online routing in faulty meshes with sub-linear comparative time and traffic ratio
DESCRIPTION
Online Routing in Faulty Meshes with Sub-linear Comparative Time and Traffic Ratio. Stefan Ruehrup Christian Schindelhauer Heinz Nixdorf Institute University of Paderborn Germany. Overview. Routing in faulty mesh networks Routing as an online problem - PowerPoint PPT PresentationTRANSCRIPT
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HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
Online Routing in Faulty Mesheswith Sub-linear Comparative Time and
Traffic Ratio
Stefan RuehrupChristian Schindelhauer
Heinz Nixdorf Institute
University of Paderborn
Germany
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
2
Overview
• Routing in faulty mesh networks
• Routing as an online problem
• Basic strategies: single-path versus multi-path
• Comparative performance measures
• Our algorithm
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
3
Online Routing in Faulty Meshes
• Mesh Network with Faulty Nodes:
• Problem: Route a message from a source node to a target
active nodeactive node
faulty nodefaulty node
s
t
targettarget
sourcesource
routing pathrouting path
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
4
Offline versus Online Routing
• Routing with global knowledge(offline) is easy
• But if the faulty parts are not known in advance?
• Online Routing:
– no knowledge about the network
– no routing tables
– only neighboring nodes can identifyfaulty nodes
s
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
5
Why Online Routing is difficult
• Faulty nodes form barriers
• barriers can be like mazes
• Online routing in a faulty network = search a point in a maze
• Related problems:
navigation in an unknown terrain, maze traversal,
graph exploration, position-based routing
perimeterperimeter
barrierbarrier
s
t
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
6
Basic Strategies: Single-path
• Barrier Traversal
– follow a straight line connecting source and target
– traverse all barriers intersecting the line
– leave at nearest intersection point
• Time and traffic: h = optimal hop-distance
p = sum of perimeters
• no parallelism, traffic-efficient
Problem: time consuming, if many barriers
s t
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
7
Basic Strategies: Multi-path
• Expanding Ring Search:
– start flooding with restricted search depth
– if target is not in reach thenrepeat with double search depth
• Time: Traffic: h = optimal hop-distance
• asymptotically time optimal
Problem: traffic overhead, if few barriers
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
8
Competitive Time Ratio
• competitive ratio:
• competitive time ratio of a routing algorithm:
–h = optimal hop-distance
– algorithm needs T rounds to deliver a message
solution of the algorithmoptimal offline solution cf. [Borodin, El-Yanif, 1998]
„“
h
T
single-path
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
9
M = # messages usedh = length of shortest pathp = sum of perimeters
• optimal (offline) solution for traffic:h messages (length of shortest path)
• this is unfair, because ...
– offline algorithm knows all barriers
– but every online algorithm has to pay exploration costs
• exploration costs: sum of perimeters of all barriers (p)
• comparative traffic ratio:
h+p
Comparative Traffic Ratio
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
10
Comparative 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 Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
11
Algorithms under Comparative Measures
Barrier Traversal (single-path)
Expanding Ring Search (multi-path)
traffictime
scenario
maze
open space
Barrier Traversal (single-path)
Expanding Ring Search (multi-path)
time ratio
trafficratio
combinedratio
Is that good?
It depends ... on the
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
12
How to beat the linear ratio
1. define a search area (including source and target)
2. subdivide the search area into squares (“frames”)
3. traverse the frames efficiently decision: traversal or flooding?
4. enlarge the search area, if the target is not reached
s t
1 23
4
barrierbarrier
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
13
Frame Multicast Problem
• Inform every node on the frame as fast as possible goal: constant competitive ratio
• Traverse and Search: frameframe
entry node starts frame traversal
entry node starts frame traversal
traversal stopped, start expanding ring searchtraversal stopped, start expanding ring search
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
14
Performance of Traverse and Search
• Traverse and Search in a mesh of size g x g
– Time: constant competitive ratio
– Traffic:
1. frame traversal
2. flooded area is quadratic in the number of barrier nodes
... but also bounded by g2
3. concurrent exploration costs a logarithmic factor
1 2 3
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
15
Recursive Traverse and Search
• Expanding ring search inside a frame:
–Subdivide the flooded area in sub-frames
– apply Traverse and Search on sub-frames
• Traffic:
1st recursion:
(g1g1-frame subdivided into g0g0-frames)
2nd recursion:
3rd recursion ...
• Time: constant factor grows exponentially in #recursions
replaced by toplevel framereplaced by toplevel frame
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
16
Overall Asymptotic Performance
• Toplevel frame = 1/4 search area, size = h2
• With an appropriate choice of g0, g1, ..., gl :
• Time:
• Traffic:
• combined comparative ratio:
• sub-linear, i.e. for all
compared to
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
17
Conclusion
• Our algorithm is
– nearly as fast as flooding ... and traffic efficient
– approaches the online lower bound for traffic
• Open question:
Can time and traffic be optimized at the same time?
... or is there a trade-off?
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Stefan Ruehrup
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
18
Thank you for your attention!
Questions ...
Thank you for your attention!
Questions ...
Stefan [email protected].: +49 5251 60-6722Fax: +49 5251 60-6482
Algorithms and ComplexityHeinz Nixdof InstituteUniversity of PaderbornFuerstenallee 1133102 Paderborn, Germany