A Decentralised Coordination Algorithm for Maximising Sensor Coverage in Large Sensor Networks

1. 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

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 Radio Link 2. Construct communication network

5. 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

6. Sensing quality is modelled by a submodular set function 1 1 3 3 2 • Examples (Guestrin 2005): • Mutual Information • Area Coverage • Entropy

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 networks e.g. Coordination of sense/sleep cycles

13. Frequency allocation is a difficult challenge for two reasons 1. Might need many frequencies Reduced bandwidth Poor approximations 2. NP-hard problem or Requires lots of resources

14. Our approach deactivates sensors to simplify the problem

15. Specifically, our approach is to make the communication graph triangle-free Triangle-free Graph (K3-minor free) Arbitrary Graph Colourable with threecolours Might need many colours Colouring is NP-hard Colouring can be foundin linear time

16. Specifically, our approach is to make the communication graph triangle-free Triangle-free Graph (K3-minor free) Arbitrary Graph Colourable with threecolours Might need many colours Colouring is NP-hard Colouring can be foundin linear time

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

18. Specifically, our approach is to make the communication graph triangle-free Triangle-free Graph (K3-minor free) Square of Triangle-free Graph Conflict Graph Communication Graph Colourable with threecolours Colourable with six colours Colouring can be foundin linear time Colouring is easy

19. However, by deactivating sensors, we lose sensing quality Sensor coverage area

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 Each iteration, activate the sensor that: • Maximises quality increase without • Creating a triangle

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 Done Can’t select any more sensors.

29. The centralised algorithm achieves at least 1/7th of the optimal quality Greedy Optimal This follows from submodularity and p-independence

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* (Nemhauser, 1978) QG ≥ 1/7 Q*

34. Using similar techniques, we created a decentralised algorithm

35. Using similar techniques, we created a decentralised algorithm Central Idea In every triangledeactivate the sensor that blocks the two with highest quality 1 2 3 4

36. Using similar techniques, we created a decentralised algorithm Sensors activate themselves asynchronously 1 2 3 4

37. Sensors check if activating themselves block sensors with higher quality Sensor checks if it is part of a triangle 1 2 3 4

38. Sensors check if activating themselves block sensors with higher quality Is the sensor part of a triangle? 1 2 Yes: we have to deactivate at least one of these No: the sensor can remain active 3 4

39. 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 Q({1, 2}) ≤ Q({2, 3}) 3 4 and Q({1, 3}) ≤ Q({2, 3})

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 ✓ Q({1, 2}) ≤ Q({2, 3}) 3 4 and ✓ Q({1, 3}) ≤ Q({2, 3})

41. 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 If so, it deactivates itself 3 4

42. Sensors check if activating themselves block sensors with higher quality 1 2 3 4

43. Sensors check if activating themselves block sensors with higher quality 1 2 ✘ Q({2, 3}) ≤ Q({3, 4}) 3 4 and ✓ 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

46. Both algorithms efficiently compute a triangle-free network Original

47. Both algorithms efficiently compute a triangle-free network Centralised

48. Both algorithms efficiently compute a triangle-free network Decentralised

49. To evaluate the algorithms, we simulated sensor deployments 0 0 0 Unit square environment 1 R 300 sensors 0 1

50. Both algorithms provide >70% sensing quality of the original deployment Loss from restricting solution ( < 20% ) Sensing Quality Loss from suboptimal solution ( < 10% ) Sensing Radius