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Algorithms for Radio Networks Winter Term 2005/2006 02 Nov 2005 3rd Lecture

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Algorithms and Complexity. Christian Schindelhauer. Theory of Wireless Routing ... Example: Edge c is not allowed in the Gabriel Graph. Theorem ... – PowerPoint PPT presentation

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Title: Algorithms for Radio Networks Winter Term 2005/2006 02 Nov 2005 3rd Lecture


1
Algorithms for Radio Networks Winter Term
2005/2006 02 Nov 2005 3rd Lecture
  • Christian Schindelhauer
  • schindel_at_upb.de

2
Theory of Wireless Routing
3
A Simple Physical Network Model
  • Homogenous Network of
  • n radio stations s1,..,sn on the plane
  • Radio transmission
  • One frequency
  • Adjustable transmission range
  • Maximum range gt maximum distance of radio
    stations
  • Inside the transmission area of sender clear
    signal or radio interference
  • Outside no signal
  • Packets of unit length

4
The Routing Problem
  • Given
  • n points in the plane, V(v1,..,vn )
  • representing mobile nodes of a mobile ad hoc
    network
  • the complete undirected graph G (V,E) as
    possible communication network
  • representing a MANET where every connection can
    be established
  • Routing problem
  • f V ? V ? N, where f(u,v) packets have to be
    sent from u to v, for al u,v ? V
  • Find a path for each packet of this routing
    problem in the complete graph
  • Let
  • The union of all path systems is called the Link
    Network or Communication Network

5
Formal Definition of Interference
  • Let Dr(u) the disk of radius u with center u in
    the plane
  • Define for an edge eu,v D(e) Dr(u) ? Dr(v)
  • The set of edges interfering with an edge e
    u,v of a communication network N is defined
    as
  • The Interference Number of an edge is given by
    Int(e)
  • The Interference Number of the Network is
    maxInt(e e ? E

6
Formal Definition of Congestion
  • The Congestion of an edge e is defined as
  • The Congestion of the path system P is defined as
  • The Dilation D(P) of a path system is the length
    of the longest path.

7
Energy
  • The energy for transmission of a message can be
    modeled by a power over the distance d between
    sender and transceiver
  • Two energy models
  • Unit energy accounts only the energy for
    upholding an edge
  • Idea messages can be aggregated and sent as one
    packet
  • Flow Energy Model every message is counted
    separately

8
Three Parts of the Routing Problem
  • Path Selection
  • select a path system P for the routing problem
  • Interference handling
  • Design a strategy that can handle the
    transmission problem of a packet along a link
  • Packet switching
  • Decide when and in which order packets are sent
    along a link

9
A Lower Bound for the Routing Time
  • A routing schedule is a timeline which describes
    for each message when it is passed along its path
    in the path system.
  • A routing schedule is valid if no interferences
    occur
  • The routing time is the number of steps of a
    routing schedule.
  • The optimal routing time for a given demand is
    the number of steps of the minimal valid routing
    schedule.
  • Theorem 1

10
A short preview to MAC
  • The Problem of Medium Access Protocols is to
    decide when to send a message over the radio
    channel.
  • If the congestion of an edge is known one can use
    the following simple probabilistic protocol
  • Activate link e with probability ?(e) where
  • Lemma

11
Proof
  • Lemma
  • Proof

12
An Upper Bound for Routing
  • This Lemma can be used to prove the following
    Theorem
  • Theorem
  • Proof omitted here.

13
Minimizing Unit Energy
  • Theorem
  • Proof
  • Only trees can optimize unit energy
  • In a graph with cycles at least one edge can be
    erased while decresing unit energy
  • Note that MST also optimizes the energy (exercise)

14
Minimizing Flow Energy
  • Definition Gabriel Graph
  • Example Edge c is not allowed in the Gabriel
    Graph
  • Theorem
  • Proof by applying the Theorem of Thales

15
Worst Construction for Interferences
  • Interference Number for n nodes n-1

16
A Measure for the Ugliness of Positions
  • For a network G(V,E) define the Diversity as
  • Properties of the diversity
  • g(V)?(log n)
  • g(V)O(n)

17
g(V) O(log n) for Random Points
  • Lemma
  • Given n points V distributed by an independent
    random process over a square, i.e. choose (x,y)
    with x and y are chosen randomly from 0,1,
  • the diversity of V ist bounded by g(V) O(log n)
    with high probability, i.e. 1-n-c for some
    constant cgt0.
  • Proof
  • Consider a grid of nk x nk cells of the unit
    square of dividing it into squares of area n-2k
  • The probability that a node is in such a cell is
    n-2k
  • The probability that one of the 8 neighbored
    cells of an non-empty cell is occupied is
    therefore at most 8 n-2k
  • The probability that all non-empty cells are
    surrounded by such empty cells is at most
  • So with probability at least 1-n-c (for c2k-2)
    the minimum distance of neighbored nodes is at
    least n-k
  • With this probability g(V) O(log n)

18
Thanks for your attention End of 3rd
lecture Next lecture Mo 09 Nov 2005, 4pm,
F1.110 Next exercise class Tu 08 Oct 2005, 1.15
pm, F2.211 or Th 10 Nov 2005, 1.15 pm, F1.110
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