1 analysis of link reversal routing algorithms srikanta tirthapura (iowa state university) and...
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Analysis of Link Reversal Routing Algorithms
Srikanta Tirthapura (Iowa State University)
andCostas Busch
(Renssaeler Polytechnic Institute)
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Wireless Ad Hoc and Sensor Networks
Nodes mightmove
Nodes mightgo to sleep
Node Failures
Underlying Communication Graph is Changing
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Algorithms for Wireless Ad Hoc and Sensor Networks
• Algorithms should be simple and distributed
• Self-stabilizing (or self-healing) in the face of failures
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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
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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
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• Link Reversal Routing
• Previous Work & Contributions
• Our Analysis• Basic Properties of Link Reversal• Full Reversal Algorithm• Partial Reversal Algorithm• Lower Bounds
•Conclusions
Talk Outline
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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
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Connection graphof a wireless network
Destination oriented,directed acyclic graph
Destination
Aim of Link Reversal
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Link Failure
node moves
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Bad node: no path to destination
Good node: at least one path to destination
A bad state A good state
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sink
Full Link Reversal Algorithm
sink
sinksink
sink
sinksink
Sinks reverse all their links#reversals = 7 time = 5
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sink
Partial Link Reversal Algorithm
sink
sink
sink
sink
Sinks reverse some of their links#reversals = 5 time = 5
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)3,4,5(
Heights and Acyclicity
),,,( 21 naaa
)7,9,4(
General height:
higher lower
Heights are ordered in lexicographic order
Observation: Directed Graph is always acyclic
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)1,4()2,3(
)3,2(
)4,5()5,2(
)6,1(
)Dest,0(
Full Link Reversal Algorithm
),( iaiNode :i
Node IDReal height (breaks ties)
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)Dest,0(
Full Link Reversal Algorithm
),( iai
Sink :i
)Dest,0(
before reversal after reversal),( iai
1)}(:max{ iNjaa ji
)1,4()2,3(
)3,2(
)4,5()5,2(
)6,1(
)1,4()2,3(
)3,2(
)4,5()5,2(
)6,3(sink
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)Dest,0( )Dest,0( )Dest,0(
)Dest,0( )Dest,0(
Full Link Reversal Algorithm)1,4(
)2,3(
)3,2(
)4,5()5,2(
)6,1(
)1,4()2,3(
)3,2(
)4,5()5,2(
)6,3(
)1,4()2,3(
)3,4(
)4,5()5,6(
)6,3(
)1,4()2,7(
)3,4(
)4,5()5,6(
)6,7(
)1,8()2,7(
)3,8(
)4,5()5,6(
)6,7(
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Partial Link Reversal Algorithm
),,( iba iiNode :i
Node IDRealheight (breaks ties)
)1,4,0()2,3,0(
)3,2,0(
)4,5,0()5,2,0(
)6,1,0(
)Dest,0,0(
memory
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Partial Link Reversal Algorithm
),,( iba ii
Sink :i
before reversal after reversal
),,( iba ii
1)}(:min{ iNjaa ji
)1,4,0()2,3,0(
)3,2,0(
)4,5,0()5,2,0(
)6,1,0(
)Dest,0,0(
)1,4,0()2,3,0(
)3,2,0(
)4,5,0()5,2,0(
)6,1,1(
)Dest,0,0(
1}and)(:min{ jiji baiNjbb
sink
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)1,4,0()2,3,0(
)3,2,0(
)4,5,0()5,2,0(
)6,1,0(
)Dest,0,0(
)1,4,0()2,3,0(
)3,2,0(
)4,5,0()5,2,0(
)6,1,1(
)Dest,0,0(
)1,4,0()2,3,0(
)3,0,1(
)4,5,0()5,0,1(
)6,1,1(
)Dest,0,0(
)1,4,0()2,1,1(
)3,0,1(
)4,5,0()5,0,1(
)6,1,1(
)Dest,0,0(
)1,2,1( )2,1,1( )3,0,1(
)4,5,0()5,0,1(
)6,1,1(
)Dest,0,0(
Partial Link Reversal Algorithm
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Deterministic Link Reversal Algorithms
h1h
2h
3h
4h5h
h
Sink :i
before reversal after reversal
h1h
2h
3h
4h5h
),,,,( 21 khhhhgh
Deterministic function
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Merits of Link Reversal
• Simple
• Distributed, Acyclic
• Self-stabilizes from a bad state to a good state
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Talk Outline
• Link Reversal Routing
• Previous Work & Contributions
• Our Analysis• Basic Properties of Link Reversal• Full Reversal Algorithm• Partial Reversal Algorithm• Lower Bounds
•Conclusions
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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.
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Previous WorkMalpani, 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
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Our Contribution
First formal performance analysisof link reversal routing algorithmsin terms of #reversals and time
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# reversals: total number of node reversalstill stabilization (work)
Time: number of parallel time steps till stabilization
Metrics
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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
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Further News
Full reversal algorithm:
#reversals and time: )( 2nO
n “bad” nodes
There are worst-cases with: )( 2n
Partial reversal algorithm:)( 2nanO
#reversals and time: There are worst-cases with: )( 2nan
a depends on the network state
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More News – Lower Bound
Any deterministic algorithm:
#reversals and time:
n bad nodes
)( 2nThere are states such that
Full reversal alg. is worst-case optimal
Partial reversal alg. is not
)( 2nO
)( 2nan
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Talk Outline
• Link Reversal Routing
• Previous Work & Contributions
• Our Analysis• Basic Properties of Link Reversal• Full Reversal Algorithm• Partial Reversal Algorithm• Lower Bounds
•Conclusions
31Bad state
dest.
Good nodes Bad nodes
Definitions
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Resulting Good state
dest.
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dest.
Good nodes
Good Nodes Never Reverse
Proof by a simple induction on distance from dest.
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Many possible reversal schedules
A
B
C
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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
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Talk Outline
• Link Reversal Routing
• Previous Work & Contributions
• Our Analysis• Basic Properties of Link Reversal• Full Reversal Algorithm• Partial Reversal Algorithm• Lower Bounds
•Conclusions
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Bad state
dest.
Good nodes Bad nodes
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Layers of bad nodes
dest.
Good nodes Bad nodes
1L 2L 3L 4L
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Layers of bad nodes
1L 2L 3L mL
dest.
A layer:
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1L 2L 3L 4L
There is an execution segment such that:Every bad node reverses exactly once
1E
dest.
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dest.
1L 2L 3L 4Lr
r
r
There is an execution segment such that:Every bad node reverses exactly once
1E
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dest.
1L 2L 3L 4Lr
r
r
r
r
There is an execution segment such that:Every bad node reverses exactly once
1E
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dest.
1L 2L 3L 4Lr
r
r
r
r
r
r
r
There is an execution segment such that:Every bad node reverses exactly once
1E
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dest.
1L 2L 3L 4L
•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 1L
r
r
r
r
r
r
r
r
r
r
r
r
1E
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dest.
2L 3L 4L
At the end of execution :1E
•The remaining bad nodes return to the same state as before the execution
•All nodes of layer become good nodes 1L
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dest.
2L 3L 4L
There is an execution such that:
Every (remaining) bad node reverses exactly once
2E
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dest.
2L 3L 4L
At the end of execution :2E
•The remaining bad nodes return to the same state as before the execution
•All nodes of layer become good nodes 2L
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dest.
3L 4L
At the end of execution :2E
•The remaining bad nodes return to the same state as before the execution
•All nodes of layer become good nodes 2L
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dest.
4L
At the end of execution :
All nodes of layer become good nodes 3L3E
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dest.
At the end of execution :
All nodes of layer become good nodes 4L4E
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1L 2L 3L mL
dest.
Reversals per node:0 0 0 0
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1L 2L 3L mL
dest.
Reversals per node:1 1 1 1
End of execution 1E
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1L 2L 3L mL
dest.
Reversals per node:1 2 2 2
End of execution 2E
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1L 2L 3L mL
dest.
Reversals per node:1 2 3 3
End of execution 3E
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1L 2L 3L mL
dest.
Reversals per node:1 2 3 m
End of execution mE
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Each node in layer reverses times iL i
Reversals per node:1 2 3 m
1L 2L 3L mL
dest.
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Reversals per node:1 2 3 m
Nodes per layer: 1n 2n 3n mn
#reversals: 11 n 22 n 33 n mnm
1L 2L 3L mL
dest.
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For bad nodes, trivial upper bound: n )( 2nO
#reversals: 11 n 22 n 33 n mnm
(#reversals and time)
1L 2L 3L mL
dest.
nn
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#reversals bound is tight
1L 2L 3L nL
dest.
Reversals per node:
1 2 3 n
#reversals: )(321 2nn
)( 2nO
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None of these reversals are performed in parallel
time bound is tight
1L 2L 12/ nLdest.
)( 2nO2/nL
12n#nodes =
#reversals in layer :2/nL
)(122
2nnn
)( 2nTime needed
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Complete Solution to Full Reversal
• Given any initial state of the Full Reversal algorithm, our analysis canpredict
– the number of reversals of each node exactly
– the time taken to convergence
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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
Partial Link Reversal Algorithm
),,( iba ii
Sink :i
before reversal after reversal
),,( iba ii
1)}(:min{ iNjaa ji
)1,4,0()2,3,0(
)3,2,0(
)4,5,0()5,2,0(
)6,1,0(
)Dest,0,0(
)1,4,0()2,3,0(
)3,2,0(
)4,5,0()5,2,0(
)6,1,1(
)Dest,0,0(
1}and)(:min{ jiji baiNjbb
sink
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Bad state
dest.
Good nodes Bad nodes
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Layers of bad nodes
dest.
Good nodes Bad nodes
1L 2L 3L 4L 5L
Nodes at layer are at distance from good nodes
iLi
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Layers of bad nodes
1L 2L 3L mL
dest.
),,( iba ii ),,( max iba i ),,( min iba i
alpha value Max alpha Min alpha
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when the network reaches a good state:
upper bound on alpha value
1L 2L 3L mL
dest.
1max a 2 3
maxa
m
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when the network reaches a good state:
upper bound on #reversals
1L 2L 3L mL
dest.
1a 2 3 m
minmax aaa
maxa mina mina
Reason: Each partial reversal increases alpha value by at least 1.
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when the network reaches a good state:
1L 2L 3L mL
dest.
1a 2 3
maxa
m
For bad nodes: n a bad node reverses atmost timesna
mina mina
#reversals and time: )( 2nanO
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#reversals bound is tight
1L 2L 3L nL
dest.
Reversals per node: 2
12
a
22
23
2n
#reversals: )( 2nan
)( 2nanO
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None of these reversals are performed in parallel
time bound is tight
1L 2L 12/ nLdest.
2/nL
2n
#nodes =
#reversals in layer :2/nL
)( 2nan
)( 2nan Time needed
)( 2nanO
12/ nL
222
na
n
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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
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Layers of bad nodes
dest.
Good nodes Bad nodes
1L 2L 3L 4L 5L
Nodes at layer are at distance from good nodes
iLi
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1L 2L 3L nL
dest.
Reversals per node:
0 1 2 1n
#reversals: )(321 2nn
Lower Bound on #reversals on worst case graphs
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None of these reversals are performed in parallel
1L 2L 12/ nLdest.
2/nL
2n
#nodes =
#reversals in layer :2/nL
)( 2n
)( 2nTime needed
12/ nL
122nn
Lower Bound on time
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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
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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/