Téléchargement

Transcription

Téléchargement
"!#%$%
'&)(*,+
-.$/%0123456
%-7
%8!9:-.$%;&<$%=?>;
@
A)BDCFEAHGICFEKJML)ABONQPMRSDTMAHUMAWVWCDPQLXYA[Z\E
][^_a`b.c:deFdfgc[hd[i<^fjkdjlf?monpb.eq`c.jkrgsKd
BtuwvxDRPQy8z|{?}~ € ‚fa`ƒl„?…:ddc‚†npfK†d_c.jknpf‡h‚ˆ%`ƒli<npb.jlc.‰eFd…‹Š%sc:npeDfKdŒŽŽŽ<
‘
†b:^h?jlc:…OŠ ‘’ @Ž“” ’/• “”<
BDCDXYX\R–\—˜UMT™BDCFPQLX\šDE6A˜G)—›X\RCDUMRSITMA
UYZ[LAœ€#ž Ÿw¡wvžg¡#¢8£¤ž¦¥a}'x
žg§w¨
ž ©‡ª«vž­¬K}a}~
NMA›TMXYAœ€8®?~KNQz~­¯°®a{?N­®~
E8C\BOZ\E±€#V³²´¬¬K}|®
UMTMX\AAœ€µ®?NQ¥a}
P‡CFL)A¶€ZYuw·gu«Ÿ¸žQ¡«t¹·gu«©'ž;Ÿ¨
º?¨¢»tŸ¼§6žgvp©½¢¤x
žK¾¿Z\u¸·gu«ŸwžÀ·<ºK£¤·gu«£¤º?¨
v¢¤·<ž¦§žgvp©½¢¤x
ža¾
BžÀÁ|uwžgx
¨¢»tŸwŸwºK¢¤vž¦·<t©½§wvž Ÿw¡ÂzÃÁ|u¸ž x¨¢¤tŸwxDxu¸v2¬­§«ºÄažgxY§tuwvY¬a}ŧt¢%Ÿ¨
xŽ¾
ÆItŸÂ¨
vºgÇ?ºK¢%£È
ÉÀÊËaÌÍ<Î/ÏwÐÒÑÔÓÕ½Ï|ÍpÖÍ<Î/ÏwÐ×Ö¹ÌØÙÔÚ6͎ϹÍ<ÎÜÛ#ÊËÞݤ߰ÚϸÎÜÐwÍ<Ìà
`<2][npffKdá)ƒâ`Fh^´ãKfjlc.jknpfQmonpb.eFdƒlƒkd)hd
è t£%u¸¨¢¤tŸ6€
Ω(f )
_npsbäsfKd›monpfK†c.jknpf
f : IN → IR≥0
“¹å”Œ‹_ npjlfgc:…çæ
Ω(f ) = {t : IN → IR≥0 | (∃c ∈ IR+ )(∃n0 ∈ IN)(∀n ≥ n0 )[t(n) ≥ c · f (n)]}
é 2][^eFnpf c.b:dá‹r sKd[ƒâ`Om%npfK†c.jknpf
è t£%u¸¨¢¤tŸ6€
ë8ì.í´îïpîað
c = 3
4n − 8 ≥ 0 ý
†2þÿƒ”`jkhdÅhd…M….„;e
íð
`__a`b.c.jkdf cݐ
g(n) = 3n2 + 4n − 8
n0 = 2 ñ‹ò
îÞï
g(n) = 3n2 + 4n − 8 ≥ 3n2
é npƒkd…2b:dƒâ`c.jknpffKdƒk…
…:sjp`f c:d…
Ω(n2 )
=
dc
⊂
ål@[_npjlf cçæ
ógôöõ­÷ø í
n ≥ 2
eFdc.c:dá½dfÞb3`fi°ƒ npb:h?b:dÅhd…\monpfK†c.jknpfK…Qhd
√
n, n lg n, (1.09)n , ( n − 2 lg n)2 , n1.09 , lg(n!), 1n
Š#`bddeD_ƒkd_ npsbähd…monpfK†c.jknpfK…
f, g, h
npfQ_ npsb.b3`jlc‚n é c:dfjlb
npsK…‚f”`<dá)_a`p…›êsK…:c.j,ãadb<nŽ…b:^_ npfK…:d…
“¹å ‘
_ npjlfgc:…çæ
ïŽü ì
ù ø;ú»õ÷øû ò õ
O(g) ⊂ O(f ) = O(h)
“
è t£%u¸¨¢¤tŸ6€
√
O(1n ) ⊂ O(n) = O(( n − 2 lg n)2 ) ⊂ O(n lg n) = O(lg(n!)) ⊂ O(n1.09 ) ⊂ O((1.09)n )
n
"!#%$%
'&)(*,+
-.$/%0123456
%-7
%8!9:-.$%;&<$%=?>;
ha2][npffKdá)sf¦d?deD_ƒkd)hdm%npfK†c.jknpfK…
c:dƒlƒkd…rgsKd
è t£%u¸¨¢¤tŸ6€
f (n) ∈
/ O(g(n))
íð
sin(n) + 1
cos(n) + 1 ñ
ù
dc
Œ
dc
f : IN → IR≥0
g : IN → IR≥0
“¹ål@_ npjlfgcçæ
g(n) ∈
/ O(f (n))
ïŽì‹ígí
ù
ükí
ÉÀÊËaÌÍ<Î/ÏwÐ×ÓÅÐÖoØÌËÞÝÔÚϸÎÜÐwÍ<Ìà
npjlcƒ”`ƒli<npb.jlc.‰eFd‹….sjŽ`fgc
ºK£¤Äatv¢»¨#"«©'žFeO„;…:c%$b:d<Š
s←0
§tuwv
i←1
§tuwv
¯
n
n2
j←1
dfM_K…3dsKhn ’ †n hd!
&
' ºK¢¤vž
¯
' ºK¢¤vž
i
s ← s+i
`<();_b.jleFdá›d`p†c:deFdf c dfDmonpfK†c.jknpfqhd
唌D_npjlf c:…çæ
n
ƒkdfKnpe
é b:d‚hd8monpjk…r sKdƒ ^fKnpfK†^*
s ← s+i
…:db3`)d?^†sc:^Ž“
è t£%u¸¨¢¤tŸ6€
P n2 P i
Pn2
4
2
n2 (n2 +1)
= n2 + n2 ý
i=1
j=1 1 =
i=1 i =
2
é 2][npffKdá)ƒâ`I†npeD_ƒkd?jlc:^)dfMc:deD_K…hd†
`ƒk†sƒ¸hd[ƒ”`ƒli<npb.jlc.‰eFddf2monpfK†c.jknpfÀhd
dc‚_b:^†jk…:^eFdfgc_nŽ…:…:j é ƒkddfQfKnpc3`c.jknpf
è t£%u¸¨¢¤tŸ6€
O
“¹ål@)_npjlf cçæ
O(n4 )
†2][npffKdá)ƒâ`I†npeD_ƒkd?jlc:^)dfMc:deD_K…hd†
`ƒk†sƒ¸hd[ƒ”`ƒli<npb.jlc.‰eFddf2monpfK†c.jknpfÀhd
dc‚_b:^†jk…:^eFdfgc_nŽ…:…:j é ƒkddfQfKnpc3`c.jknpf
è t£%u¸¨¢¤tŸ6€
Ω
“¹ål@[_npjlf cçæ
n
n
ƒkd[_ƒlsK…ä….jleD_ƒkdeFdfgc
ƒkd[_ƒlsK…ä….jleD_ƒkdeFdfgc
Ω(n4 )
+f'ð/ï-,´ükí.ï /
h ^ƒk^eFdfgc:…›rgs npfM„
ø . ülï õ ð úo÷ø íd…:cäsc.jlƒkdƒknpb:…:r s npfMfKd)†npffa`10kcä_a`p…‚êDƒ”`Ž`fK†dƒkdfKnpe é b:dO
eFdc.c.b3`?“npjk†jµdf'rgsKnpj6†dƒâ`Q†npfK….jk….c:dŽ“+fÅc3` é ƒkd
`s½hdDc3`jlƒlƒkd
d…:cjlfjlc.jâ`ƒkdeFdfgcI†b:^^Ž“2|npb:…:r s npfÃ`1.npsc:d
sf
n+1
j3$eFd‹^ƒk^eFdfgc)`sÀc3` é ƒkd
`s­hdOc3`jlƒlƒkd
c3` é ƒkd
`sFhd c3`jlƒlƒkd
^ƒk^eFdfgc
7
2n 7
npfD†np_jkd8ƒkd…
n
1
n
dc›_ npsb^4;jlc:db‚sf­h^ é npb:hdeFdf c5npf¦†b:^dOsf¦fKnps6<d
`s
^ƒk^eFdfgc:…6hd ƒ”`fK†jkdfDc3` é ƒkd
`sFhK`fK…#ƒkd8fKnps6<d
`s
êDsf­`sc.b:dOhdc3`jlƒlƒkd‹Œ
_sjk…
•
:9;
npfF`1.npsc:d8ƒkdµfKnps6<dƒ
dc:†Ž“8f¦†npfK…:jkh6$b:dOr sKdh”`1.npsc:dbsf‡^ƒk^eFdfgc)`sÀc3` é ƒkd
`s­hdOc3`jlƒlƒkd
†npfK…:npeDeFdqsfKdDsfjlc:^FhdDb:d…:…3npsb:†dDƒknpb:…:r s jlƒf „­`M_a`p…hdDh^ é npb:hdeFdfgcIdc
†np_jkdbäƒkd…
7
ƒkd[fKnps6<d
`sÀc3` é ƒkd
`s‡hd4 jkdf c‚fKnpc.b:dc3` é ƒkd
`sˠnpsb3`fgc
“8fQ_a`p…:…:d)`jlfK…:j¸h sfÀc3` é ƒkd
`sÀhd)c3`jlƒlƒkdq@
n
^ƒk^eFdf c:…dc‚`1.npsc:dbäƒkd[fKnps6<d
`s´“
ha(fI…:s__nŽ…3`f c«r s npfDh^eq`b.b:d`<d† sfOc3` é ƒkd
`s<;jkhd hd6c3`jlƒlƒkd
hd)ƒ”`1çnpsc‚h ^ƒk^eFdf c:…
“a][^c3`jlƒlƒkdá=<nŽ…‚†
`ƒk†sƒk…
“|唌I_ npjlfgc:…çæ
1
n+1
n
sfjlc:^…O….jlfKnpfԊ¤_ npsb
^4Ž`ƒlsKdá8ƒâ`‚†npfK…3npeDeq`c.jknpfq`eFnpb.c.jkd
"!#%$%
'&)(*,+
-.$/%0123456
%-7
%8!9:-.$%;&<$%=?>;
‘
è t£%u|¨¢¤tŸ€
>öî õ ú ì î î ø õäópø ð%ì ø;@›ó í @ /ð ïA, ú ü ú ð .Bø í´î @ ü,ï
ù
ò õ ? ò g
ò
ù
ú î
î õ ülí J
ï
í
,
ì
í
q
´
í
a
î
ð
ì õ ÷ø¿û ø î2ð/ï-,´ükí.ï
õ @ ä
õ ó
óA. ò ó
ò
ý:K|ò ë ø;úâõ÷ø íäükí ó í´ìöî ú í´ì óA. , ì ó íq
í´îað¿ï‹í ø ü ú
i
ó í
î
î @
õöõ í D
ý C ø <
ù ùò õ ò õäø
òFE
‚
í
/
ð
ï
ü
¤
k
ü
í
,
ì í
ø2ó
ú
2i ó-. ò ó ñ
í ø ï ø
qíäïIH
2 +1 ý
2i−1 +1 ú/ô
ò
ò
ì
ð
a
î
ð
ð
ï
ü
í
î
ï
ä
ì
. .
@%@ø
ø .öõó ù ø;ú»õ 6
ó¸û B ò ú ï õ3õ í%P
ý MON
(γ −1)×(2i−1 −1) @ ..óŽú õ ò
U,´ì:í ó í @ ì ..ópú ð õ ï @%@ø ø ü .öõ
ü
ükí õ
î
í.ï ü¤ü,ïpì õ‹õ ú ðäï ø @ íFî
γ ò øVB ø ó ò
ò
ù øgõ
ò
,
ì
í
q
´
í
a
î
ð
J
Y
ópø¦óA. ò ó
i−1
i
Z
ûí õ ð @ í
÷øû ò î
ò
,ð ú í´îað¿í´î
ù
ì.íîïpîað
ÉÀÊËaÌÍ<Î/ÏwÐ\×Ó/]|Ï_^gÎoÍa`6Ù
(γ − 1) × (2
ù
ïpìígí
ù
ükí
ð#ïG
ì´ð ú|ó í
ïpì‚ïIH ø ð#íð¿í<ï ?
ò
ò
γù
üú í´îað%ì.ïFL¤îí ø îí @ î õ 5FïŽð ú î
ò
ò
ò
ø ð ñ (γ −1)×(2i −(2i−1 +1)) =
î ,.ïpî ÷ø í ú 4
ü Tï ø ð ÷ø í
ówû ïpRì Qgí´îð«í´S
ñ
î
Q<ïŽX
ü WMülï @ î õ 5FïŽð ú î
ò ú õ<.
ò
ò
ò
− 1) + γ ≥ 2 + 1.
γ = 3 ýD[ò
î @
ü,ï
@
ò
î õ 5FïŽð ú îQïF
ò
ò
ò
ì´ð ú [
í í õ ð ó ïŽî õ
Θ(1) ý
ËaÌUb¸Ï^gÖdc?Ë̽Ý\ÔÚϸÎ"иÍ<Ìà
`<fesKdƒlƒkdOd….cƒâ`‹_b.jlfK†jl_a`ƒkd†
`b3`p†c:^b.jk…:c.jkrgsKdOhd…‚`ƒli<npb.jlc.‰eFd…J<npb3`p†d…hgFål@_ npjlfgcçæ
è t£%u¸¨¢¤tŸ6€
iwîí
ð ,j<ð ú í
ïpì ó í k
õ ò ü ø ð ú ò î'í õ *
õ @l ò ú õöø6@%@ í õ3õöú T õ‹÷ø;ú îí õ ò îaðDHïGDï úâõ ì.í4 ú»õ í´î ÷ø í õ ð ú ò î ý
ù
é 2__ƒljkr sKdá`sMeFnpcmÜjleDjlc3`c.jknpf ƒ ”`ƒli<npb.jlc.‰eFdOhdons;p eq`fM_npsbƒâ`‹†npeD_b:d…:….jknpf¦hdc:d?c:dŽ“][npffKdá
sf‡`b é b:d)hd†n hK`i<ddcäsc.jlƒljk…:dá ’ ƒkd_ npsb‚†n;hdbä†deFnpc
“|唌‹_ npjlfgc:…çæ
è t£%u¸¨¢¤tŸ6€
ï*T´ì .
. ÷ø í´î @ í ó í @l ï @ø îí
a
î
ð
,lü í õ
îð
õ ò
ù?ò õ3õ´ú
ñ ó ò
K
ó í õ
ükíðoð%ì:í õ
í õ ðXY
>
ë ü ø õöú í ø ì õ
M<qsrtñ
qvu4tñxwyqIrztñ{<qsrtñ}|&qIrztñJ~mqt<ý
0
í
K
@
0
1
T
I
1
0
Ó/]|Ï_^gÎoÍa`6Ù
ËãË
Ú}^<Ï]^<Ö¹ÙÞÙ
1
0
1
0
1
A
M
N
O
íOì .öõöø ü,ð/ïpîaðäí õ ð*Y)€
€
€ €a€ €a€a€a€-€
€
ò ó
rar
r r
r
rararar-rar ý
ÉÀÊËaÌÍ<Î/ÏwЂ
Ö͎ÎÜϸЄƒ¿ØÐÖ¹ÙÞÎÜÛ#ÊË
݁
ÚϸÎ"иÍ<Ìà
s__ nŽ…:dár sKd[_npsb8b:^…:npsKh?b:d[sfQ†db.c3`jlfM_b:n é ƒ3$eFd_a`b_b:npiŽb3`eDeq`c.jknpf‡h?„ fa`eDjkr sKd
c3` é ƒkd
`s
t
….sb
n
ƒljliŽfKd…‚dc
m
†npƒknpffKd…‚dcnpff<npsK…ähnpffKdƒâ`Ib:^†sb.b:dfK†d
_ db.eFdc.c3`fgchd)ƒkd[b:deD_ƒljlb
“
ïpì,ì.í õ
npfMh^´ãKfjlcäsf
t[i, j] = max(t[i, j − 1] + 3, t[i − j, j − 1])
`<2][npffKdá)df2_K…:dsKhn ’ †n;hd[sfKdmo`F…npfQ†npb.b:d†c:dhd›b:deD_ƒljlbä†d›c3` é ƒkd
`s½Š¤fKdo<npsK…8_b:^n ††s_ dá_a`p…hd…
Ž`ƒkdsb:…m%b:npfgc.j3$b:d´“w唌‹_npjlf c:…çæ
è t£%u¸¨¢¤tŸ6€
§tuwv
j←1
¯
m
' ºK¢¤vž
"!#%$%
'&)(*,+
-.$/%0123456
%-7
%8!9:-.$%;&<$%=?>;
§tuwv
i←1
¯
n
•
' ºK¢»vž
t[i, j] ← max(t[i, j − 1] + 3, t[i − j, j − 1])
î
ì ì.ï ú ðï øgõöõöú
ö
ì
í´ìIü ú Qpîí
ïpìü ú Qpîí
|
ù?ò ø
ù ò @†.:ó
ù
ý
siŽi<^b:dá8sfKdµm/`F…npf‹h ^†npfKnpeDjk…:db¿hdµƒ d…._a`p†d ’ eF^eFnpjlb:d8_ npsb«†dc¿`ƒli<npb.jlc.‰eFddcwhnpffKdá8ƒkdµ_K…:dsKhn ’
é 
†n;hd)†npb.b:d…._ npfKhK`fgc
“w唌‹_npjlf c:…çæ
è t£%u¸¨¢¤tŸ6€
î‡ì.í4
|
|
§tuwv
î
ù
ü,ï @ í‹ülíOð/ï-,ülí.ï ø
i←n
¯
1
' ºK¢»vž
ù
ïpì
ø îí
õ í ø ükí
c[i] ← max(c[i] + 3, c[i − j])
ï
íIï ú î õöúó í
ù õöõ
Θ(nm)
í õ ï @ í5W
ù
2¸d&¿b:nm%d…3…:dsbˆ
ó í
Θ(n)
õ í õk@ ò ü ò îKîí õ ñ
cñ
÷ø¿û ò î­ì.í4
ù
ü ú ð @ ‡qí
ò
í õ ï @ í
ù
ý
‰›jlƒlƒkd…Jwd…3`f c‹ŠŠ
’"‘• @

c:dƒ/“ ‘•  ’/•A‹ @Ž@Od;c
“ • @ • Œ<
õ´ø ú ðmY

Documents pareils