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ใบรับรองวิทยานิพนธ
บัณฑิตวิทยาลัย มหาวิทยาลัยเกษตรศาสตร
วิศวกรรมศาสตรมหาบัณฑิต (วิศวกรรมคอมพิวเตอร)
ปริญญา
วิศวกรรมคอมพิวเตอร
ภาควิชา
วิศวกรรมคอมพิวเตอร
สาขา
เรื่อง
การเขารหัสเครือขายเชิงเสนสําหรับปญหาการสื่อสารระหวางตนทางปลายทางแบบหลายคู
Linear Network Coding for the Multiple Source-Sink Pair Communication Problem
นามผูวิจัย นายมุนนิ ทร เอี่ยมโอภาส
ไดพิจารณาเห็นชอบโดย
อาจารยที่ปรึกษาวิทยานิพนธหลัก
(
ผูชวยศาสตราจารยจิตรทัศน ฝกเจริญผล, Ph.D.
)
(
ผูชวยศาสตราจารยชัยพร ใจแกว, Ph.D.
)
(
ผูชวยศาสตราจารยภุชงค อุทโยภาศ, Ph.D.
)
อาจารยที่ปรึกษาวิทยานิพนธรวม
หัวหนาภาควิชา
บัณฑิตวิทยาลัย มหาวิทยาลัยเกษตรศาสตรรับรองแลว
(
รองศาสตราจารยกัญจนา ธีระกุล, D.Agr.
คณบดีบัณฑิตวิทยาลัย
วันที่
เดือน
พ.ศ.
)
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Munin Eamopas 2011: Linear Network Coding for the Multiple Source-Sink Pair
Communication Problem. Master of Engineering (Computer Engineering), Major
Field: Computer Engineering, Department of Computer Engineering. Thesis Advisor:
Assistant Professor Jittat Fakcharoenphol, Ph.D. 28 pages.
Network coding is an another technique to improve efficiency of the network. This
thesis has the objectives to study and to improve the network coding, and focus on the multiple
source-sink pair communication problem
Recently, Iwama et al. present an algorithm for k source-sink pair network coding,
when k is some fixed constant. The running time of the algorithm depends on the upper bound
on number of vertices performing encoding operations. Iwama et al prove that ℑ 3k encoding
vertices are sufficient to obtain the network coding whose operated on field ℑ . In this work,
we improve the bound to k 2|ℑ|2k and this tighter bound, in turns, improves the running time of
the algorithm proposed by Iwama et al. The techniques used are elementary linear algebra and
Dilworthps theorem.
/
Studentps signature
Thesis Advisorps signature
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18
ND@W@[b !?B@@@ ≺ ?BY >? u "! v V?XVb N! u ≺ v Z @@B@ v \WO\?N u
N!\? u [AAF v
MNO 1 D#$:6$
"#!$: k 2 ℑ k ?
C A# >#:6 k
C ′ ⊆ C 867:6"
k + 1 $!&:6 C ′ ⊆ C 867:6"
k ℑ + 1
]F^BBY[BZ\??N >?N]F^B Dilworth (1950)
#?BVBA?a>bNN>bBA?B@W"\?Nb !?
?Y\@@Ba>bB@B?WO k + 1 N @B>b? k ℑ k + 1 V?B
\@@B>bB?WO>bB@B?WO k ℑ k + 1 N@Ba>b? k + 1
MNO 2 D#$:6$
"#!$ : k 2 ℑ 2k A# >#:6"$
"#!$ C ′ 867 !=!>8$:& A> C ′ F867"
k + 1 $!&F867"
k
k ℑ +1
(D NB Q![ VC#@ k @ no!? ℑ ?Y Q!N" \?@A? ℑ k " @B>?@ k 2 ℑ 2k "!N @Bb >?B Q!@bO@B?@ k 2 ℑ k "!N
]FaBB 1 "? @Bb C ′ #VBY
!NBYN"?VDBB@Ba>b?@ k >b?@ k ℑ k bO
@B Q!@"! @W!BV>?!?!
\?>?@#!!\@!B"!
19
4 a>b? k+1 B@B Q!@
MNO 3 ?$# C F"$
"#!$867:6 !=!>8$:& A>:6"
: k + 1 $!!"#!$%!&"' σ 867?(#$
"#!$ Q $
>:6!"#!$
%!&"' σ ′ 867?(#$
"#!$ Q-1 $
A>"#:;9''89
#$:& σ
(D V>?Va>b v0, v1, J, vk @B@#!b[ u0, u1, J, uk @!? ?CB
4 "!@B σ (vi ) = (ai , bi ) N@#!" !>?@NQ!@@#! k
NO@W ri BV
k
u 0 = ∑ ri u i
\? ?bB>? v0 NV
i =1
k
@#!B v0 !B\ Ku0a0 − ∑ ri u i a0 @!BbB>? vi
i =1
k
\ a0ri 1 ≤ i ≤ k "!Q!@@#!B!B\N>? v1 WO vk + ∑ riuia0
i=1
?Y@WV@B!BV σ′(v0) = (0,b0) "! σ′(v0 ) = (ai + a0ri ,b0 )
@ 1 ≤ i ≤ k >?BQ!@#!\!@ σ \?
DBB[>bYNOY"!?"a>b\@\?N!B
>?V>bN@BQ!@#!"! Q!>?XV>b?
>?VBBYN@NDNDBB>b@BB 2 >?
MNO 4 D#:6%;$
v1,v2 867
- v1 ≺ v2 A>
- :6"#:;"#8#'$:&%& u A>
- :6 !=!>8$:& 7%& E(v1 ) = E(v2 ) A>
20
- XL(v1 ) ≠ 0 :&7 L %&#(&7:"#8#'" v2
>:6!"#!$%!&"'
σ′(v2 ) = (0, B) :&7 σ (v2 ) = ( A, B) A> σ ′(w) = σ (w)
8QR w∈V −{v1 , v2 } 867"#:;9''89
#$:& σ
(D !Bb\ α B v1 N\?@#!BN!B\ α ⋅ u "!@#!
b>? v2 N@B[ u1 (1 + α ⋅ e L (v1 )) "!@ ?bB>? v2
@#!BN!B\ − A ⋅ u1 (1 + α ⋅ eL (v1 ) ) !V α = A (1 − A ⋅ eL (v1 ) ) N\?
α ⋅ u = A ⋅ u1 (1 + α ⋅ e L (v1 ) ) bO!Q!!B"!\\? >?@W α
\?@ A ⋅ e L (v1 ) ≠ 1 B[BYW A ⋅ e L (v1 ) = 1 N@N\?
X L (v1 ) = E (v1 ) − I L (v1 )
= E (v1 ) − E (v 2 ) ⋅ A ⋅ e L (v1 )
= 0
(4)
bO?"\B XL(v1 ) ≠ 0
NBYN"?V!B>?V>bY W Q!>?@BG?B !!B"! YN@B
G?B#
MNO 5 D#:6%;$
v1,v2 867
- v1 ≺ v2 A>
- !=!>8"8@%;:68S8
6' $!&"6'9
# E (v1 ) = δ ⋅ E (v2 )
$!67'8#'($!&8" )"$
v2 A> !=!>8?$:"$
v1 ,
E ′(v1 ) >':68S8
6' E (v 2 ) ';
(D @V L @b v2 I L (v1 ) = 0 !BbB v2 \@@BQ!
V?X E (v1 ) #"! ?YN"?#NVDBB I L (v1 ) ≠ 0 @@ V
σ (v 2 ) = (a 2 , b2 ) @ND Q!#
21
E (v1 ) = X L (v1 ) + I L (v1 ) = δ ⋅ E (v 2 )
(5)
I L (v1 ) = e L (v1 ) ⋅ a 2 ⋅ E (v 2 )
(6)
?Y XL(v1) N @BG?B E(v2) @!BbB>? v2 I L′ (v1 ) N@B?!B\" G?@# XL(v1) Y\@!B"!" V? NO
@WA\? E ′(v1 ) @B?!B\" @BG?B E (v2 ) # !BY@W#N\?V?B
N]F^BB 5 Y V@W!B"!>?V>b\?
>?B Q!@BG?@ W !Y@BG?B#N@B
!B V!? \N"?@W!B>bV@#!
!@B!B\ @B \?
MNO 6 D#:6"$
"#!$ C = {v1,J, vh+1 } 867
- v1 ≺ v2 ≺ J ≺ vh+1
- !=!>8" vh+1 ≠ 0 A>
- :6"#:;"#8#'$:&%& u A>
- vi ∈ C D#$ !=!>8 E (vi ) ≠ 0 !=!>8" vl+1 δ i δ i ⋅ E (vi ) = E (vh+1 )
- eL(i+1)(vi )ai+1 ≠ 0 $! 1 ≤ i ≤ k :&7?$# L(i) F#(&7:"#8#'"$
vi
A> σ (vi ) = (ai , bi ) $!8QR vi ∈ C A>
- !"#:;8" "$
vi %& γ (i ,1) y1 + γ (i , 2) y 2 + + γ (i ,i ) yi :&7
γ (i ,i ) ≠ 0 A> γ (i ,1) , , γ ( i ,i ) ∈ ℑ A> ! y1 , y 2 , , y i 8@$:
@W
$!!"#!$%!&"' σ A>%?
R γ 1′ , γ 2′ , γ l′ ' ∈ ℑ >:6!"#!$%!&"' σ ′ 867
σ ′(vi ) = (ai′ , bi′ ) $!8QR vi ∈ C − {v h +1 } A> σ ′(v h +1 ) = (a h′ +1 , bh +1 ) A> σ ′( w) = σ ( w)
$! w∈V / C 8678?$#"#:;"#"'8 ti 67'9 (γ 1′ y1 + γ 2′ y2 + + γ l′' yl ) ⋅ ei (vh+1 )
22
(D N#N]F^BBY?#N>?A (induction) h
B@#!j h = 1 ND#>? v1 "! v2 W E (v1 ) ≠ 0 N!B
\B B v1 V@#!B! i !B\ γ 1′ y1 ⋅ ei (v2 ) # δ 1 ⋅ E (v1 ) = E (v2 ) ?Y
!B>?V B =
γ 1′
δ 1 y1
N\?!B@#! @B VDBB E (v1 ) = 0 N!BV E (v1 ) ≠ 0 >?@BB
!Bb\ + eL (v1 ) B>? v1 !BBY\@@BQ!V?X@#!B!N
Q![ 0 "!\?
σ ′(v1 ) = (a1 + e L 2 (v1 ) , b1 )
(7)
HY@ ?bB v2 NV Q! v1 !B[
E ′(v1 ) = X L ( 2 ) (v1 ) = −e L ( 2 ) (v1 ) ⋅ a 2 ⋅ E (v 2 )
(8)
"!@#!! B i !B\ − a 2 ⋅ u ⋅ ei (v2 ) NY@!BbB v1 \
− e L ( 2) (v1 ) @#!! B i N!B\ + a 2 ⋅ u ⋅ ei (v 2 ) bON! ?b
B v2 "!D σ ′′(v1 ) = σ (v1 ) "!\? E (v1 ) ≠ 0 bO@W!B
\??B"?\ "!
5 >b? p+1 %D#EWD(&%v
23
YA Y@@ j]F^B[N@ h = p N"?[NDB h = p+1
ND>? vp "! vp+1 V>b? p+1 ?CB 5 W E (v p ) = 0 N!B >?B?BY@#!jV
E (v p ) = −e L ( p +1) (v p ) ⋅ a p +1 ⋅ E (v p +1 ) ≠ 0
?V (γ 1′
y1 + γ 2′ y 2 + + γ ′p' y p ) ⋅ ei (v p +1 )
γ ′p
B i B >?!B\
δ p yp
[!B"!@#!B! N\?@#!B! B i !B\
@ φ [ BOY
!B@#!B B
(φ + γ ′p' y p ) ⋅ ei (v p +1 )
(9)
y1 , y 2 ,..., y p −1
Y?YN!
(γ 1′ y1 + γ 2′ y 2 + + γ ′p'−1 y p −1 − φ ) ⋅ ei (v p +1 ) = (γ 1′ y1 + γ 2′ y 2 + + γ ′p'−1 y p −1 − φ ) ⋅ δ p ⋅ ei (v p )
= (γ 1′′y1 + γ 2′′ y2 + + γ ′p′'−1 y p −1 ) ⋅ ei (v p )
(10)
bON@@ jB]F^B[NB h = p "?@W!BBYN>? v1, J,
vp-1 \? "!#N]F^BBY[N
N]F^BBYV@W ?B>?V>b"!!B
@?!B"!@#!B?N ?BY\?
MNO 7 D#$:6!"#!$%!&"' σ 867?(#$
"#!$ Q $
A>:6 C "
$
"#!$867:6 !=!>8$:& B7:6"
'#' k ℑ k + 1 A# >:6!
"#!$%!&"' σ ′ 867:6!"#:;9''8$:& σ '?(#$
"#!$#' Q $
(D N @#!@@B\?@BA? ℑ k #" ?YN @Bb
C ′ ∈ C k + 1 B"? v1 , v 2 , , v k +1 @ v1 ≺ v 2 ≺ ≺ v k +1 "!
V Li [@b>? vi @B#>? vi,vj B@B X L (vi ) ≠ 0 @W!?
j
>?\?>?]F^BB 4 NDDBB XLj(vi) = 0 V yi [ @#!
24
@>? vi Nb C ′′ ∈ C ′ B @\V]FaBB 6 >?
@N C ′′ = {y1 } "!V y1′ = y1 NYbY?BY
1. V j [?BA?B vj \@#V C ′′ W yj TG&
y1′ , y 2′ , , y ′j −1 A?bY
2. "?@B y ′′ B[ y1′ , y 2′ ,, y ′j −1 '&$%O
y j = γ ( j ,1) y1′ + γ ( j , 2) y ′2 + + γ ( j , j −1) y ′j −1 + y ′′
γ ( j ,1) , γ ( j , 2) , , γ ( j , j −1) ∈ F # vj V C ′′ "!V
y ′j = y ′′
! 1
N @#!!BY[ V k @ @WbY\?@BA? k !NYN @#!BOY k B@B#@
NZb D = C ′′ + {v j } @\" !V]F^BB 6 ?N
#"! "!VB eL (i +1) (vi ) ⋅ ai +1 ≠ 0 Y[N X L (i +1) (vi ) = 0 "!NB
E (vi ) = E (vi +1 ) V\? e L ( i +1) (vi ) ⋅ ai +1 = 1
@@Bb B ]F^BB 6 "! @W ?B>? vj V>?
BY\@[>?"!V@#!B!!B\
(γ 1′ y1 + γ 2′ y 2 + + γ ′j '−1 y j −1 ) ⋅ ei (v j )
(11)
bON]F^BB 6 @W!B>? v1, v2 ,J, vj-1 B@B
!B"!B@#!B!! ?BY\?
N]F^BB 3 "!]F^BB 7 @WAN>?\?Y
VDB>b"!a>b"! V]F^BN!WON>?Y@?V
MNO 2 $:6!"#!$%!&"'867?(#9
#
'?(#$
"#!$: k 2 ℑ 2k $
>:6!"#!$%!&"'6A$B7867?(#
':6$
"#!$';9: k 2 ℑ 2k >?
25
(D @B>?@ k 2 ℑ 2k N]F^BB 2 N\?@B>ba>b
bO @\V ]F^BB 3 ]F^BB 7 ?Y@W
V@B@B>?\@ k 2 ℑ 2k \?
%DH
BB?N?N>?VNBY[g N
Iwama et al. >?@W!?>? VDBB@B@#! OY @#! B
@B?BY\? N?@B @B@#!@Y \Z @
BBY>?N[ @B Q!B@# NDVDBB
Q!\@@?N@WB?NN>?B"
BY\?B
26
'&'
NBYB!?B?N?>?N?@B[ ℑ 3k V!
2k
k 2 ℑ >? >?V!BD "!]F^B Dilworth V#N@W# B ? N ? V@ bO B ?N ? B " OY BY V ! O @ V D B@B !!#V!V!?!\?
'
VNBYB"B?N?N>?V@Y \@\?
A!O@ "!O@V@>?OWO!FD? N@W
g!O@V@BC"!V!V!BY\?B
27
'&"""
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Fortune, S., J. Hopcroft, and J. Wyllie. 1980. The directed subgraph homeo-morphism problem.
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Iwama, K., H. Nishimura, M. Paterson, R. Raymond, and S. Yamashita. 2008. Polynomial-time
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Jaggi, S., P. Sanders, P. A. Chou, M. Effros, S. Egner, and L. Tolhuizen. 2005. Polynomial time
algorithms for multicast network code construction. IEEE Transactions on
Information Theory. Vol. 51. No 6
Lehman, A. R. and E. Lehman. 2003. Complexity classification of network information flow
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Li, S-Y. R., R. W. Yeung, and N. Cai. 2003. Linear network coding. IEEE Transactions on
Information Theory
Wang, C.C. and N. B. Sheoff. 2007. Beyond the butterfly a graph-theoretic characterization of the
feasibility of network coding with two simple unicast sessions. IEEE International
Symposium on Information Theory
28
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GOF
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Q!?B?/Q!
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16 @F 2528
C\ N?A@
G.. (G@\nne) @!F G \?A>DE GOF CG@@ @!F G (.G. 2551)