mobihoc03_slides

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Worst-Case Optimal and

Average-Case Efficient

Geometric Ad-Hoc Routing

Fabian Kuhn

Roger Wattenhofer

Aaron Zollinger

Geometric Routing s

?

?

?

s

MobiHoc 2003 t

2

Greedy Routing

• Each node forwards message to “best” neighbor t s

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Greedy Routing

• Each node forwards message to “best” neighbor

?

t s

• But greedy routing may fail: message may get stuck in a “dead end”

• Needed: Correct geometric routing algorithm

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What is Geometric Routing?

• A.k.a. location-based, position-based, geographic, etc.

• Each node knows its own position and position of neighbors

• Source knows the position of the destination

• No routing tables stored in nodes!

• Geometric routing is important:

– GPS/Galileo, local positioning algorithm, overlay P2P network, Geocasting

– Most importantly: Learn about general ad-hoc routing

MobiHoc 2003 5

Related Work in Geometric Routing

Kleinrock et al.

Kranakis, Singh,

Urrutia

Various

1975ff

CCCG

1999

MFR et al.

Face

Routing

Geometric Routing

First correct proposed algorithm

Bose, Morin,

Stojmenovic, Urrutia

Karp, Kung

Kuhn, Wattenhofer,

Zollinger

Kuhn, Wattenhofer,

Zollinger

DialM

1999

GFG First average-case efficient algorithm (simulation but no proof)

MobiCom

2000

GPSR

DialM

2002

MobiHoc

2003

A new name for GFG

AFR First worst-case analysis. Tight

(c 2 ) bound.

GOAFR Worst-case optimal and averagecase efficient, percolation theory

MobiHoc 2003 6

Overview

• Introduction

– What is Geometric Routing?

– Greedy Routing

• Correct Geometric Routing: Face Routing

• Efficient Geometric Routing

– Adaptively Bound Searchable Area

– Lower Bound, Worst-Case Optimality

– Average-Case Efficiency

– Critical Density

– GOAFR

• Conclusions

MobiHoc 2003 7

Face Routing

• Based on ideas by [Kranakis, Singh, Urrutia CCCG 1999]

• Here simplified (and actually improved)

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Face Routing

• Remark: Planar graph can easily (and locally!) be computed with the Gabriel Graph, for example

Planarity is NOT an assumption

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Face Routing s t

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Face Routing s t

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Face Routing s t

MobiHoc 2003 12

Face Routing s t

MobiHoc 2003 13

Face Routing s t

MobiHoc 2003 14

Face Routing s t

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Face Routing s t

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Face Routing Properties

• All necessary information is stored in the message

– Source and destination positions

– Point of transition to next face

• Completely local:

– Knowledge about direct neighbors‘ positions sufficient

– Faces are implicit

“Right Hand Rule”

• Planarity of graph is computed locally (not an assumption)

– Computation for instance with Gabriel Graph

MobiHoc 2003 17

Overview

• Introduction

– What is Geometric Routing?

– Greedy Routing

• Correct Geometric Routing: Face Routing

• Efficient Geometric Routing

– Adaptively Bound Searchable Area

– Lower Bound, Worst-Case Optimality

– Average-Case Efficiency

– Critical Density

– GOAFR

• Conclusions

MobiHoc 2003 18

Face Routing

• Theorem: Face Routing reaches destination in O(n) steps

• But: Can be very bad compared to the optimal route

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Bounding Searchable Area s

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20

Adaptively Bound Searchable Area

What is the correct size of the bounding area?

– Start with a small searchable area

– Grow area each time you cannot reach the destination

– In other words, adapt area size whenever it is too small

→ Adaptive Face Routing AFR

Theorem: AFR Algorithm finds destination after O(c 2 ) steps, where c is the cost of the optimal path from source to destination.

Theorem: AFR Algorithm is asymptotically worst-case optimal.

[Kuhn, Wattenhofer, Zollinger DIALM 2002]

MobiHoc 2003 21

Overview

• Introduction

– What is Geometric Routing?

– Greedy Routing

• Correct Geometric Routing: Face Routing

• Efficient Geometric Routing

– Adaptively Bound Searchable Area

– Lower Bound, Worst-Case Optimality

– Average-Case Efficiency

– Critical Density

– GOAFR

• Conclusions

MobiHoc 2003 22

GOAFR – Greedy Other Adaptive Face Routing

• AFR Algorithm is not very efficient (especially in dense graphs)

• Combine G reedy and ( O ther A daptive) F ace R outing

– Route greedily as long as possible

– Overcome “dead ends” by use of face routing

– Then route greedily again

• Similar as GFG/GPSR, but adaptive

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Early Fallback to Greedy Routing?

• We could fall back to greedy routing as soon as we are closer to t than the local minimum

• But:

• “Maze” with 

(c 2 ) edges is traversed

(c) times

 

(c 3 ) steps

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GOAFR Is Worst-Case Optimal

• GOAFR traverses complete face boundary:

Theorem: GOAFR is asymptotically worst-case optimal.

• Remark: GFG/GPSR is not

– Searchable area not bounded

– Immediate fallback to greedy routing

• GOAFR’s average-case efficiency?

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Average Case

• Not interesting when graph not dense enough

• Not interesting when graph is too dense

• Critical density range (“percolation”)

– Shortest path is significantly longer than Euclidean distance too sparse critical density

MobiHoc 2003 too dense

26

Critical Density: Shortest Path vs. Euclidean Distance

• Shortest path is significantly longer than Euclidean distance

• Critical density range mandatory for the simulation of any routing algorithm (not only geometric)

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Randomly Generated Graphs: Critical Density Range

1.5

1.4

1.3

1.2

1.1

1

0

1.9

1.8

1.7

1.6

Connectivity

Greedy success

Shortest Path Span

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 critical

5 10 15

Network Density [nodes per unit disk]

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Average-Case Performance: Face vs. Greedy/Face

20

18

16

14

12

10

8

6

4

2

0

0

Face Routing critical

5

Greedy/Face

15

Network Density [nodes per unit disk]

MobiHoc 2003

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

29

Simulation on Randomly Generated Graphs

6

5

4

3

2

1

0

9

8

7

GFG/GPSR

2

GOAFR+

4 critical

6 8 10

Network Density [nodes per unit disk]

12

0.5

0.4

0.3

0.2

0.1

0

1

0.9

0.8

0.7

0.6

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Conclusion

Greedy Routing (MFR)

Face Routing

GFG/GPSR

AFR

GOAFR

Correct

Routing

Worst-Case

Optimal

Avg-Case

Efficient

Comprehensive

Simulation

(  )

 

MobiHoc 2003 31

Questions?

Comments?

Demo?

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