1. Minimizing interference for the highway model in Wireless Ad-hoc and Sensor Networks Haisheng Tan, Tiancheng, Lou, Francis C.M. Lau, YuexuanWang, Shiteng Chen CS, The University of Hong Kong, Hong Kong, China ITCS, Tsinghua University, Beijing, China Jan. 25 th, SOFSEM, 2011

2. Outline Introduction Problem Definitions Minimizing the Average Interference Minimizing the Maximum Interference Discussions and Future work Q & A

3. Introduction Wireless Ad hoc and Sensor Networks

4. Introduction Wireless Ad hoc and Sensor Networks Environmental monitoring, intrusion detection, health care, etc. Smart Earth (IBM), Sense China …

5. Introduction Energy !

6. Introduction Energy ! Interference

7. Introduction Energy ! Interference Receiver-centric interference transmission radius of u

8. Problem Definitions the average interference of a graph G the maximum interference of a graph G

9. Problem Definitions the average interference of a graph G the maximum interference of a graph G Problems: Given nodes arbitrarily deployed along a 1D line (the highway model) Connected Min-Avg or Min-max interference The optimal solution is actually a spanning tree.

10. Observations

11. small node degrees

12. Observations small node degrees sparse topology

13. Observations small node degrees sparse topology Nearest Neighbor Forest (each node is connected to its nearest neighbor)

14. Observations small node degrees  sparse topology  Nearest Neighbor Forest (each node is connected to its nearest neighbor)  a) b) c)

15. Minimizing the Average Interference In 2D networks: an asymptotically optimal algorithm with an approximation ratio of O(log n) (Moscibroda et al. 2005)

16. Minimizing the Average Interference In 2D networks: an asymptotically optimal algorithm with an approximation ratio of O(log n) (Moscibroda et al. 2005) In the highway model (Our work): a polynomial-time exact algorithm

17. Minimizing the Average Interference In 2D networks: an asymptotically optimal algorithm with an approximation ratio of O(log n) (Moscibroda et al. 2005) In the highway model (Our work): 1. No-cross property

18. Minimizing the Average Interference In 2D networks: an asymptotically optimal algorithm with an approximation ratio of O(log n) (Moscibroda et al. 2005) In the highway model (Our work): 1. No-cross property when |ac| <=|bc|+|cd| 

19. Minimizing the Average Interference In the highway model: 2. Calculate the total interference via the interference created by each node

20. Minimizing the Average Interference In the highway model: 2. Calculate the total interference via the interference created by each node

21. Minimizing the Average Interference In the highway model: 2. Calculate the total interference via the interference created by each node Independent sub-problems

22. Minimizing the Average Interference Two questions: How to divide the whole line into sub-segments How to connect the nodes inside each segment

23. Minimizing the Average Interference Two questions: How to divide the whole line into sub-segments How to connect the nodes inside each segment Functions for DP F(s,t), s<t, which is short for Compute the minimum total interference created by the nodes from s+1 to t-1, such that

24. Minimizing the Average Interference Two questions: How to divide the whole line into sub-segments How to connect the nodes inside each segment Functions for DP F(s,t), s<t, which is short for OR

25. Minimizing the Average Interference Two questions: How to divide the whole line into sub-segments How to connect the nodes inside each segment Functions for DP F(s,t), s<t, which is short for OR

26. Minimizing the Average Interference Functions for DP G(s,t), s<t Compute the minimum total interference created by nodes from s +1 to t-1, such that

27. Minimizing the Average Interference Functions for DP G(s,t), s<t

28. Minimizing the Average Interference Functions for DP G(s,t), s<t

29. Minimizing the Average Interference Functions for DP G(s,t), s<t The minimum average interference

30. Minimizing the Average Interference Correctness Verified by the brute-force search running in time the maximum node degree

31. Minimizing the Average Interference Correctness Verified by the brute-force search running in time Time complexity: the maximum node degree

32. Minimizing the Average Interference Correctness Verified by the brute-force search running in time Time complexity: (the numbers are the interference created by the nodes) the maximum node degree

33. Minimizing the Average Interference Correctness Verified by the brute-force search running in time Time complexity: (the numbers are the interference created by the nodes) Can we do better ?? Y! the maximum node degree

34. Minimizing the Maximum Interference Harder!! No-cross property: still holds

35. Minimizing the Maximum Interference Harder!! No-cross property: still holds Independent sub-segments: not found 

36. Minimizing the Maximum Interference Harder!! No-cross property: still holds Independent sub-segments: not found  In 2D networks: NP-hard (Buchin 2008) Bounded in

37. Minimizing the Maximum Interference Harder!! No-cross property: still holds Independent sub-segments: not found  In 2D networks: NP-hard (Buchin 2008) Bounded in In 1D networks: An appr. with ratio (von Richenbach, et al. 2005) A sub-exponential-time exact algorithm (Our work )

38. Minimizing the Maximum Interference Check whether the min-max can be k, where 1<k<n

39. Minimizing the Maximum Interference Check whether the min-max can be k, where 1<k<n A skeleton : Record the nodes from s to t that can interfere with nodes outside [s,t] with their transmission radii

40. Minimizing the Maximum Interference Check whether the min-max can be k, where 1<k<n A skeleton : Record the nodes from s to t that can interfere with nodes outside [s,t] with their transmission radii

41. Minimizing the Maximum Interference Check whether the min-max can be k, where 1<k<n A skeleton : Record the nodes from s to t that can interfere with nodes outside [s,t] with their transmission radii

42. Minimizing the Maximum Interference Functions: boolean F*(s,t), which is short for

43. Minimizing the Maximum Interference Functions: boolean F*(s,t), which is short for OR

44. Minimizing the Maximum Interference Functions: boolean F*(s,t), which is short for OR

45. Minimizing the Maximum Interference Functions: boolean G*(s,t)

46. Minimizing the Maximum Interference Functions: boolean G*(s,t)

47. Minimizing the Maximum Interference Functions: boolean G*(s,t)

48. Minimizing the Maximum Interference Functions: boolean G*(s,t) Check the whole line

49. Minimizing the Maximum Interference Time complexity # of the different valid skeletons for a segment from s to t, where s>0 and t<n-1:

50. Minimizing the Maximum Interference Time complexity # of the different valid skeletons for a segment from s to t, where s>0 and t<n-1: Time complexity:

51. Minimizing the Maximum Interference Time complexity # of the different valid skeletons for a segment from s to t, where s>0 and t<n-1: Time complexity: Can we do better? No idea yet 

52. Discussion and Future work Planarity Multiple optimal spanning trees the min-max for the 6-node exponential chain

53. Discussion and Future work Planarity Multiple optimal spanning trees Is min-max in 1D NP-hard? How about 3D networks? How to design efficient approximations to minimize the maximum in 2D networks? How to tackle interference minimization with other network properties, such as small node degree and spanner? …

54. Q & A Thanks!