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[]\OT1/cmr/m/n/10 Assume the op-po-site, \OT1/cmr/m/it/10 i.e. $\OML/cmm/m/it/1
0 ^^Z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ and $\OML/cmm/m/it
/10 ^^Z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ are in-com-pa-ra
-ble. Then $\OML/cmm/m/it/10 T\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^Z[]\OT1/cmr/m
/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ))$
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[]\OT1/cmr/bx/n/10 (The cost/weight of a rec-on-cil-i-a-tion) \OT1/cmr/m/n/10 L
et $\U/euf/m/n/10 R \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 G; G[]; S; ^^^; ^^Z; ^^
N\OT1/cmr/m/n/10 )$
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[]\OT1/cmr/bx/n/10 (Minimum/optimal rec-on-cil-i-a-tion) \OT1/cmr/m/n/10 Let $\
U/euf/m/n/10 R \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 G; G[]; S; ^^^; ^^Z; ^^N\OT1
/cmr/m/n/10 )$
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[]\OT1/cmr/bx/n/10 (Minimal rec-on-cil-i-a-tion) \OT1/cmr/m/n/10 A rec-on-cil-i
-a-tion $\U/euf/m/n/10 R \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 G; G[]; S; ^^^; ^^
Z; ^^N\OT1/cmr/m/n/10 )$
 []


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\OT1/cmr/m/it/10 Then there ex-ists a unique (up to $\OML/cmm/m/it/10 ^^N$\OT1/
cmr/m/it/10 ) min-i-mal rec-on-cil-i-a-tion $\U/euf/m/n/10 R \OT1/cmr/m/n/10 = 
(\OML/cmm/m/it/10 G; G[]; S; ^^^; ^^Z; ^^N\OT1/cmr/m/n/10 )$
 []

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[]\OT1/cmr/m/n/10 Let $\U/euf/m/n/10 R \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 G; G
[]; S; ^^^; ^^Z; ^^N\OT1/cmr/m/n/10 )$ be any rec-on-cil-i-a-tion. Let $\OML/cm
m/m/it/10 dist[]\OT1/cmr/m/n/10 (\U/euf/m/n/10 R\OT1/cmr/m/n/10 ) =
 []

[12] [13] [14] [15] [16]
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[]\OT1/cmr/m/n/10 Now, let $\OML/cmm/m/it/10 e$ \OT1/cmr/m/n/10 be a non-leaf e
dge, $\OML/cmm/m/it/10 m \OT1/cmr/m/n/10 = \OML/cmm/m/it/10 max\OT1/cmr/m/n/10 
(\OML/cmm/m/it/10 min\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 (\OML/c
mm/m/it/10 e[]; \U/euf/m/n/10 R[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]\OT1/cmr/
m/n/10 ))\OML/cmm/m/it/10 ; F\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]; \U/euf/m/n/
10 R[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 )))\OML/cmm/m/it/10 
; \OT1/cmr/m/n/10 0)$,
 []


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\OT1/cmr/m/it/10 and $\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e; \U
/euf/m/n/10 R[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T \OT1/cmr/m/n/10 0$\OT1/cm
r/m/it/10 . By as-sign-ing an ex-tra loss to $\OML/cmm/m/it/10 e$ \OT1/cmr/m/it
/10 we ob-tain $\U/euf/m/n/10 R[]$\OT1/cmr/m/it/10 . Then $\OML/cmm/m/it/10 !\O
T1/cmr/m/n/10 (\OML/cmm/m/it/10 c\OT1/cmr/m/n/10 (\U/euf/m/n/10 R[]\OT1/cmr/m/n
/10 )) =
 []

[17] [18]
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[]\OT1/cmr/m/it/10 Let $\U/euf/m/n/10 R$ \OT1/cmr/m/it/10 be a rec-on-cil-i-a-t
ion with non-extended losses, $\OML/cmm/m/it/10 t[]$ \OT1/cmr/m/it/10 ($\OML/cm
m/m/it/10 i \OT1/cmr/m/n/10 = 1\OML/cmm/m/it/10 ; [] ; k$\OT1/cmr/m/it/10 )
 []

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[21] [22] [23] [24] [25]
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\OT1/cmr/m/sc/10 ExtendOneLossIntoNonFreeTree$\OT1/cmr/m/n/10 (\U/euf/m/n/10 R\
OML/cmm/m/it/10 ; l\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 ex-tends loss $\OML/cmm/
m/it/10 l$ \OT1/cmr/m/it/10 into a non-free tree. 
 []

[26]
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[]\OT1/cmr/m/it/10 Procedure \OT1/cmr/m/sc/10 Ex-tend-Loss-esIntoNon-Free-Trees
 \OT1/cmr/m/it/10 does not change
 []


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[]\OT1/cmr/m/n/10 Let $\OML/cmm/m/it/10 e[]; e[] \OMS/cmsy/m/n/10 2 \OML/cmm/m/
it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^Z\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t
[]\OT1/cmr/m/n/10 ))$ be sib-lings, $\OML/cmm/m/it/10 e[]$ \OT1/cmr/m/n/10 thei
r par-ent, and $\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]; \U/euf
/m/n/10 R[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^U \OML/cmm/m/it/10 F\OT1/cmr/m/
n/10 (\OML/cmm/m/it/10 e[]; \U/euf/m/n/10 R[]\OT1/cmr/m/n/10 )$.
 []

[27]
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[]\OT1/cmr/m/n/10 From the Cases 1 and 2, we have that if $\OML/cmm/m/it/10 F\O
T1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]; \U/euf/m/n/10 R[]\OT1/cmr/m/n/10 ) \OMS/cm
sy/m/n/10 ^^T \OT1/cmr/m/n/10 0$, then $\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 (\OML
/cmm/m/it/10 e[]; \U/euf/m/n/10 R[]\OT1/cmr/m/n/10 ) \OML/cmm/m/it/10 >
 []


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\OT1/cmr/m/n/10 0$, $\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]; \
U/euf/m/n/10 R[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T \OT1/cmr/m/n/10 0$, and 
if $\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]; \U/euf/m/n/10 R[]\
OT1/cmr/m/n/10 ) \OML/cmm/m/it/10 > \OT1/cmr/m/n/10 0$, then $\OML/cmm/m/it/10 
F\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]; \U/euf/m/n/10 R[]\OT1/cmr/m/n/10 ) \OML
/cmm/m/it/10 > \OT1/cmr/m/n/10 0$, $\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 (\OML/cmm
/m/it/10 e[]; \U/euf/m/n/10 R[]\OT1/cmr/m/n/10 ) \OML/cmm/m/it/10 >
 []

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[]$\OML/cmm/m/it/10 F\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e; \U/euf/m/n/10 R\OT1/c
mr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T \OT1/cmr/m/n/10 0  =[]\OMS/cmsy/m/n/10 )  \U/e
uf/m/n/10 L[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e; \U/euf/m/n/10 R[]\OT1/cmr/m/n
/10 ) = \U/euf/m/n/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e; \U/euf/m/n/10 R\OT1
/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ \U/euf/m/n/10 D\OT1/cmr/m/n/10 (\OML/cmm/m/i
t/10 e; \U/euf/m/n/10 R\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 
max\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 min\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 F\OT1
/cmr/m/n/10 (\OML/cmm/m/it/10 e[]; \U/euf/m/n/10 R\OT1/cmr/m/n/10 )\OML/cmm/m/i
t/10 ; F\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[]; \U/euf/m/n/10 R\OT1/cmr/m/n/10 )
)\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0)$\OT1/cmr/m/it/10 ; 
 []

[32]
Overfull \hbox (25.86867pt too wide) in paragraph at lines 144--144
[]\OT1/cmr/m/n/10 So, by \OT1/cmr/m/sc/10 Ex-tend-Loss-esIntoNon-Free-Trees \OT
1/cmr/m/n/10 we ob-tain $\U/euf/m/n/10 R[]$\OT1/cmr/m/n/10 , and by $[](\U/euf/
m/n/10 R[]\OT1/cmr/m/n/10 )$,
 []

[33]) (./conclusion.tex [34]
File: figures/Fig8.eps Graphic file (type eps)
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Missing character: There is no � in font cmr10!
Missing character: There is no � in font cmr10!

Underfull \vbox (badness 10000) has occurred while \output is active []

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Underfull \vbox (badness 10000) has occurred while \output is active []

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[38] (./main.aux) ) 
Here is how much of TeX's memory you used:
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Output written on main.dvi (38 pages, 160740 bytes).
