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

2. Wireless Ad Hoc and Sensor Networks
Node Failures
Nodes might
go to sleep
Nodes might
move

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. 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. 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 in terms of work and time

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

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

9. Link Failure
node moves

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

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

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

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

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

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

16. Full Link Reversal Algorithm

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

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

19. Partial Link Reversal Algorithm

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

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

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

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

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

26. 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

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

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

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

30. Definitions
dest.
Good nodes
Bad nodes
Bad state

31. Resulting Good state
dest.

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

33. Many possible reversal schedules

34. 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

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

36. Bad state
dest.
Good nodes
Bad nodes

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

38. Layers of bad nodes
dest.
A layer:

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

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

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

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

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

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

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

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

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

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

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

50. dest.
Reversals per node:

51. dest.
Reversals per node:
End of execution

52. dest.
Reversals per node:
End of execution

53. dest.
Reversals per node:
End of execution

54. dest.
Reversals per node:
End of execution

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

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

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

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

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

60. 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

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

62. Partial Link Reversal Algorithm
sink
Sink
before reversal
after reversal

63. Bad state
dest.
Good nodes
Bad nodes

64. Layers of bad nodes
dest.
Good nodes
Bad nodes
Nodes at layer
are at distance from good nodes

65. Layers of bad nodes
dest.
alpha value
Max alpha
Min alpha

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

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

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

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

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

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

72. Layers of bad nodes
dest.
Good nodes
Bad nodes
Nodes at layer
are at distance from good nodes

73. Layers of bad nodes
dest.
for any height function g, there is
an initial assignment of heights such that……

74. when the network reaches a good state:
dest.
lower bound
on reversals
per node

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

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

77. 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

78. Open Problems
Improve worst-case performance of partial link reversal algorithm
Analyze randomized algorithms
Analyze average-case performance