1. 1 Analysis of Link Reversal Routing Algorithms Srikanta Tirthapura (Iowa State University) and Costas Busch (Renssaeler Polytechnic Institute)

2. 2 Wireless Ad Hoc and Sensor Networks Nodes might move Nodes might go to sleep Node Failures Underlying Communication Graph is Changing

3. 3 Algorithms for Wireless Ad Hoc and Sensor Networks Algorithms should be simple and distributed Self-stabilizing (or self-healing) in the face of failures

4. 4 Research Goal Design Algorithms for which we can –Prove convergence –Analyze performance –Predict behavior on large scale networks Complementary to evaluation through simulation and experiments

5. 5 Link Reversal Algorithms Very simple Been around for 20 years Gafni-Bertsekas –Full Reversal Algorithm –Partial Reversal Algorithm Our Contribution: First formal performance analysis of link reversal

6. 6 Link Reversal Routing Previous Work & Contributions Our Analysis Basic Properties of Link Reversal Full Reversal Algorithm Partial Reversal Algorithm Lower Bounds Conclusions Talk Outline

7. 7 Distributed Dynamic Graph Problem Communication Graph: –Vertices = Computers (perhaps mobile) –Edges = Wireless communication links Task: Maintain a distributed structure on this graph –Routing –Leader Election Issues: –Deal with node and link failures –Acyclicity

8. 8 Connection graph of a wireless network Destination oriented, directed acyclic graph Destination Aim of Link Reversal

9. 9 Link Failure node moves

10. 10 Bad node: no path to destination Good node: at least one path to destination A bad stateA good state

11. 11 sink Full Link Reversal Algorithm sink Sinks reverse all their links #reversals = 7time = 5

12. 12 sink Partial Link Reversal Algorithm sink Sinks reverse some of their links #reversals = 5 time = 5

13. 13 Heights and Acyclicity General height: higherlower Heights are ordered in lexicographic order Observation: Directed Graph is always acyclic

14. 14 Full Link Reversal Algorithm Node Node IDReal height (breaks ties)

15. 15 Full Link Reversal Algorithm Sink before reversalafter reversal sink

16. 16 Full Link Reversal Algorithm

17. 17 Partial Link Reversal Algorithm Node Node ID Real height (breaks ties) memory

18. 18 Partial Link Reversal Algorithm Sink before reversalafter reversal sink

19. 19 Partial Link Reversal Algorithm

20. 20 Deterministic Link Reversal Algorithms Sink before reversalafter reversal Deterministic function

21. 21 Merits of Link Reversal Simple Distributed, Acyclic Self-stabilizes from a bad state to a good state

22. 22 Talk Outline Link Reversal Routing Previous Work & Contributions Our Analysis Basic Properties of Link Reversal Full Reversal Algorithm Partial Reversal Algorithm Lower Bounds Conclusions

23. 23 Previous Work Gafni and Bertsekas: 1981 Designed First Reversal Algorithms Proof of stability (eventual convergence) Park and Corson: INFOCOM 1997 TORA – Temporally Ordered Routing Alg. - Variation of partial reversal - Deals with partitions Corson and Ephremides: Wireless Net. Jour. 1995 LMR – Lightweight Mobile Routing Alg.

24. 24 Previous Work Malpani, Welch and Vaidya.: DIAL-M 2000 Distributed Leader election based on TORA (partial) proof of stability Experimental work and surveys: Broch et al.: MOBICOM 1998 Samir et al.: IC3N 1998 Perkins: “Ad Hoc Networking”, Rajamaran: SIGACT news 2002 Intanagonwiwat, Govindan, Estrin: MOBICOM 00 “Directed Diffusion” – Sensor network routing Similar to the TORA algorithm

25. 25 Our Contribution First formal performance analysis of link reversal routing algorithms in terms of #reversals and time

26. 26 # reversals: total number of node reversals till stabilization (work) Time: number of parallel time steps till stabilization Metrics

27. 27 The Good News The work and time taken depend only on the number of nodes which have lost their paths to destination Algorithm is Local

28. 28 Further News Full reversal algorithm: #reversals and time: “bad” nodes There are worst-cases with: Partial reversal algorithm: #reversals and time: There are worst-cases with: depends on the network state

29. 29 More News – Lower Bound Any deterministic algorithm: #reversals and time: bad nodes There are states such that Full reversal alg. is worst-case optimal Partial reversal alg. is not

30. 30 Talk Outline Link Reversal Routing Previous Work & Contributions Our Analysis Basic Properties of Link Reversal Full Reversal Algorithm Partial Reversal Algorithm Lower Bounds Conclusions

31. 31 Bad state dest. Good nodes Bad nodes Definitions

32. 32 Resulting Good state dest.

33. 33 dest. Good nodes Good Nodes Never Reverse Proof by a simple induction on distance from dest.

34. 34 Many possible reversal schedules A B C

35. 35 Schedule of Reversals is NOT important Lemma: For all executions of any deterministic reversal algorithm starting from the same initial state –# of reversals is the same –Final state is the same For upper bounds and lower bounds, we can choose a “convenient” execution schedule

36. 36 Talk Outline Link Reversal Routing Previous Work & Contributions Our Analysis Basic Properties of Link Reversal Full Reversal Algorithm Partial Reversal Algorithm Lower Bounds Conclusions

37. 37 Bad state dest. Good nodes Bad nodes

38. 38 Layers of bad nodes dest. Good nodes Bad nodes

39. 39 Layers of bad nodes dest. A layer:

40. 40 There is an execution segment such that: Every bad node reverses exactly once dest.

41. 41 dest. r r r There is an execution segment such that: Every bad node reverses exactly once

42. 42 dest. r r r r r There is an execution segment such that: Every bad node reverses exactly once

43. 43 dest. r r r r r r r r There is an execution segment such that: Every bad node reverses exactly once

44. 44 dest. The remaining bad nodes return to the same state as before the execution At the end of execution : All nodes of layer become good nodes r r r r r r r r r r r r

45. 45 dest. At the end of execution : The remaining bad nodes return to the same state as before the execution All nodes of layer become good nodes

46. 46 dest. There is an execution such that: Every (remaining) bad node reverses exactly once

47. 47 dest. At the end of execution : The remaining bad nodes return to the same state as before the execution All nodes of layer become good nodes

48. 48 dest. At the end of execution : The remaining bad nodes return to the same state as before the execution All nodes of layer become good nodes

49. 49 dest. At the end of execution : All nodes of layer become good nodes

50. 50 dest. At the end of execution : All nodes of layer become good nodes

51. 51 dest. Reversals per node:

52. 52 dest. Reversals per node: End of execution

53. 53 dest. Reversals per node: End of execution

54. 54 dest. Reversals per node: End of execution

55. 55 dest. Reversals per node: End of execution

56. 56 Each node in layer reverses times Reversals per node: dest.

57. 57 Reversals per node: Nodes per layer: #reversals: dest.

58. 58 For bad nodes, trivial upper bound: #reversals: (#reversals and time) dest.

59. 59 #reversals bound is tight dest. Reversals per node: #reversals:

60. 60 None of these reversals are performed in parallel time bound is tight dest. #nodes = #reversals in layer : Time needed

61. 61 Complete Solution to Full Reversal Given any initial state of the Full Reversal algorithm, our analysis can predict –the number of reversals of each node exactly –the time taken to convergence

62. 62 Talk Outline Link Reversal Routing Previous Work & Contributions Our Analysis Basic Properties of Link Reversal Full Reversal Algorithm Partial Reversal Algorithm Lower Bounds Conclusions

63. 63 Partial Link Reversal Algorithm Sink before reversalafter reversal sink

64. 64 Bad state dest. Good nodesBad nodes

65. 65 Layers of bad nodes dest. Good nodesBad nodes Nodes at layer are at distance from good nodes

66. 66 Layers of bad nodes dest. alpha value Max alphaMin alpha

67. 67 when the network reaches a good state: upper bound on alpha value dest.

68. 68 when the network reaches a good state: upper bound on #reversals dest. Reason: Each partial reversal increases alpha value by at least 1.

69. 69 when the network reaches a good state: dest. For bad nodes: a bad node reverses at most times #reversals and time:

70. 70 #reversals bound is tight dest. Reversals per node: #reversals:

71. 71 None of these reversals are performed in parallel time bound is tight dest. #nodes = #reversals in layer : Time needed

72. 72 Lower Bound for any Deterministic Algorithm For any deterministic reversal algorithm, there is an initial assignment of heights such that: Nodes at a distance of d from a good node have to reverse d times

73. 73 Layers of bad nodes dest. Good nodesBad nodes Nodes at layer are at distance from good nodes

74. 74 dest. Reversals per node: #reversals: Lower Bound on #reversals on worst case graphs

75. 75 None of these reversals are performed in parallel dest. #nodes = #reversals in layer : Time needed Lower Bound on time

76. 76 Conclusions We gave the first formal performance analysis of deterministic link reversal algorithms Worst case performance-wise –Full Link Reversal is optimal (surprisingly) –Partial Link Reversal is not Good News: The time and work to stabilization depend only on the number of bad nodes Bad News: There is an inherent lower bound on efficiency of link reversal algorithms

77. 77 Open Problems Improve worst-case performance of partial link reversal algorithm Analyze randomized algorithms Analyze average-case performance A Preliminary version of this work appeared in SPAA 2003 (Symposium on Parallelism in Algorithms and Architectures) Full version available at: http://www.eng.iastate.edu/~snt/