Competitive Time and Traffic Analysis of Position-based Routing using a Cell-Structure
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Transcript of Competitive Time and Traffic Analysis of Position-based Routing using a Cell-Structure
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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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Stefan Rührup 2
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
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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Stefan Rührup 3
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and Complexity
Part I
Topology Control for Position-based Routing
Part I
Topology Control for Position-based Routing
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Stefan Rührup 4
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
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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Stefan Rührup 5
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
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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Stefan Rührup 6
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and Complexity
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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Stefan Rührup 7
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
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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Stefan Rührup 8
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and ComplexityThe Cell Structure
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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Stefan Rührup 9
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and ComplexityThe Cell Structure
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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Stefan Rührup 10
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and ComplexityThe Cell Structure
v
node cell link cell barrier cell
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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Stefan Rührup 11
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
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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Stefan Rührup 12
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and Complexity
node cell link cell barrier cell
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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Stefan Rührup 13
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and Complexity
Part II
Performance Measures and Algorithms
Part II
Performance Measures and Algorithms
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Stefan Rührup 14
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
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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Stefan Rührup 15
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and ComplexityHow difficult is a scenario?
barrier
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Stefan Rührup 16
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and ComplexityHow difficult is a scenario?
perimeterperimeter
barrier
perimeter (p) = number of border cells
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Stefan Rührup 17
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and ComplexityWhat would the best algorithm do?
length of shortest barrier-free cell path (h)
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Stefan Rührup 18
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and Complexity
• 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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Stefan Rührup 19
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and Complexity
• 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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Stefan Rührup 20
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
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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Stefan Rührup 21
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
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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Stefan Rührup 22
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
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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Stefan Rührup 23
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
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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Stefan Rührup 24
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
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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Stefan Rührup 25
HEINZ NIXDORF INSTITUTEUniversity of Paderborn, Germany
Algorithms and Complexity
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