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


Bounding searchable area
Bounding Searchable Area

s

t

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


Questions comments demo

???

Questions?Comments?Demo?


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