Worst-Case Optimal and Average-Case Efficient

Geometric Ad-Hoc Routing

Fabian KuhnRoger Wattenhofer

Aaron Zollinger

MobiHoc 2003 2

Geometric Routing

t

s

s

???

MobiHoc 2003 3

Greedy Routing

• Each node forwards message to “best” neighbor

t

s

MobiHoc 2003 4

Greedy Routing

• Each node forwards message to “best” neighbor

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

t

?s

MobiHoc 2003 5

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 6

Related Work in Geometric Routing

Kleinrock et al. Various 1975ff

MFR et al.

Geometric Routing proposed

Kranakis, Singh, Urrutia

CCCG 1999

Face Routing

First correct algorithm

Bose, Morin, Stojmenovic, Urrutia

DialM 1999

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

Karp, Kung MobiCom 2000

GPSR A new name for GFG

Kuhn, Wattenhofer, Zollinger

DialM 2002

AFR First worst-case analysis. Tight (c2) bound.

Kuhn, Wattenhofer, Zollinger

MobiHoc 2003

GOAFR Worst-case optimal and average-case efficient, percolation theory

MobiHoc 2003 7

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 8

Face Routing

• Based on ideas by [Kranakis, Singh, Urrutia CCCG 1999]• Here simplified (and actually improved)

MobiHoc 2003 9

Face Routing

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

Planarity is NOT an assumption

MobiHoc 2003 10

Face Routing

s t

MobiHoc 2003 11

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

MobiHoc 2003 15

Face Routing

s t

MobiHoc 2003 16

Face Routing

s t

MobiHoc 2003 17

• 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

• Planarity of graph is computed locally (not an assumption)– Computation for instance with Gabriel Graph

Face Routing Properties

“Right Hand Rule”

MobiHoc 2003 18

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 19

Face Routing

• Theorem: Face Routing reaches destination in O(n) steps• But: Can be very bad compared to the optimal route

MobiHoc 2003 20

Bounding Searchable Area

ts

MobiHoc 2003 21

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(c2) 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 22

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 23

GOAFR – Greedy Other Adaptive Face Routing

• AFR Algorithm is not very efficient (especially in dense graphs)• Combine Greedy and (Other Adaptive) Face Routing

– Route greedily as long as possible

– Overcome “dead ends” by use of face routing

– Then route greedily again

• Similar as GFG/GPSR, but adaptive

MobiHoc 2003 24

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 (c2) edges is traversed (c) times (c3) steps

MobiHoc 2003 25

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?

MobiHoc 2003 26

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 too densecritical density

MobiHoc 2003 27

• Shortest path is significantly longer than Euclidean distance

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

Critical Density: Shortest Path vs. Euclidean Distance

MobiHoc 2003 28

Randomly Generated Graphs: Critical Density Range

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

0 5 10 15

Network Density [nodes per unit disk]

Sh

ort

est

Pa

th S

pa

n

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fre

qu

en

cy

Greedy success

Connectivity

Shortest Path Span

critical

MobiHoc 2003 29

Average-Case Performance: Face vs. Greedy/Face

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15

Network Density [nodes per unit disk]

Pe

rfo

rma

nce

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fre

qu

en

cy

bett

erw

orse

Face Routing

Greedy/FaceRouting

critical

MobiHoc 2003 30

Simulation on Randomly Generated Graphs

GFG/GPSR

GOAFR+

bett

erw

orse

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10 12

Network Density [nodes per unit disk]

Pe

rfo

rma

nce

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fre

qu

en

cy

critical

MobiHoc 2003 31

Conclusion

Greedy Routing (MFR) ()

Face Routing

GFG/GPSR

AFR

GOAFR

CorrectRouting

Avg-CaseEfficient

Worst-CaseOptimal

ComprehensiveSimulation

Questions?Comments?

Demo?