online routing in faulty meshes with sub-linear comparative time and traffic ratio

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

Stefan Ruehrup



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Overview

• Routing in faulty mesh networks

• Routing as an online problem

• Basic strategies: single-path versus multi-path

• Comparative performance measures

• Our algorithm

Stefan Ruehrup



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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

Stefan Ruehrup



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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

Stefan Ruehrup



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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

Stefan Ruehrup



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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

Stefan Ruehrup



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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

Stefan Ruehrup



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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

Stefan Ruehrup



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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

Stefan Ruehrup



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Comparative Ratios

• measure for time efficiency:

competitive time ratio

• measure for traffic efficiency:

comparative traffic ratio

• Combined comparative ratio

time efficiency and traffic efficiency

Stefan Ruehrup



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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

Stefan Ruehrup



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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

Stefan Ruehrup



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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

Stefan Ruehrup



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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

Stefan Ruehrup



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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

Stefan Ruehrup



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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

Stefan Ruehrup



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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?

Stefan Ruehrup



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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

online routing in faulty meshes with sub-linear comparative time and traffic ratio

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