Online Routing in Faulty Meshes with Sublinear Comparative Time and Traffic Ratio

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Online Routing in Faulty Mesheswith Sub-linear
Comparative Time and Traffic Ratio

• Stefan Ruehrup
• Christian Schindelhauer
• Heinz Nixdorf Institute
• University of Paderborn
• Germany

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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

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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

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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
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Why Online Routing is difficult
barrier

• 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

s
perimeter
t
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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

s
t
Problem time consuming, if many barriers
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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
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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 algorithm
optimal offline solution
cf. Borodin, El-Yanif, 1998
T
h
single-path
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Comparative Traffic Ratio

• optimal (offline) solution for traffich
  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

M messages used h length of shortest path p
sum of perimeters
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Comparative Ratios

• measure for time efficiency competitive time
  ratio
• measure for traffic efficiency comparative
  traffic ratio
• Combined comparative ratiotime efficiency and
  traffic efficiency

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Algorithms under Comparative Measures
traffic
time
Barrier Traversal (single-path)
Expanding Ring Search (multi-path)
Is that good?
scenario
It depends ...
on the
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How to beat the linear ratio

• define a search area (including source and
  target)
• subdivide the search area into squares (frames)
• traverse the frames efficiently ? decision
  traversal or flooding?
• enlarge the search area, if the target is not
  reached

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3
1
2
s
t
barrier
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Frame Multicast Problem

• Inform every node on the frame as fast as
  possible goal constant competitive ratio
• Traverse and Search

frame
traversal stopped, start expanding ring search
entry node starts frame traversal
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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
• frame traversal
• flooded area is quadratic in the number of
  barrier nodes... but also bounded by g2
• concurrent exploration costs a logarithmic factor

3
1
2
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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 (g1?g1-frame subdivided into
  g0?g0-frames)
• 2nd recursion
• 3rd recursion ...
• Time constant factor grows exponentially in
  recursions

replaced by toplevel frame
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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
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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?

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Thank you for your attention! Questions ...
Stefan Ruehrup sr_at_upb.de Tel. 49 5251
60-6722 Fax [phone]
Algorithms and Complexity Heinz Nixdof
Institute University of Paderborn Fuerstenallee
11 33102 Paderborn, Germany