Analysis of Link Reversal Routing Algorithms

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Analysis of Link Reversal Routing Algorithms
Srikanta Tirthapura (Iowa State University)
and Costas Busch (Renssaeler Polytechnic
Institute)
2
Wireless Ad Hoc and Sensor Networks
Node Failures
Nodes might go to sleep
Nodes might move
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

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

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Link Reversal Algorithms

• Very simple
• Been around for 20 years
• Gafni-Bertsekas
• Full Reversal Algorithm
• Partial Reversal Algorithm
• Our ContributionFirst formal performance
  analysis of link reversal

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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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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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Aim of Link Reversal
Connection graph of a wireless network
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Link Failure
node moves
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A bad state
A good state
Bad node no path to destination
Good node at least one path to destination
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Full Link Reversal Algorithm
sink
Sinks reverse all their links
reversals 7
time 5
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Partial Link Reversal Algorithm
sink
Sinks reverse some of their links
time 5
reversals 5
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Heights and Acyclicity
General height
higher
lower
Heights are ordered in lexicographic
order Observation Directed Graph is always
acyclic
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Full Link Reversal Algorithm
Node
Node ID
Real height
(breaks ties)
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Full Link Reversal Algorithm
sink
Sink
before reversal
after reversal
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Full Link Reversal Algorithm
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Partial Link Reversal Algorithm
Node
Node ID
memory
Real height
(breaks ties)
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Partial Link Reversal Algorithm
sink
Sink
before reversal
after reversal
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Partial Link Reversal Algorithm
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Deterministic Link Reversal Algorithms
Sink
before reversal
after reversal
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)

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

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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
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Our Contribution
First formal performance analysis of link
reversal routing algorithms in terms of
reversals and time
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Metrics
reversals total number of node reversals till
stabilization (work)
Time number of parallel time steps till
stabilization
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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
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
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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
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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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Definitions
dest.
Good nodes
Bad nodes
Bad state
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Resulting Good state
dest.
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Good Nodes Never Reverse
dest.
Good nodes
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
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Layers of bad nodes
dest.
A layer
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dest.
41
r
r
dest.
r
42
r
r
dest.
r
r
r
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r
r
dest.
r
r
r
r
r
r
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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
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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.
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There is an execution such that
Every (remaining) bad node reverses exactly once
dest.
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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

• The remaining bad nodes return to the
• same state as before the execution

dest.
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At the end of execution
All nodes of layer become good nodes
dest.
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At the end of execution
All nodes of layer become good nodes
dest.
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dest.
Reversals per node
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dest.
Reversals per node
End of execution
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dest.
Reversals per node
End of execution
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dest.
Reversals per node
End of execution
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dest.
Reversals per node
End of execution
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dest.
Reversals per node
Each node in layer reverses times
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dest.
Reversals per node
Nodes per layer
reversals
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dest.
For bad nodes, trivial upper bound
(reversals and time)
reversals
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reversals bound is tight
dest.
Reversals per node
reversals
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time bound is tight
nodes
dest.
reversals in layer
Time needed
None of these reversals are performed in
parallel
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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

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Partial Link Reversal Algorithm
sink
Sink
before reversal
after reversal
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Bad state
dest.
Good nodes
Bad nodes
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Layers of bad nodes
dest.
Good nodes
Bad nodes
Nodes at layer are at distance from
good nodes
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Layers of bad nodes
dest.
alpha value
Max alpha
Min alpha
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when the network reaches a good state
dest.
upper bound on alpha value
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when the network reaches a good state
dest.
upper bound on reversals
Reason Each partial reversal increases alpha
value by at least 1.
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when the network reaches a good state
dest.
a bad node reverses at most times
For bad nodes
reversals and time
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reversals bound is
tight
dest.
Reversals per node
reversals
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time bound is tight
dest.
reversals in layer
nodes
Time needed
None of these reversals are performed in
parallel
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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
Nodes at layer are at distance from
good nodes
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Lower Bound on reversals on worst case graphs
dest.
Reversals per node
reversals
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Lower Bound on time
dest.
reversals in layer
nodes
Time needed
None of these reversals are performed in
parallel
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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/