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PPT – Algorithms for Radio Networks Winter Term 2005/2006 02 Nov 2005 3rd Lecture PowerPoint presentation | free to download - id: 275ae4-ZDc1Z

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Algorithms for Radio Networks Winter Term

2005/2006 02 Nov 2005 3rd Lecture

- Christian Schindelhauer
- schindel_at_upb.de

Theory of Wireless Routing

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

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

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

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.

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

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

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

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

Proof

- Lemma
- Proof

An Upper Bound for Routing

- This Lemma can be used to prove the following

Theorem - Theorem
- Proof omitted here.

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)

Minimizing Flow Energy

- Definition Gabriel Graph
- Example Edge c is not allowed in the Gabriel

Graph - Theorem
- Proof by applying the Theorem of Thales

Worst Construction for Interferences

- Interference Number for n nodes n-1

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)

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)

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