≤ Q({3, 4})

44

Sensors check if activating themselves block sensors with higher quality

1 2

3 4

45

The algorithm terminates when the sensor is no longer part of a triangle

Done

49

Both algorithms efficiently compute a triangle-free network

Original

50

Both algorithms efficiently compute a triangle-free network

Centralised

51

Both algorithms efficiently compute a triangle-free network

Decentralised

52

0

0

0

0

To evaluate the algorithms, we simulated sensor deployments

1

1

Unit squareenvironment

R300 sensors

53

Both algorithms provide >70% sensing quality of the original deployment

0.1 0.2 0.3 0.4 0.50.600000000000001

0.700000000000001

0.800000000000001

0.900000000000001

1

OptimalCentralisedDecentralised

Loss from restricting solution( < 20% )

Loss from suboptimalsolution( < 10% )Se

nsin

g Q

ualit

y

Sensing Radius

55

0

0

0

0

We also considered a dynamic environment, where sensors can fail

1

1

R

When a sensor fails:Centralised: rerun algorithm with remaining sensors

Decentralised: rerun algorithm if a neighbour fails

56

Both algorithms achieve a coverage over time close to the optimal

0.10 0.20 0.30 0.40 0.500

500

1000

1500One at a timeCentralisedDecentralisedAll active

Cove

rage

x T

ime

Sensing Radius

Upper bound on achievable performance

57

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