HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Algorithms for Radio Networks
Winter Term 2005/2006
02 Nov 2005
3rd Lecture
Christian Schindelhauer
[email protected]
1
HEINZ NIXDORF INSTITUTE
Theory of Wireless Routing
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
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2
HEINZ NIXDORF INSTITUTE
A Simple Physical Network Model
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Homogenous Network of
– n radio stations s1,..,sn on the plane
Radio transmission
– One frequency
– Adjustable transmission range
• Maximum range > maximum distance
of radio stations
• Inside the transmission area of sender:
clear signal or radio interference
• Outside: no signal
– Packets of unit length
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3
HEINZ NIXDORF INSTITUTE
The Routing Problem
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
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
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Formal Definition of Interference
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Let Dr(u) the disk of radius u
with center u in the plane
Define for an edge e={u,v}
D(e) = Dr(u)  Dr(v)
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (LZW)“
benötigt.
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 max{|Int(e} | e  E}
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5
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Formal Definition of Congestion
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
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.
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6
HEINZ NIXDORF INSTITUTE
Energy
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
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
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HEINZ NIXDORF INSTITUTE
Three Parts of the Routing Problem
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
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
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8
HEINZ NIXDORF INSTITUTE
A Lower Bound for the Routing Time
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
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
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9
HEINZ NIXDORF INSTITUTE
A short preview to MAC
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
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
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10
HEINZ NIXDORF INSTITUTE
Proof
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Lemma
Proof:
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11
HEINZ NIXDORF INSTITUTE
An Upper Bound for Routing
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
This Lemma can be used to prove the following Theorem
Theorem
Proof omitted here.
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12
HEINZ NIXDORF INSTITUTE
Minimizing Unit Energy
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
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)
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13
HEINZ NIXDORF INSTITUTE
Minimizing Flow Energy
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Definition Gabriel Graph
Example: Edge c is not allowed in the Gabriel Graph
Zur Anzeige wird der QuickTime™
Dekompressor „TIFF (LZW)“
benötigt.
Theorem
Proof by applying the Theorem of Thales
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14
HEINZ NIXDORF INSTITUTE
Worst Construction for Interferences
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Interference Number for n nodes = n-1
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A Measure for the Ugliness of Positions
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
For a network G=(V,E) define the Diversity as
Properties of the diversity:
– g(V)=(log n)
– g(V)=O(n)
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HEINZ NIXDORF INSTITUTE
g(V) = O(log n) for Random Points
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
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 c>0.
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 c=2k-2) the minimum distance of
neighbored nodes is at least n-k
– With this probability g(V) = O(log n)
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HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Christian Schindelhauer
Thanks for your attention
End of 3rd lecture
Next lecture:
Next exercise class:
Mo 09 Nov 2005, 4pm, F1.110
Tu 08 Oct 2005, 1.15 pm, F2.211 or
Th 10 Nov 2005, 1.15 pm, F1.110
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