Routing Algorithms for Mobile Ad Hoc Networks Costas Busch (RPI) - - PowerPoint PPT Presentation

Analysis of Link Reversal Routing Algorithms for Mobile Ad Hoc Networks

Costas Busch (RPI) Srikanth Surapaneni (RPI) Srikanta Tirthapura (Iowa State University)

Talk Outline

Link Reversal Routing Previous Work & Contributions Analysis of Full Reversal Algorithm Analysis of Partial Reversal Algorithm Analysis of Deterministic Algorithms Conclusions

Connection graph

• f a mobile network

Destination oriented, acyclic graph Destination

Link Reversal Routing

Link Failure

node moves

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

sink

Full Link Reversal Algorithm

sink sink sink sink sink sink

Sinks reverse all their links

#reversals = 7 time = 5

sink

Partial Link Reversal Algorithm

sink sink sink sink

Sinks reverse some of their links

#reversals = 5 time = 5

) 3 , 4 , 5 , , ( Heights

) , , , (

2 1 n

a a a 

) 7 , 9 , 4 , , ( General height: higher lower Heights are ordered in lexicographic order

) 1 , ( ) 2 , ( ) 3 , ( ) 4 , ( ) 5 , ( ) 6 , ( ) Dest , (

Full Link Reversal Algorithm ) , ( i ai Node

: i

Node ID Real height (breaks ties)

) 1 , ( ) 2 , ( ) 3 , ( ) 4 , ( ) 5 , ( ) 6 , ( ) Dest , (

Full Link Reversal Algorithm ) , ( i ai Sink

: i

) 1 , ( ) 2 , ( ) 3 , ( ) 4 , ( ) 5 , ( ) 6 , 1 ( ) Dest , (

before reversal after reversal ) , ( i ai  1 )} ( : max{     i N j a a

j i

) 1 , ( ) 2 , ( ) 3 , ( ) 4 , ( ) 5 , ( ) 6 , ( ) Dest , ( ) 1 , ( ) 2 , ( ) 3 , ( ) 4 , ( ) 5 , ( ) 6 , 1 ( ) Dest , ( ) 1 , ( ) 2 , ( ) 3 , 2 ( ) 4 , ( ) 5 , 2 ( ) 6 , 1 ( ) Dest , ( ) 1 , ( ) 2 , 3 ( ) 3 , 2 ( ) 4 , ( ) 5 , 2 ( ) 6 , 3 ( ) Dest , ( ) 1 , 4 ( ) 2 , 3 ( ) 3 , 4 ( ) 4 , ( ) 5 , 2 ( ) 6 , 3 ( ) Dest , (

Full Link Reversal Algorithm

Partial Link Reversal Algorithm ) , , ( i b a

i i

Node

: i

Node ID Real height (breaks ties)

) 1 , 4 , ( ) 2 , 3 , ( ) 3 , 2 , ( ) 4 , 5 , ( ) 5 , 2 , ( ) 6 , 1 , ( ) Dest , , (

memory

Partial Link Reversal Algorithm ) , , ( i b a

i i

Sink : i before reversal after reversal ) , , ( i b a

i i

 

1 )} ( : min{     i N j a a

j i

) 1 , 4 , ( ) 2 , 3 , ( ) 3 , 2 , ( ) 4 , 5 , ( ) 5 , 2 , ( ) 6 , 1 , ( ) Dest , , ( ) 1 , 4 , ( ) 2 , 3 , ( ) 3 , 2 , ( ) 4 , 5 , ( ) 5 , 2 , ( ) 6 , 1 , 1 ( ) Dest , , (

1 } and ) ( : min{       

j i j i

b a i N j b b

) 1 , 4 , ( ) 2 , 3 , ( ) 3 , 2 , ( ) 4 , 5 , ( ) 5 , 2 , ( ) 6 , 1 , ( ) Dest , , ( ) 1 , 4 , ( ) 2 , 3 , ( ) 3 , 2 , ( ) 4 , 5 , ( ) 5 , 2 , ( ) 6 , 1 , 1 ( ) Dest , , ( ) 1 , 4 , ( ) 2 , 3 , ( ) 3 , , 1 ( ) 4 , 5 , ( ) 5 , , 1 ( ) 6 , 1 , 1 ( ) Dest , , ( ) 1 , 4 , ( ) 2 , 1 , 1 (  ) 3 , , 1 ( ) 4 , 5 , ( ) 5 , , 1 ( ) 6 , 1 , 1 ( ) Dest , , ( ) 1 , 2 , 1 (  ) 2 , 1 , 1 (  ) 3 , , 1 ( ) 4 , 5 , ( ) 5 , , 1 ( ) 6 , 1 , 1 ( ) Dest , , (

Partial Link Reversal Algorithm

Deterministic Link Reversal Algorithms

h

1

h

2

h

3

h

4

h

5

h h Sink : i before reversal after reversal h

1

h

2

h

3

h

4

h

5

h ) , , , , (

2 1 k

h h h h g h   

Deterministic function

Interesting measures: #reversals: total number of node reversals (work) Time: time needed to reach a good state (stabilization time)

Talk Outline

Link Reversal Routing Previous Work & Contributions Analysis of Full Reversal Algorithm Analysis of Partial Reversal Algorithm Analysis of Deterministic Algorithms Conclusions

Previous Work

Gafni and Bertsekas: IEEE Tsans. on Commun. 1981

• Introduction of the problem
• First proof of stability

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.

Previous Work

Malpani, Welch and Vaidya.: Workshop on Discr. Alg.

And methods for mobile comput. and commun. 2000

• Leader election based on TORA
• (partial) proof of stability

Experimental work and surveys: Broch et al.: MOBICOM 1998 Samir et al.: IC3N 1998 Perkins: “Add Hoc Networking”, Ad. Wesley 2000 Rajamaran: SIGACT news 2002

Contributions

First formal performance analysis

• f link reversal routing algorithms

in terms of #reversals and time

Contributions

Full reversal algorithm:

#reversals and time:

) ( 2 n O

n bad nodes

There are worst-cases with:

) ( 2 n 

Partial reversal algorithm:

) (

2

n a n O  



#reversals and time: There are worst-cases with:

) (

2

n a n   





a

depends on the network state

Contributions

Any deterministic algorithm:

#reversals and time:

n bad nodes

) ( 2 n 

There are states such that

Full reversal is worst-case optimal Partial reversal is not!

) ( 2 n  ) (

2

n a n   



Talk Outline

Link Reversal Routing Previous Work & Contributions Analysis of Full Reversal Algorithm Analysis of Partial Reversal Algorithm Analysis of Deterministic Algorithms Conclusions

Bad state dest. Good nodes Bad nodes

Resulting Good state dest.

For any execution of the full reversal algorithm:

• #reversals is the same
• Final state is the same

(this holds for any deterministic algorithm)

Bad state dest. Good nodes Bad nodes

Layers of bad nodes dest. Good nodes Bad nodes

1

L

2

L

3

L

4

L

Layers of bad nodes

1

L

2

L

3

L

m

L dest. A layer:

There is an execution such that: Every bad node reverses exactly once

1

L

2

L

3

L

4

L

1

E

dest.

dest.

1

L

2

L

3

L

4

L

r r r

There is an execution such that: Every bad node reverses exactly once

1

E

dest.

1

L

2

L

3

L

4

L

r r r r r

There is an execution such that: Every bad node reverses exactly once

1

E

dest.

1

L

2

L

3

L

4

L

r r r r r r r r

There is an execution such that: Every bad node reverses exactly once

1

E

dest.

1

L

2

L

3

L

4

L

• 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

1

L

r r r r r r r r r r r r

1

E

dest.

2

L

3

L

4

L

At the end of execution :

1

E

• The remaining bad nodes return to the

same state as before the execution

• All nodes of layer become good nodes

1

L

dest.

2

L

3

L

4

L There is an execution such that: Every bad node reverses exactly once

2

E

dest.

2

L

3

L

4

L

At the end of execution :

2

E

• The remaining bad nodes return to the

same state as before the execution

• All nodes of layer become good nodes

2

L

dest.

3

L

4

L

At the end of execution :

2

E

• The remaining bad nodes return to the

same state as before the execution

• All nodes of layer become good nodes

2

L

dest.

4

L

At the end of execution : All nodes of layer become good nodes

3

L

3

E

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

4

L

4

E

1

L

2

L

3

L

m

L dest. Reversals per node: 0



1

L

2

L

3

L

m

L dest. Reversals per node: 1 1 1 1



End of execution

1

E

1

L

2

L

3

L

m

L dest. Reversals per node: 1 2 2 2



End of execution

2

E

1

L

2

L

3

L

m

L dest. Reversals per node: 1 2

3 3



End of execution

3

E

1

L

2

L

3

L

m

L dest. Reversals per node: 1 2

3

m



End of execution

m

E

Each node in layer reverses times

i

L

i

Reversals per node: 1 2

3

m



1

L

2

L

3

L

m

L dest.

Reversals per node: 1 2

3

m Nodes per layer:

1

n

2

n

3

n

m

n #reversals:

1

1 n 

2

2 n 

3

3 n 

m

n m

   

  

1

L

2

L

3

L

m

L dest.

For bad nodes, trivial upper bound:

n

) ( 2 n O

#reversals:

1

1 n 

2

2 n 

3

3 n 

m

n m

   



(#reversals and time)

1

L

2

L

3

L

m

L dest. n n

#reversals bound is tight

1

L

2

L

3

L

n

L

dest.

Reversals per node: 1 2

3 n

#reversals:

) ( 3 2 1

2

n n       

) ( 2 n O

None of these reversals are performed in parallel time bound is tight

1

L

2

L

1 2 /  n

L

dest.

) ( 2 n O

2 / n

L

1 2  n

#nodes =

#reversals in layer :

2 / n

L ) ( 1 2

2 2

n n          ) ( 2 n  Time needed

Talk Outline

Link Reversal Routing Previous Work & Contributions Analysis of Full Reversal Algorithm Analysis of Partial Reversal Algorithm Analysis of Deterministic Algorithms Conclusions

Bad state dest. Good nodes Bad nodes

Layers of bad nodes dest. Good nodes Bad nodes

1

L

2

L

3

L

4

L

5

L Nodes at layer are at distance from good nodes

i

L

i

Layers of bad nodes

1

L

2

L

3

L

m

L dest. ) , , ( i b a

i i

) , , ( max i b a

i

) , , ( min i b a

i

alpha value

max

a

min

a

min

a

when the network reaches a good state: upper bound

• n alpha value

1

L

2

L

3

L

m

L dest. 1

max 

a 2 

3 

max

a m 

min

a

min

a

when the network reaches a good state: upper bound

• n reversals

per node

1

L

2

L

3

L

m

L dest. 1 



a 2 

3 

m 

min max

a a a  



max

a

min

a

min

a

when the network reaches a good state:

1

L

2

L

3

L

m

L dest. 1 



a 2 

3 

max

a m  For bad nodes: n a bad node reverses at most times n a 

 min

a

min

a #reversals and time: ) (

2

n a n O  



#reversals bound is tight

1

L

2

L

3

L

n

L

dest.

Reversals per node: 1 



a 2 

3 

n  #reversals:

) (

2

n a n   



) (

2

n a n O  



None of these reversals are performed in parallel time bound is tight

1

L

2

L

1 2 /  n

L

dest.

2 / n

L

2 n

#nodes =

#reversals in layer :

2 / n

L ) (

2

n a n   



) (

2

n a n   



Time needed

) (

2

n a n O  



1 2 /  n

L

      



2 2 n a n

Talk Outline

Link Reversal Routing Previous Work & Contributions Analysis of Full Reversal Algorithm Analysis of Partial Reversal Algorithm Analysis of Deterministic Algorithms Conclusions

Layers of bad nodes dest. Good nodes Bad nodes

1

L

2

L

3

L

4

L

5

L Nodes at layer are at distance from good nodes

i

L

i

Layers of bad nodes

1

L

2

L

3

L

m

L dest. for any height function g, there is an initial assignment of heights such that……

when the network reaches a good state: lower bound

• n reversals

per node

1

L

2

L

3

L

m

L dest. 1 2 1  m

1

L

2

L

3

L

n

L

dest.

Reversals per node: 1 2

1  n #reversals:

) ( 3 2 1

2

n n       

Lower Bound on #reversals

None of these reversals are performed in parallel

1

L

2

L

1 2 /  n

L

dest.

2 / n

L

2 n

#nodes =

#reversals in layer :

2 / n

L ) ( 2 n  ) ( 2 n  Time needed

1 2 /  n

L

      1 2 2 n n

Lower Bound on time

Talk Outline

Link Reversal Routing Previous Work & Contributions Analysis of Full Reversal Algorithm Analysis of Partial Reversal Algorithm Analysis of Deterministic Algorithms Conclusions

• We gave the first formal performance

analysis of deterministic link reversal algorithms Open problems:

• Improve worst-case performance
• f partial link reversal algorithm
• Analyze randomized algorithms
• Analyze average-case performance