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 1

2. This work is about constructing large sensor networks Frequency assignment problem Maintain good sensor quality Efficient (polynomial time) algorithms 2

3. These networks consist of many resource constrained sensing devices 3 Sensor 1. Deployment

4. These networks consist of many resource constrained sensing devices 4 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 11 3 3 2 5

6. Sensing quality is modelled by a submodular set function Examples (Guestrin 2005): Mutual Information Area Coverage Entropy 11 3 3 2 6

7. Frequency allocation is one of the key challenges Equivalent to (multi-agent) graph colouring 7 Communication graph

8. Frequency allocation is one of the key challenges 8 Communication graph

9. Frequency allocation is one of the key challenges Garbled Reception 9 Colouring the communication graph is not sufficient

10. Frequency allocation is one of the key challenges 10 We need to consider the conflict graph (Square of the communication graph)

11. Frequency allocation is one of the key challenges 11 We need to consider the conflict graph (Square of the communication graph)

12. 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

13. 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

14. Our approach deactivates sensors to simplify the problem 14

15. 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

16. 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

17. 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

18. 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

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

20. However, by deactivating sensors, we lose sensing quality 20 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 21

22. Therefore, we developed two efficient approximate algorithms Arbitrary GraphTriangle-free Graph 22

23. The centralised algorithm iteratively selects sensors that improve quality Creating a triangle Each iteration, activate the sensor that: without Maximises quality increase 23

24. The centralised algorithm iteratively selects sensors that improve quality 24

25. The centralised algorithm iteratively selects sensors that improve quality 25 Step 1

26. The centralised algorithm iteratively selects sensors that improve quality 26 Step 2

27. The algorithm terminates when no remaining sensor can be activated Can’t add: creates triangle! Can’t select any more sensors. 27

28. The algorithm terminates when no remaining sensor can be activated Done Can’t select any more sensors. 28

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

30. 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

31. 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

32. 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

33. 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)

34. Using similar techniques, we created a decentralised algorithm 34

35. 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

36. Using similar techniques, we created a decentralised algorithm Sensors activate themselves asynchronously 12 34 36

37. Sensor checks if it is part of a triangle Sensors check if activating themselves block sensors with higher quality 12 34 37

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 12 34 No: the sensor can remain active 38

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 12 34 39

40. 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

41. 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

42. Sensors check if activating themselves block sensors with higher quality 12 34 42

43. 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

44. Sensors check if activating themselves block sensors with higher quality 12 34 44

45. The algorithm terminates when the sensor is no longer part of a triangle Done 45

46. By deactivating sensors, the network can become disconnected 46

47. Radio communication range To attempt to reconnect the network, we boost the radio signals Edge iff sensor is in range 47

48. Boost radio range until triangle is created, then reduce Triangle! To attempt to reconnect the network, we boost the radio signals 48

49. Both algorithms efficiently compute a triangle-free network Original 49

50. Both algorithms efficiently compute a triangle-free network Centralised 50

51. Both algorithms efficiently compute a triangle-free network Decentralised 51

52. 0 0 0 0 To evaluate the algorithms, we simulated sensor deployments 1 1 Unit square environment R 300 sensors 52

53. 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

54. 0 0 0 0 We also considered a dynamic environment, where sensors can fail 1 1 R Battery B ΔB ∝ -R 2 54

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 55

56. Both algorithms achieve a coverage over time close to the optimal Coverage x Time Sensing Radius Upper bound on achievable performance 56

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