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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

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This work is about constructing large sensor networks

Frequency assignment problem

Maintain good sensor quality

Efficient (polynomial time) algorithms

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These networks consist of many resource constrained sensing devices

Sensor

1. Deployment

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These networks consist of many resource constrained sensing devices

2. Construct communication network

Radio Link

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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

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2

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Sensing quality is modelled by a submodular set function

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

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2

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Frequency allocation is one of the key challenges

Equivalent to (multi-agent) graph colouring

Communication graph

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Frequency allocation is one of the key challenges

Communication graph

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Frequency allocation is one of the key challenges

Garbled Reception

Colouring the communication graph is not sufficient

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Frequency allocation is one of the key challenges

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

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Frequency allocation is one of the key challenges

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

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The frequency allocation is one of the key challenges

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

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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

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Our approach deactivates sensors to simplify the problem

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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)

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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)

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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)

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Colourable with sixcolours

Colouring is easy

Square of Triangle-free Graph

Communication Graph

Conflict Graph

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However, by deactivating sensors, we lose sensing quality

Sensor coveragearea

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However, by deactivating sensors, we lose sensing quality

Sensing quality is given by submodular function

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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

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Therefore, we developed two efficient approximate algorithms

Arbitrary Graph Triangle-free Graph

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The centralised algorithm iteratively selects sensors that improve quality

• Creating a triangle

Each iteration, activate the sensor that:

without

• Maximises quality increase

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The centralised algorithm iteratively selects sensors that improve quality

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The centralised algorithm iteratively selects sensors that improve quality

Step 1

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The centralised algorithm iteratively selects sensors that improve quality

Step 2

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The algorithm terminates when no remaining sensor can be activated

Can’t add:creates triangle!

Can’t select any more sensors.

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The algorithm terminates when no remaining sensor can be activated

DoneCan’t select any more sensors.

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

This follows from submodularity and p-independence

Greedy Optimal

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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

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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

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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

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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)

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Using similar techniques, we created a decentralised algorithm

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Using similar techniques, we created a decentralised algorithm

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

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3 4

Central Idea

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Using similar techniques, we created a decentralised algorithm

Sensors activate themselves asynchronously 1 2

3 4

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Sensor checks if it is part of a triangle

Sensors check if activating themselves block sensors with higher quality

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3 4

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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

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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

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3 4

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and✓

✓

Sensors check if activating themselves block sensors with higher quality

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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

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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

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Sensors check if activating themselves block sensors with higher quality

1 2

3 4

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and✘

✓

Sensors check if activating themselves block sensors with higher quality

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3 4

Q({2, 3}) ≤ Q({3, 4})

Q({2, 4}) ≤ Q({3, 4})

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Sensors check if activating themselves block sensors with higher quality

1 2

3 4

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

Done

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Both algorithms efficiently compute a triangle-free network

Original

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Both algorithms efficiently compute a triangle-free network

Centralised

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Both algorithms efficiently compute a triangle-free network

Decentralised

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0

0

0

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To evaluate the algorithms, we simulated sensor deployments

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1

Unit squareenvironment

R300 sensors

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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

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OptimalCentralisedDecentralised

Loss from restricting solution( < 20% )

Loss from suboptimalsolution( < 10% )

Sens

ing

Qua

lity

Sensing Radius

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0

0

0

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We also considered a dynamic environment, where sensors can fail

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1

R

When a sensor fails:

Centralised: rerun algorithm with remaining sensors

Decentralised: rerun algorithm if a neighbour fails

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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

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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

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