Competitive Time and Traffic Analysis of Position-based Routing using a Cell-Structure

Stefan Rührup 1

HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany

Algorithms and Complexity

Competitive Time and Traffic Analysisof Position-based Routing

using a Cell-Structure

Stefan Rührup and Christian Schindelhauer

Heinz Nixdorf Institute

University of Paderborn

Germany

IEEE WMAN‘05

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Algorithms and ComplexityOutline

• Part I: Topology control for position-based routing

– Position-based routing: greedy forwarding and recovery

– Topology issues in position-based routing

– Abstracting from graph theory: the cell structure approach

• Part II: Performance measures and algorithms

– Competitive performance measures

– Single-path versus multi-path routing strategies

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Part I

Topology Control for Position-based Routing

Part I

Topology Control for Position-based Routing

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Algorithms and ComplexityPosition-based routing in a nutshell

Given: Source, location of the destination

Task: Deliver a message to the destination

Assumptions:

• A node can determine its own position

• Each node knows the positions of the neighbors

• The position of the target is known

transmission range

source

target (x,y)

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Algorithms and ComplexityGreedy forwarding and recovery (1)

• With position informationone can forward a message in the "right" direction(greedy forwarding)

Example:

s

t

no routing tables, no flooding!

transmissionrange

progress boundary (circle around the

destination)

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barrierbarrier??

Greedy forwarding and recovery (2)

• Greedy forwarding is stopped by barriers (local minima)• Recovery strategy: Traverse the border of a barrier

... until a forwarding progress is possible (right-hand rule)

transmissionrange

s

t

greedy

recoverygreedy

routing time depends on the size of barriers!

right-hand ruleneeds planartopology!

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Algorithms and ComplexityThe Cell Structure

transmission radius(Unit Disk Graph)

v

Define a grid consisting of l l squaresDefine a grid consisting of l l squares

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transmission radius(Unit Disk Graph)

v

nodes exchange beacon messages node v knows positions of ist neighbors

nodes exchange beacon messages node v knows positions of ist neighbors

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v

node cell link cell barrier cell

each node classifies the cells in ist transmission range

each node classifies the cells in ist transmission range

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v


each node includes the classification in its beacon messages (only constant overhead)

each node includes the classification in its beacon messages (only constant overhead)

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Algorithms and ComplexityRouting based on the Cell Structure

• Routing based on the cell structure uses cell pathscell path = sequence of orthogonally neighboring cells

• Paths in the original network (here: unit disk graph) and cell paths are equivalent up to a constant factor

• no planarization strategy needed(required for recovery using the right-hand rule)

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Routing based on the Cell Structure

v

virtual forwarding using cellsvirtual forwarding using cells

w

physical forwarding from v to w, if visibility range is exceeded

physical forwarding from v to w, if visibility range is exceeded

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Part II

Performance Measures and Algorithms

Part II

Performance Measures and Algorithms

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Algorithms and ComplexityPerformance Measures

• barriers make routing difficult• what is the worst case scenario?

it depends ...

• how difficult is a scenario?• what would the best algorithm do?

comparative ratios

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Algorithms and ComplexityHow difficult is a scenario?

barrier

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Algorithms and ComplexityHow difficult is a scenario?

perimeterperimeter

barrier

perimeter (p) = number of border cells

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Algorithms and ComplexityWhat would the best algorithm do?

length of shortest barrier-free cell path (h)

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• competitive ratio:

• competitive time ratio of a routing algorithm– h = length of shortest barrier-free path– algorithm needs T rounds to deliver a message

Competitive Ratio

solution of the algorithm

optimal offline solution cf. [Borodin, El-Yanif, 1998]

h

T

single-path

„“

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• optimal (offline) solution for traffic:h messages (length of shortest path)

• this is unfair, because ...– offline algorithm knows the barriers– but every online algorithm has to pay

exploration costs• exploration costs:

sum of perimeters of all barriers (p)

• comparative traffic ratio cf. [Koutsoupias, Papadimitriou 2000]

Comparative Ratios

M = # messages usedh = length of shortest pathp = sum of perimeters

h+p

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Algorithms and ComplexityComparative Ratios

• measure for time efficiency:

competitive time ratio

• measure for traffic efficiency:

comparative traffic ratio

• Combined comparative ratio

time efficiency and traffic efficiency

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Algorithms and ComplexityAlgorithms under Comparative Measures

• Sinlge-path strategies:no parallelism, traffic-efficient (time = traffic)example: GuideLine/Recovery– follow a guide line connecting source and target– traverse all barriers intersecting the guide line

Time and Traffic:

• Multi-path strategies: speed-up by parallel exploration, increasing trafficexample: Expanding Ring Search– start flooding with restricted search depth– if target is not in reach then

repeat with double search depth

Time: Traffic:

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Algorithms and ComplexityAlgorithms under Comparative Measures

GuideLine/Recovery (single-path)

Expanding Ring Search (multi-path)

traffictime

scenario

maze

open space

GuideLine/Recovery (single-path)

Expanding Ring Search (multi-path)

time ratio

trafficratio

combinedratio

Is that good?

It depends ... on the

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Algorithms and ComplexityThe Alternating Algorithm

... uses a combination of both strategies:

1. i = 1

2. d = 2i

3. start GuideLine/Recovery with time-to-live = d3/2

4. if the target is not reached thenstart Flooding with time-to-live = d

5. if the target is not reached theni = 2 · igoto line 2

Combined comparative ratio:

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Algorithms and ComplexityConclusion

• cell structure abstracts from graph theoretical issues

• neighborhood information (= cell classification) causes only constant overhead in beacon messages

• implicit planarization,well-suited for position-based routing

• comparative performance measuresin relation to the difficulty of the scenario (optimal distance & perimeter of barriers)

• time and traffic efficiency

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Thank you for your attention!

Questions ...

Thank you for your attention!

Questions ...

Stefan Rü[email protected].: +49 5251 60-6722Fax: +49 5251 60-6482

Algorithms and ComplexityHeinz Nixdof InstituteUniversity of PaderbornFürstenallee 1133102 Paderborn, Germany

Competitive Time and Traffic Analysis of Position-based Routing using a Cell-Structure

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