AdHoc Networks Beyond Unit Disk Graphs

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Ad-Hoc Networks Beyond Unit Disk Graphs
Fabian Kuhn Roger Wattenhofer Aaron Zollinger
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Overview

• Introduction
• Graph Models for Mobile Ad-Hoc Networks
• Quasi Unit Disk Graphs
• Related Work
• Volatile Memory Routing
• Flooding
• Lower Bound for Message Complexity
• Topology for Optimal Flooding
• Greedy Flooding
• Volatile Memory Routing Algorithms
• Combining Greedy and Flooding
• Geometric Routing for
• How to obtain a planar graph
• Optimality of AFR/GOAFR

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Mobile Ad-Hoc Networks

• Mobile Devices communicating via radio
• Network without centralized control (base
  station)
• We consider the abstraction level of graphs

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Graph Models for Ad-Hoc Networks

• Simple Models
• Unit Disk Graph is most widely applied model
• Underlying assumptionAll nodes are in R2, have
  exactly the same transmission range (normalized
  to one), and there are no obstacles.
• Far from realityBUT There are numerous
  theoretical results.

• Realistic Models
• We need more general graph models
• However, arbitrary graphs are too general to
  obtain strong results for routing, etc.
• We need something between UDG and arbitrary
  graphs
• general enough to model reality as close as
  possible
• restrictive enough to allow useful theoretical
  results

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Quasi Unit Disk Graph

• Definition Unit Disk Graph
• Edge between u and v if u-v1
• No edge between u and v if u-vgt1
• Definition Quasi Unit Disk Graph
• Edge between u and v if u-vd
• No edge between u and v if u-vgt1
• May have an edge if dltu-vlt1

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Related Work

• The Quasi Unit Disk Graph model is not
  newBarrière, Fraigniaud, and Narayanan have
  shown that correct geometric routing is possible
  if .Dial-M 2001 and Wireless
  Networks Journal Vol. 3(2) 2003
• Other generalizations of the unit disk graph have
  been proposed, e.g. (r,s)-civilized graphs by
  Krumke, Marathe, and RaviDial-M 1998 and
  Wireless Networks Journal Vol. 7(6) 2001

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Volatile Memory Routing

• We want to consider routing without routing
  tables
• We need to allow nodes to temporarily store some
  information
• Volatile Memory Routing AlgorithmFor each
  message, each node is allowed to temporarily
  store O(log n) bits.(temporary while the
  message has not reached the destination)

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Overview

• Introduction
• Graph Models for Mobile Ad-Hoc Networks
• Quasi Unit Disk Graphs
• Related Work
• Volatile Memory Routing
• Flooding
• Lower Bound for Message Complexity
• Topology for Optimal Flooding
• Greedy Flooding
• Volatile Memory Routing Algorithms
• Combining Greedy and Flooding
• Geometric Routing for
• How to obtain a planar graph
• Optimality of AFR/GOAFR

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Message Complexity Lower Bound

• Lower Bound Graph is a Quasi Unit Disk Graph
  (parameter d)
• To find destination t, all vertical chains (all
  nodes) have to be visited
• Length of one chain c
• Optimal path has length O(c)
• There are O(c2/d2) nodes! O(c2/d2) messages

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Flooding

• Flooding on the Quasi UDG gives unbounded message
  complexity
• We need a subgraph on which flooding is efficient
  (a kind of topology control)
• Desired properties
• Nodes form a dominating set
• O(A/d2) nodes per area A
• O(A/d2) edges per area A
• Optional spanner

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Topology Control I

• Construct a Minimal Independent Set (MIS)!
  dominating set and O(A/d2) nodes per area A
• If we make all 2- and 3-hop connections, we have
  a spanner, but too many nodes and edges
• Solution Choose only a subset of the 2- and
  3-hop connections (virtual edges of length 3)

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Topology Control II
Place grid over the nodes (cell size 6)
Add another grid shifted by (3,3)
Add another grid shifted by (3,0)
Add another grid shifted by (0,3)
Each virtual edge is completely covered by a
cell of at least one of the grids
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Topology Control III

• In each cell, we calculate a spanner of the nodes
  (MIS) and virtual edges lying completely inside
  the cell! Applying a randomized construction of
  Linial and Saks (SODA 91) yields a
  O(log(1/d2))-spanner with O(1/d2) virtual edges.
• Combining all local spanners gives a
  O(log(1/d2))-spanner with O(A/d2) virtual edges
  per area A.! Backbone Graph

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Flooding on the Backbone Graph

• Flooding/Echo with exponentially growing TTL on
  the Backbone Graph gives
• O(log(1/d2)c) time and O(c2/d2) message
  complexity in the synchronous model
• O(log(1/d2)log3(c/d)c) time and
  O(log3(c/d)c2/d2) message complexity in the
  asynchronous model (using a synchronizer
  described by Awerbuch and Peleg, FOCS 90)
• Geometric Flooding/Echo uses disks with
  exponentially growing radius instead of TTL
• O(c2/d2) time and message complexity (synchronous
  and asynchronous)

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Overview

• Introduction
• Graph Models for Mobile Ad-Hoc Networks
• Quasi Unit Disk Graphs
• Related Work
• Volatile Memory Routing
• Flooding
• Lower Bound for Message Complexity
• Topology for Optimal Flooding
• Greedy Flooding
• Geometric (Volatile Memory) Routing Algorithms
• Combining Greedy and Flooding
• Geometric Routing for
• How to obtain a planar graph
• Optimality of AFR/GOAFR

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Geometric Routing

• A.k.a. location-based, position-based,
  geographic, etc.
• Each node knows its own position and position of
  neighbors
• Source knows the position of the destination
• No routing tables in the nodes, all routing
  information is in the message!
• Volatile Geometric Routing
• Geometric Routing O(log n) bits per message in
  each node (while message is on the way from s to
  t)

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Geometric Routing
???
t
s
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Greedy Routing

• Each node forwards message to best neighbor

t
s
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Greedy Routing

• Each node forwards message to best neighbor
• But greedy routing may fail message may get
  stuck in a dead end
• Needed Correct geometric routing algorithm

t
?
s
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Greedy Flooding

• Straight-forward idea to make greedy routing
  correct(i.e. always find the destination)!
  combine greedy and flooding
• We want to keep worst-case optimality
• Apply geometric flooding with exponentially
  increasing radius and the right criterion to fall
  back from flooding to greedy

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Greedy Flooding, Fall Back Criterion

• Flooding phase with radii r0, r1, where rir02i
• Flooding starts at node u, node vi is best
  (closest to destination t) node for radius ri
• Go back to greedy if u-t - vi-t qri (q is
  a predefined constant)
• Message and time complexity O(c2/d2)
• Simulations on UDG suggest that the algorithm is
  efficient in the average case

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Overview

• Introduction
• Graph Models for Mobile Ad-Hoc Networks
• Quasi Unit Disk Graphs
• Related Work
• Volatile Memory Routing
• Flooding
• Lower Bound for Message Complexity
• Topology for Optimal Flooding
• Greedy Flooding
• Geometric (Volatile Memory) Routing Algorithms
• Combining Greedy and Flooding
• Geometric Routing for
• How to obtain a planar graph
• Optimality of AFR/GOAFR

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Questions?Comments?

• ???