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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
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This work is about constructing large sensor networks Frequency assignment problem Maintain good sensor quality Efficient (polynomial time) algorithms 2
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These networks consist of many resource constrained sensing devices 3 Sensor 1. Deployment
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These networks consist of many resource constrained sensing devices 4 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 11 3 3 2 5
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Sensing quality is modelled by a submodular set function Examples (Guestrin 2005): Mutual Information Area Coverage Entropy 11 3 3 2 6
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Frequency allocation is one of the key challenges Equivalent to (multi-agent) graph colouring 7 Communication graph
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Frequency allocation is one of the key challenges 8 Communication graph
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Frequency allocation is one of the key challenges Garbled Reception 9 Colouring the communication graph is not sufficient
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Frequency allocation is one of the key challenges 10 We need to consider the conflict graph (Square of the communication graph)
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Frequency allocation is one of the key challenges 11 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 12 Multi-agent graph colouring occurs often in sensor networks e.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 approximations Requires lots of resources or 13
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Our approach deactivates sensors to simplify the problem 14
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Specifically, our approach is to make the communication graph triangle-free Colourable with three colours Colouring can be found in linear time Might need many colours Colouring is NP-hard Arbitrary Graph Triangle-free Graph (K 3 -minor free) 15
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Specifically, our approach is to make the communication graph triangle-free Colourable with three colours Colouring can be found in linear time Might need many colours Colouring is NP-hard Arbitrary Graph Triangle-free Graph (K 3 -minor free) 16
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Specifically, our approach is to make the communication graph triangle-free Triangle-free Graph (K 3 -minor free) Colourable with three colours Colouring can be found in linear time 17
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Specifically, our approach is to make the communication graph triangle-free Colourable with three colours Colouring can be found in linear time Triangle-free Graph (K 3 -minor free) 18 Colourable with six colours 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 coverage area 19
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However, by deactivating sensors, we lose sensing quality 20 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 21
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Therefore, we developed two efficient approximate algorithms Arbitrary GraphTriangle-free Graph 22
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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 23
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The centralised algorithm iteratively selects sensors that improve quality 24
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The centralised algorithm iteratively selects sensors that improve quality 25 Step 1
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The centralised algorithm iteratively selects sensors that improve quality 26 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. 27
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The algorithm terminates when no remaining sensor can be activated Done Can’t select any more sensors. 28
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The centralised algorithm achieves at least 1/7 th of the optimal quality This follows from submodularity and p-independence Greedy Optimal 29
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The centralised algorithm achieves at least 1/7 th of the optimal quality 30 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 1/7 th of the optimal quality 31 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 1/7 th of the optimal quality 32 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 1/7 th of the optimal quality 33 Greedily maximising submodular function subject to p-independence constraint Q G ≥ 1/(1+p) Q* Q G ≥ 1/7 Q* (Nemhauser, 1978)
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Using similar techniques, we created a decentralised algorithm 34
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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 12 34 35 Central Idea
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Using similar techniques, we created a decentralised algorithm Sensors activate themselves asynchronously 12 34 36
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Sensor checks if it is part of a triangle Sensors check if activating themselves block sensors with higher quality 12 34 37
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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 12 34 No: the sensor can remain active 38
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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 12 34 39
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and ✓ ✓ Sensors check if activating themselves block sensors with higher quality 12 34 Q({1, 2}) ≤ Q({2, 3}) Q({1, 3}) ≤ Q({2, 3}) 40 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 12 34 41 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 12 34 42
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and ✘ ✓ Sensors check if activating themselves block sensors with higher quality 12 34 Q({2, 3}) ≤ Q({3, 4}) Q({2, 4}) ≤ Q({3, 4}) 43
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Sensors check if activating themselves block sensors with higher quality 12 34 44
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The algorithm terminates when the sensor is no longer part of a triangle Done 45
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By deactivating sensors, the network can become disconnected 46
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Radio communication range To attempt to reconnect the network, we boost the radio signals Edge iff sensor is in range 47
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Boost radio range until triangle is created, then reduce Triangle! To attempt to reconnect the network, we boost the radio signals 48
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Both algorithms efficiently compute a triangle-free network Original 49
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Both algorithms efficiently compute a triangle-free network Centralised 50
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Both algorithms efficiently compute a triangle-free network Decentralised 51
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0 0 0 0 To evaluate the algorithms, we simulated sensor deployments 1 1 Unit square environment R 300 sensors 52
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Both algorithms provide >70% sensing quality of the original deployment Loss from restricting solution ( < 20% ) Loss from suboptimal solution ( < 10% ) Sensing Quality Sensing Radius 53
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0 0 0 0 We also considered a dynamic environment, where sensors can fail 1 1 R Battery B ΔB ∝ -R 2 54
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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 55
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Both algorithms achieve a coverage over time close to the optimal Coverage x Time Sensing Radius Upper bound on achievable performance 56
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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 57
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