Lineær netværkskodning: Hvordan man løser et lineært netværkskodningsproblem med Gröbner-baser

79 Pages·2012·0.684 MB·Danish

by  Thomas Vestergaard

Lineær netværkskodning
Hvordan man løser et lineært netværkskodningsproblem
med Gro¨bner-baser
Thomas Vestergaard
Specialeafhandling i Matematik
Vejleder: Diego Ruano
Aalborg Universitet
Institut for Matematiske Fag
Fredrik Bajers Vej 7 G • 9220 Aalborg Øst • Danmark

Institut for Matematiske Fag
Fredrik Bajers Vej 7G
9220 Aalborg Øst
Telefon [phone]
Fax [phone]
http://www.math.aau.dk
Synopsis:
Form˚alet med denne specialeafhandling
er at forst˚a og forklare artiklen Aspects
of random network coding af Lektor
ved Aalborg Universitet Olav Geil og
Titel: Lineær netværkskodning
tidligere Phd-studerende Casper Thom-
sen, og herunder Gr¨obnerbasisteori. Et af
Projektperiode: 1.sep. 2011-6.jan.2012
hovedresultaterne er Fodaftryksgrænsen,
som er med til at give et kriterium for,
Projektgruppe: G4-103
hvorn˚ar et lineært netværkskodnings-
problem er løseligt. For at bevise dette
Skrevet af:
kriterium benyttes flere store sætninger
fraGr¨obnerbasisteorien, hvilketinkluderer
resulater som Dicksons lemma, Hilberts
basissætning, Buchbergers kriterium og
Thomas Vestergaard
Buchbergers algoritme. Til at eksempli-
ficere resultater fra Gr¨obnerbasisteorien
anvendes computerprogrammet Singular.
Vejleder: Diego Ruano
Efter at have opbygget en værkstøjskasse
Oplagstal: 4 til forst˚aelse af begrebet netværkskodning,
beskrives først id´een med netværk-
Sidetal: 79 skodning, hvorefter en mere teoretisk
gennemgang følger. Det vil herefter
Bilagsantal og –art: Ingen blive vist, hvordan man sikrer en høj
sandsynlighed for at alle modtagere i et
Afsluttet: 6. jan. 2012. netværk f˚ar al den ønskede information. I
afsnittet der følger præsenteres endvidere
en algoritme til bestemmelse af flows ud
fra et givet flowsystem til et netværk. Som
anvendelse og illustration af begreberne
tages udgangspunkt i Butterfly-netværket.
Projektrapportens indhold er frit tilgængeligt, offentliggørelse er tilladt med kildeangivelse.
3

Kapitel 1
Abstract
Network Coding Theory is one of the newest research areas within mathematics end engineering.
Forthelast20yearstheresearchhasbeenmovingforward,andtheideaisusedmanyapplications
from the everyday life. One example is wireless cellphones. The cellphones are connected in a data
transmitting network. The objective of this thesis is to understand and explain the article Aspects
of random network coding by associate professor at Aalborg University H. Olav Geil and former
Phd. student Casper Thomsen.
A network or more specifically a computer network is a collection of computers connected, so they
can exchange data. Network coding is quite simply the transmission of data from a source to a
certain amount of receivers, and transmission must be noiseless. That is, there can not be any
data loss on the way through the network. Data transmitted over point-to-point distances in a
network, is best illustrated by a directed graph G(V,E). Here V is the nodes and E is the edges.
By S = {s ,...,s } ⊆ V, we mean the senders and by R = {r ,...,r } ⊆ V, we mean the
1 |S| 1 |R|
recievers of the network. One of the most simple network examples is shown on the frontpage of
the report. It is called a Butterfly network because of its structure, which is similar to the body
of a butterfly. The article Aspects of random network coding and therefore the report focuses on
linear networkcoding.Networkcodingislinear,whenthemessagesenttakesvaluesinafinalfield
F , and when the coding and decoding is done linearly over F .
q q
Now, a natural question to ask regarding a network is whether a given network actually works
and transmits every piece of information to all recievers, and if the network does it effectively.
Effectively refers to if the transmission of data is done within a standard time limit. To answer
these questions we will turn to the theory of a Groebner basis. A Groebner basis G is a particular
kind of generating subset of an ideal I in a polynomial ring k[x ,...,x ], and they have many
1 n
applications. Groebnerbases was firstly introduced by professor of Computer Mathematics Bruno
Buchberger, who namedthem after his advisorWolfgang Groebner.The reason why they are such
apowerfultoolinbothmathematicsandengineeringis,thattheyarerelativelyeasytounderstand.
But most importantly they solve complicated problems regarding final fields.
The first three chapters of the thesis are the abstract, the preface, and the introduction, and
their meaning is obvious. The first real chapter of the thesis is called Groebner basis theory,
and it holds several key results. The first section of chapter 4 holds some general information on
ideals in a polynomial ring k[x ,...,x ]. We then proceed by exploring monomial orderings. Such
1 n
5
orderings are important for submitting the division algorithm on polynomials, and the division
algorithm plays an important role in solving, what is known as the ideal membership problem: If
f ,...,f ∈ k[x ,...,x ], how do we decide if a polynomial f ∈ k[x ,...,x ] is contained in the
1 s 1 n 1 n
ideal (cid:104)f ,...,f (cid:105). The answer is stated in Buchbergers Criterion, section 4.5, which has to do with
1 s
something called S-polynomials.
Another problem concerning polynomial ideals is the ideal description problem: Is it possible to
write every polynomial ideal as (cid:104)f ,...,f (cid:105) for some f ,...,f ∈k[x ,...,x ]. The answer to this
1 s 1 s 1 n
is yes. In section 4.3 the first key result is presented. It is Dickson’s Lemma, which solves the ideal
descriptionproblemforaspecialkindofpolynomialideal.ThelemmaextendstoTheHilbertBasis
theorem,whichsolvestheproblemforallpolynomialideals.AsabonusTheHilbertBasisTheorem
gives rise to Groebner bases. Section 4.5 and 4.6 explores the applications of Groebner bases and
gives an algorithm for producing Groebner bases. Sections 4.7 and 4.8 explores the relationsship
between ideals and varieties, while sections 4.9 and 4.10 involves functions between varieties. The
goal of these sections are also to establish the Footprint Bound in the last section of this chapter.
This specific bound holds the key to linear network coding and marks the end of the chapter.
Chapter 5 is all about linear network coding. The chapter begins with a short introduction to
the topic, then a more theoretical approach follows. The section following shows how to ensure,
that there is a high probability that all receivers in a network gets all information. Afterwards an
algorithmfordeterminingtheflowfromagivenflowsystemispresented.Resultswillbeillustrated
through the Butterfly network. In the end we will see what else can be said about the probability
of succes.
6
Kapitel 2
Forord
DetteerenspecialeafhandlingindenforhovedretningenDiskretMatematikvedInstitutforMatem-
atiske Fag, Aalborg Universitet. Specialet er udarbejdet i perioden 1. sep. 2011 - 6. jan. 2012.
M˚alet med specialet er at forst˚a og forklare artiklen [2010 Geil & Thomsen] og herunder at forst˚a
ogforklareGr¨obnerbasisteoriognetværkskodning.Derforudsættesetbasaltkendskabtilgrafteori,
abstrakt algebra og lineær algebra. Specialet er et forsøg p˚a at kaste lys over netværkskodning for
almindelige studerende og ikke-professorere.
I rapporten er vigtige begreber skrevet med kursiv første gang de nævnes. Litteraturhenvisninger
angivessom(udgivelses˚ar,Forfatter(e))oglitteraturlistenkanfindesbagerstirapporten.Istarten
af hvert afsnit anføres, hvilke kilder afsnittet er baseret p˚a. Der gøres endvidere opmærksom p˚a,
atdervednoglesætningererudeladtbeviser,ogdervilistedetværeenhenvisningtilstedet,hvor
beviset kan findes. Henvisninger i parentes som (4.28) henviser til et ligningsnummer, mens sæt-
ningermv.henvisestilsomf.eks.sætning4.31.Tilnogleeksemplererbrugtcomputerprogrammet
Singular og Singular-koden vil være inkluderet. For mere information om programmet henvises til
http://www.singular.uni-kl.de. I nogle af eksemplerne vil Singular-koden være inkluderet.
Til slut vil jeg gerne takke min vejleder Diego Ruano for et stort engagement og faglig vejledning
gennem hele processen. Endvidere skal der lyde en tak til Martin Steffano for kyndig vejledning
indenfor særligt Gr¨obnerbasisteorien.
Thomas Vestergaard
Aalborg d. 6. jan. 2012
7

Indhold
1 Abstract 5
2 Forord 7
3 Indledning 11
4 Gr¨obnerbasisteori 13
4.1 Idealer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Monomiel ordning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 Monomielle idealer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 Hilberts Basissætning og Gr¨obnerbaser . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.5 Gr¨obnerbasis og dets egenskaber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.6 Buchbergers algoritme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.7 Hilberts Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.8 Radikale idealer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.9 Affine varieteters indbyrdes korrespondence . . . . . . . . . . . . . . . . . . . . . . 48
4.10 Polynomielle kvotientringe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.11 Fodaftryksgrænsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Random Network Coding 60
5.1 Introduktion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 Network coding problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3 Linear network coding for multicast . . . . . . . . . . . . . . . . . . . . . . . . . . 66
9
5.4 Flow-algoritme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5 Vilk˚arlig lineær netværkskodning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
10