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
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Face Routing s t
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Face Routing s t
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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
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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
(  )







 
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