Efficient Operation of Wireless
Packet Networks Using Network
Coding
Desmond S. Lun, Muriel M´edard, and Ralf Koetter
指導老師:許子衡 老師
學生:羅英辰
學號:M97G0216
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Introduction
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Lossless Networks
There is a significant effect on many network
problems, particularly multicast, where it has
been given the name, the “wireless multicast
advantage”.
 Minimum-energy broadcast problem

 Wireline：Various
minimum-weight spanning tree
algorithm.<NP-complete>
 Multicast Incremental Power(MIP)
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Random multicast connections.
 Random wireless networks of varying size.
 Reductions ranging from 13% to 49% .

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A.Model
A directed hypergraph H=(N,A)
A is the set of hyperarcs.
N is the set of nodes.
 A hyperarc is a pair (i,J).

i
are start nodes, which are included N.
J
are end nodes, which are non-empty subset of N.
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(i,J) represents a lossless broadcast link from i
to nodes in the non-empty set J.
 zi,J is the average rate at which packets are
injected and received on hyperarc (i,J).
 The rate vector z is called the coding subgraph
and can be varied within a constraint set Z
dictated to us by lower layers.

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B.Single multicast connections

A multicast of rate arbitrarily close to R is
achievable with coding from source node s to
t T
sink nodes in the set T.
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From such a coding subgraph, the connection
can be straightforwardly achieved using the
decentralized, random coding schemes or by
modifying the deterministic coding schemes.
 To establish a minimum-cost multicast
connection in a lossless wireless packet
network.

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
These two problems：
 Subgraph
selection program
 Coding program

Subgraph selection program：
f is a cost function.
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We have Z  0.1 A
And f z   (i , J )A ai , J zi , J
•ai,J is the cost per unit rate.
We have flows x(1)
and x(2) of unit size from
t1 to t2.
. zi , J  max

(1)
( 2)
x
,
x
jJ iJj  jJ iJj
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

It achieves the optimal cost of 5/2.
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

The constraint set not only necessarily induces a
degree of coupling among separate links because of
contention for the wireless medium, it is also usually
difficult to describe.
Heuristic approach is to find a set of feasible
constraint sets {Z1, Z2, . . . , ZN} (for example, each
Zn could correspond to the links established by some
set of noninterfering transmitters)
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N
N
n 1
n 1
Z    n Z n ,   0,   n  1
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The constraint set Z poses less of a problem in
low energy systems because nodes seldom
transmit and there is less contention.
 If energy is the most limiting constraint and we
wish to achieve minimum-energy multicast
without regard for throughput or bandwidth,
then Z can be dropped altogether.

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 ai,J
represents the energy required to transmit
to nodes in J from node I for some fixed time
interval.
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