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Worst-Case Optimal and Average-Case Efficient Geometric Ad-Hoc Routing. Fabian Kuhn Roger Wattenhofer Aaron Zollinger. s. Geometric Routing. ? ? ?. t. s. Greedy Routing. Each node forwards message to “best” neighbor. t. s. Greedy Routing. - PowerPoint PPT Presentation
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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
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Face Routing
s t
MobiHoc 2003 13
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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• 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
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0 5 10 15
Network Density [nodes per unit disk]
Sh
ort
est
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th S
pa
n
0
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1
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qu
en
cy
Greedy success
Connectivity
Shortest Path Span
critical
MobiHoc 2003 29
Average-Case Performance: Face vs. Greedy/Face
0
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0 5 10 15
Network Density [nodes per unit disk]
Pe
rfo
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qu
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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
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0 2 4 6 8 10 12
Network Density [nodes per unit disk]
Pe
rfo
rma
nce
0
0.1
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0.9
1
Fre
qu
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cy
critical
MobiHoc 2003 31
Conclusion
Greedy Routing (MFR) ()
Face Routing
GFG/GPSR
AFR
GOAFR
CorrectRouting
Avg-CaseEfficient
Worst-CaseOptimal
ComprehensiveSimulation
Questions?Comments?
Demo?