worst case optimal and average case efficient geometric ad hoc routing
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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.

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worst case optimal and average case efficient geometric ad hoc routing

Worst-Case Optimal and Average-Case Efficient Geometric Ad-Hoc Routing

Fabian Kuhn

Roger Wattenhofer

Aaron Zollinger

geometric routing

s

Geometric Routing

???

t

s

MobiHoc 2003

greedy routing
Greedy Routing
  • Each node forwards message to “best” neighbor

t

s

MobiHoc 2003

greedy routing1
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

what is geometric routing
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

overview
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

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

MobiHoc 2003

face routing1
Face Routing
  • Remark: Planar graph can easily (and locally!) be computed with the Gabriel Graph, for example

Planarity is NOT an assumption

MobiHoc 2003

face routing2
Face Routing

s

t

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face routing3
Face Routing

s

t

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face routing4
Face Routing

s

t

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face routing5
Face Routing

s

t

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face routing6
Face Routing

s

t

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face routing7
Face Routing

s

t

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face routing8
Face Routing

s

t

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face routing properties

“Right Hand Rule”

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

MobiHoc 2003

overview1
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

face routing9
Face Routing
  • Theorem: Face Routing reaches destination in O(n) steps
  • But: Can be very bad compared to the optimal route

MobiHoc 2003

adaptively bound searchable area
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

overview2
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

goafr greedy other adaptive face routing
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

early fallback to greedy routing
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

goafr is worst case optimal
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

average case
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

too dense

MobiHoc 2003

critical density shortest path vs euclidean distance
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)

MobiHoc 2003

randomly generated graphs critical density range
Randomly Generated Graphs: Critical Density Range

1.9

1

0.9

1.8

Connectivity

0.8

1.7

0.7

1.6

0.6

Greedy success

1.5

Shortest Path Span

Frequency

0.5

1.4

0.4

Shortest Path Span

1.3

0.3

1.2

0.2

1.1

0.1

critical

1

0

10

15

0

5

Network Density [nodes per unit disk]

MobiHoc 2003

average case performance face vs greedy face
Average-Case Performance: Face vs. Greedy/Face

20

1

worse

18

0.9

16

0.8

14

0.7

12

0.6

Performance

10

0.5

Frequency

8

0.4

Face Routing

6

0.3

better

4

0.2

2

0.1

critical

Greedy/FaceRouting

0

0

0

5

10

15

Network Density [nodes per unit disk]

MobiHoc 2003

simulation on randomly generated graphs
Simulation on Randomly Generated Graphs

9

1

GFG/GPSR

worse

0.9

8

0.8

7

0.7

6

0.6

Frequency

5

0.5

Performance

0.4

4

GOAFR+

0.3

3

0.2

better

2

0.1

critical

1

0

0

2

4

6

8

10

12

Network Density [nodes per unit disk]

MobiHoc 2003

conclusion

CorrectRouting

Worst-CaseOptimal

Avg-CaseEfficient

ComprehensiveSimulation

Conclusion

MobiHoc 2003

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