1. Data Transmission and Base Station Placement for Optimizing Network Lifetime. E. Arkin, V. Polishchuk, A. Efrat, S. Ramasubramanian,V. PolishchukA. EfratS. Ramasubramanian J. TaheriJ. Taheri, J. Mitchell, S. SankararamanJ. MitchellS. Sankararaman 1

2. Motivation. H & K algorithm for finding a Maximum Matching. Basic principles. Computing an optimal forwarding protocol. Optimizing the location of the base station. 2

3. Motivation: 3

4. Motivation. Hopcroft & Karp algorithm for finding a Maximum Matching. Basic principles. Computing an optimal forwarding protocol. Optimizing the location of the base station. 4

5. Preliminaries A matching in a bipartite graph is a subset of edges such that every vertex is incident to at most one edge in the subset. 5

6. Maximum matching A maximum cardinality matching in G can be found in time. 6

7. An augmenting path Let M be a Matching, P is an augmenting path of M if: P is an odd length path. P starts and ends with an unmatched vertex. Every other edge is not in M. 7

8. Hopcroft & Karp algorithm U V 8 back

9. H & K First BFS stage U V 9

10. - A subset of unmatched vertices U V 10

11. H & K First DFS stage U V 11

12. Second iteration stage U V 12

13. Second iteration stage U V 13

14. H & K Runtime Lemma 1: – The length of the shortest augmenting path increases in each phase. 14

15. H & K Runtime Lemma 2: – Let M* be a maximum matching. – Let M be a matching. – If M* have K edges more than M, there are K vertex disjoint augmenting paths for M. 15

16. Lemma 2 – cont. – In the union of M* and M we can have: 1. even length circles. 2. even length paths. 3. odd length paths. 16

17. Even length circle and path. U V 17

18. An odd length path U V 18

19. An odd length path U V 19

20. Lemma 2- cont. We have to consider only the third group which are augmenting paths. – Each of the other groups doesn’t change the balance between M and M* edges. All these paths are vertex disjoint and contains one more M*’s edge than M’s. – The degree of each vertex in the union of M and M* is at most 2. – There is no augmenting path for M*. The number of such paths is k. 20

21. The Runtime of H & K After phases: – By lemma 1- all the paths of length less than are already discovered. – By lemma 2- we have more vertex disjoint augmenting paths. – Therefore, we get – In each step the remaining number of augmenting paths is decreased by 1. – After at most phases we will get M*. 21

22. The Runtime of H & K Hence, there are at most phases. Each phase takes. In total we get. (BFS+DFS) 22

23. -Matching Let be a vector of integers with a component,, associated with each vertex. A - matching is a subset of edge such that each vertex is incident to at most edges of. b b b b(v) 23

24. Extension of the H & K algorithm We extend the H & K algorithm for finding a maximum cardinality - matching. A vertex is “exposed” if it’s degree in M is less than. An “exposed” vertex is equivalent to unmatched vertex in the original H & K. The extended H & K takes also. b b(v) 24

25. Maintaining a maximum - matching Let M be a maximum matching. is increased by 1. We can update M efficiently to get the maximum - matching for the modified. If an augmenting path exists, it must start from v. Therefore one BFS+DFS phase is enough. b b b b b(v) 25

26. Maintaining a maximum - matching Let M be a maximum matching. is decreased by 1. If M is still Maximum matching, we done. Otherwise remove any edge (v,u) from M. If an augmenting path exists, it must start from u. Therefore one BFS+DFS phase is enough. b b b(v) 26

27. Motivation. H & K algorithm for finding a Maximum Matching. Basic principles. Computing an optimal forwarding protocol. Optimizing the location of the base station. 27

28. Multi hop network topology 28

29. Price-part 1 Which is more expensive? 29

30. Which is more expensive? Price-part 2 30

31. More realistic model 31

32. Simultaneously If we want that the cameras will be able to communicate they have to be awake in same time. 32

33. Lifetime 33

34. Definitions of lifetime The lifetime: – Starts from: The system starts operating. – Until: One sensor is dead. Some percentage of the sensors is dead. All sensors in some specific subset are dead. 34

35. Definitions : a set of n points in the plane, each point represents a sensor. : a point in the plane, represents the base station. 35

36. The sensors The sensors are responsible for monitoring and gathering information. Each sensor can pass it’s information directly to the base station or vs. another sensor. The sensors can’t split their information. 36

37. The base station The base station is responsible for processing the data gathered from the sensors. 37

38. Energy Each sensor has a battery with limited capacity. For simplicity we assume that the capacity is 1. No cost for receiving. For sending: – Transmitting data a distance D requires- energy when. 38

39. The followers Followers are sensors that passing their information vs. other sensors. The energy that required for such a sensor to pass it’s information defined by when is the Euclidian distance between the follower to the receiving sensor. 39

40. The leaders Passes information directly to the base station. Can pass also information from other sensors. Required energy: – is the distance between to the base station. – is the degree. 40

41. The 2-tree g is the root. Leaders are on the first level. Followers are on the second. The links from followers to leaders and from leaders to the base station are the edges. g Level 1: 41

42. Motivation. H & K algorithm for finding a Maximum Matching. Basic principles. Computing an optimal forwarding protocol. Optimizing the location of the base station. 42

43. Optimal forwarding protocol The problem: Find such that and 43

44. Splitting to followers and leaders Case 1: – S can’t transmit to g. – S have to be a follower. Case 2: – S can transmit directly to g. – S have to be a leader. 44

45. Capacity Capacity of a leader- l: Capacity of a follower- f: 45

46. The graph We define: We have an edge between and iff. 46

47. Crucial observation Finding the optimal protocol is equivalent to computing maximum cardinality matching in G. A reminder. 47

48. Running time According to Hopcroft-Karp – By exploiting the geometry of the problem - 48

49. An improved analysis We can reduce the time of the BFS to. As before BFS+DFS phase invoked times. – DFS takes time. – Improved BFS takes time. In total we have time. 49

50. The improved BFS From followers to leaders: – Start from all unmatched followers. – Find all leaders that these followers can transmit to. – The method of finding the leaders contains: Defining a unit disk for each leader. For each unmatched follower searching for the legal leaders. 50

51. The improved BFS For each unmatched follower, f: – Check whether the disk centered at one of the leaders contains f. – If there is such disk: Add the corresponding edge to the matching and remove it’s disk. Go back to the first line. – Otherwise, continue to the next unmatched follower. 51

52. The improved BFS From leaders to followers : – Follow the matched edges from the reached vertices of L. 52

53. The improved BFS- Complexity Discovering and deleting each disk takes time ( Efrat et al ). We can discover up to n disks, and each discovered disk can be deleted only once. Therefore, the total complexity of “BFS+DFS” level is. 53

54. The continues model Each sensor continuously transmitting data. We will describe a subroutine for the decision problem. We use that subroutine for solving the optimization problem. 54

55. The decision problem-definitions We have: – t, the checked lifetime. – Location of g. If a sensor sends the data of k other sensors, it is spending per unit time. 55

56. The decision problem-constraints The problem: for the given check whether and. If yes, there exists a protocol in which each sensor continuously transmits data to g at a fixed rate. Since t is a constant the problem is analogues to the previous one and therefore can be solved in time. 56

57. An applet run the applet 57

58. Motivation. H & K algorithm for finding a Maximum Matching. Basic principles. Computing an optimal forwarding protocol. Optimizing the location of the base station. 58

59. Optimal location of the base station The problem: – Finding a location g for the base station and the tree T, maximizing the number of sensors that can transmit their data to g. 59

60. Optimal location of the base station Another version: Finding a location g for the base station and the tree T, maximizing the network lifetime. 60

61. The first version solution Observation: It is enough to consider only a polynomial number of candidate locations. We consider circles of Radii centered at each sensor of S. The circle is the boundary of the region to which the sensor can transmit its data and the data from i-1 additional sensors. 61

62. The first version solution – cont. g can be a vertex of the arrangement of the circles. If not, one can move g while not intersecting any circle without changing the transmission ability. 62

63. The first version solution – cont. There are vertices in the arrangement. For each vertex, we invoke the algorithm for the Optimal forwarding protocol problem. We find the vertex,v*, maximizing the number of sensors that can transmit their data to g. v* can be found in. 63

64. An improved analysis One solution relays on the Previous one. Start for an arbitrary vertex,p. Compute the optimal protocol Tp.(g=p) Move g to an adjacent vertex,q. S1 S2 S3 S4 64

65. An improved analysis – cont. As g is moved from p to q, the only changes in the graph concern to at most 2 sensors. They change their capacity by 1. Thus, the optimal protocol can be obtained from by one BFS+DFS phase, which takes. S1 S2 S3 S4 65

66. The improved running time We can go over all the vertices of the arrangement by moving between adjacent vertices. Therefore, v* can be found in. 66

67. The second version solution Given a lifetime t, we can decide in if there exists a g satisfying the problem conditions. We assume that t* is known and refer to this problem as a decision problem. – we can determine if a given t is lager, smaller or equal to t*. 67

68. A solution using Parametric Search We increase t from 0 until it exceeds t*, while keeping track of the vertices of the arrangement using a parallel sorting networks. Using this technic, v* can be found in. 68

69. Approximation algorithm For maximization problems p < 1 69

70. - approximation Let s’ be a sensor farthest from g. We can set t to be so all the sensors transmit their data directly to g. Either s’ or its leader has to transmit to distance at least. Hence,. Therefore, we get a approximation. 70

71. - approximation a b [a,b]- an interval for t. a/b approximation approximation 71

72. - approximation a b [a,b]- an interval for t. We use a binary search. We start from an - approximation If we take enough steps of the binary search we can get the needed approximation. 72

73. - approximation In we can get an - approximation for the maximum lifetime. 73

74. Thank you 74