INF4705 Analyse et conception d @algorithmes, Examen

"!#%$%
'&)(*,+
-.$/%01234 5#67128:9;&<$>=@?8-?
A
B)CEDGFBIHJDGF7KML)BCONQPMRSETMBIUMBWVWDEPQLXYB[Z\F
][^_a`b.c8deGdfgc[hd[i<^fjkdj>fmlonpb.eq`c.jkrgs7d
CtuwvxERPQy{z}|m~€  ‚ƒfa`„>…m†8ddcƒ‡npf7‡d_c.jknpfˆhƒ‰%`„>i<npb.j>c.ŠeGd†Œ‹%‚sc8npeEf7dŽŽ<‘
’
‡b8^hmj>c8†O‹ ’“ A”• “/– ”•<‘
B˜—\ZYV™B[Pšy{RPQZ\F™UMRyy{›˜X\›
UYZ[LBœ#Ÿž uw¡£¢"¤;¥¦žˆ§7¨\©ª7«€¬@¢¦žgvM§7~~a¨
NMB˜TMXYBœ{­¯®aN:~a~±°²§Ÿ­¯NQ³~
UMTMX\BBœ;§aNQ³a~
PˆDGL)B´ZYuwµgu£«@žQ¡£tŸµgu£¶'ž¯«€·
ªm·¢¸t€«º¹»žgvp¶¼¢¦x
ž7½;Z\u@µgu£«wž:µ<ª7¥¦µgu£¥¦ªm·
v¢¦µ<ž¾¹žgvp¶¼¢¦x
ža½
Cž:¿}uwžgx
·¢¸t€«w«wª7¢¦vž¾µ<t¶¼¹wvž «w¡º¨¼¿}u@ž x·¢¦t«wxE¹tuwvqu£«À·
t7·
ª7¥{¡£ž¾³€Á¹t€¢¦«€·
x<½
Â:ÃÄaÅÆ<Ç/ÈwÉËÊÍÌαÉϟÐoÑÅÄÓÒÕÔÍÖÈ@Ç×ÉwÆ<ÅØ
`<‘ÁÙk§Ú¹t€¢¦«€·
x
یÜ#Ým_„>jkr s7dÞ'„kd†Q`ߏ`fgc3`i<d†ˆdcM„kd†Mj>f7‡npfgß<^fjkdf c8†ˆb8d†._€d‡c.j,lo†Mhd†:`fa`„>…m†8d†Mc.Š7^npb.jkr s7dà
deE_j>b.jkr s7ddcŠ … áb.jkhd”
á‘ÁÙk§'¹t¢%«€·
xÛ#Ü#Ým_„>jkrgs7dÞdf:ß<n†eGnpc8†„Õ`Ehmj,âŸ^b8df7‡d)dfgc.b8d„kd†f7npc3`c.jknpf7†
‡‘ÁÙk§'¹t¢%«€·
xÛ#Ü#Ým_„>jkrgs7dއdrgs7d†8npf cã„kd†ã‡„Õ`p†8†3d†ãhd‡npeE_„kdäݯj>c8^
P
dc
O()
NP
”
dc
Ω()
”
Â:ÃÄaÅÆ<Ç/ÈwÉæåçÌαÐ/è}È@égÇoÆ<ê»ëìÄaÅ\íãîÕÄï;ÖðÐ/È@é<πƏÇ×È@ÉæíÄñè}é<Ï ÖòêÄóÒ¦ôõÖÈ@Ç×ÉwÆ<ÅØ
`<‘ÁÙ¦­Œ¹t¢%«€·Ûa][d†8†.j>f7dÞð„/ö•`b.áb8dð‡npb.b8d†8_€npf7h7`f c;÷ƒsf7d;lonpsj>„>„kdòdf_b8nlonpf7hdsb#hmsOib3`_Š7dðnpb.jkdf c8^{†.sj>ßp`f c
à
df¾‡npeEeGdf7ø
`fgc`sQ†8npeEeGdc
1
dcãdf:‡npf7†.jkh^b3`f cƒ„kd†ãdfml/`fgc8†hŸö sf:†8npeEeGdcƒdfQnpb8hmb8d)fgseG^b.jkr s7d”
4
2
1
3
5
á‘ÁÙ¦­ˆ¹t¢%«€·Û#]˜npff7dބkd_a`b8‡npsb8†ƒdfM_b8^ “ npb8hmb8dOhd„/ö•`b.áb8d)npác8dfgsÁ÷Œ„/öù^c3`_€d)_b8^‡^hdfgc8d”
"!#%$%
'&)(*,+
-.$/%01234 5#67128:9;&<$>=@?8-?

‡‘ÁÙ¦­¼¹»t¢%«€·Ûð][d†8†.j>f7dÞE„/ö•`b.áb8dE‡npb.b8d†8_€npf7h7`f c÷\sf7dlonpsj>„>„kdEdfÁ„Õ`b.i<dsb)hmsÁib3`_Š7dG÷\„/öù^c3`_€dE`<‘äàŸdf
‡npeEeGdf7ø
`f c[`s¾†8npeEeGdc
1
dcƒdfQ‡npf7†.jkh^b3`f cƒ„kd†ãdfml/`fgc8†hŸö sf:†8npeEeGdcƒdf:npb8hmb8d)fgseG^b.jkr s7d”
ha‘ÁÙk§ú¹t€¢%«€·
xÛð][npff7dÞEsf7dOlo`pønpfÁhdJh^c8db.eEj>f7db)†.j#sf¾ib3`_Š7dJf7npf±npb.jkdf c8^Ed†.c˜‡npff7däÝdàŸdfˆc8deE_7†
„>j>f7^
`j>b8dOh7`f7†„Õ`Jc3`j>„>„kd)hmsMib3`_Š7d”
d‘ÁÙk§¾¹t€¢¦«€·
x
Û@ûórgs7npj€†8db.cò„kd‡
`„k‡s„ hŸö sf7d˜á€npb.f7d˜h7`f7†òsf2`„>i<npb.j>c.ŠeGd[hd˜†8^_a`b3`c.jknpfMdcðhŸöù^ߏ`„>sa`c.jknpf
_b8npib8d†3†.j>ß<dq‹Õü×áb3`f7‡8Š “ `f7h “ á npsf7h¯ý‘þ
Â:ÃÄaÅÆ<Ç/ÈwÉæÿçÌαÐ/è}È@égÇoÆ<ê»ëìÄaÅYÖ»égÈ »Ï ðÇ"Ð×Ç"ÅƏÄaűҸԺÖÈwÇ×ÉwÆ<ÅØ
s7d„>„kdÁd†.cq„Õ`¼hmj,â
`<‘ÁÙ¦­ç¹»t¢%«€·Û
dig`p†€þ
ˆnpfgc8d{`b.„knúdcYsfó`„>i<npb.j>c.ŠeGd
@`p†
^b8df7‡dÁdfgc.b8d¾sfó`„>i<npb.j>c.ŠeGd
!#"$% &' ()+*,"$-.)/.012-.3546"”
‡‘ÁÙ¦­ˆ¹t¢%«€·Û#Ü#ݯ_„>jkr s7dއd)r s}öùd†.c„/ö798":<;=#>.?@!,".-BAC ?-ä”
ha‘ÁÙ>³º¹t¢%«€·
xÛ{‚ED7f'hdJb8^†3nps7hmb8dJsf±_b8npá„G
F eGdGhdEh^‡jk†.jknpf͋o`s7r s7d„npf±b8^_€npf7hÁ_a`bOnpsj?Hf7npf‘äàŸß<nps7†
`
ß<dÞO†8nps7†ã„Õ`Œeq`j>fJI
K ZML¾sf:`„>i<npb.j>c.ŠeGd¾
npf c8dN{
`b.„kn 1 “ ‡npb.b8d‡cƒájÕ`jk†8^_ npsbã„Õ`Œb8^_€npf7†3dü"npsj,ý3à
KPO L:sfˆ`„>i<npb.j>c.ŠeGdQˆ
npfgc8d{
`b.„kn 32 “ ‡npb.b8d‡cájÕ`jk†8^_ npsb„Õ`Œb8^_€npf7†3d˜ü×f7npfmý3à
K CRˆ
L sf:`„>i<npb.j>c.ŠeGdQˆ
npfgc8d{
`b.„kn 43 “ ‡npb.b8d‡cãf7npfQájÕ`jk†8^”
4
Üf:sc.j>„>jk†3`f c‡dsmÝ “ ‡j/à7‡npf7‡dß<dÞsf¾`„>i<npb.j>c.ŠeGdN@
`p†C
dig`p†`…g`f csf7d)_b8npáa`áj>„>j>c8^hd)†.s7‡‡:F †^ig`„kd
÷ 3 _€dsMj>eE_ npb.c8d„/öùdäÝdeE_„Õ`j>b8d”(
S s7†.c.jTaD dÞO‡dc.c8d_b8npáa`áj>„>j>c8^hd)†8s7‡‡:F †
”
á‘ÁÙ¦­ˆ¹t¢%«€·Û#Ü#ݯ_„>jkr s7dއd)r s}öùd†.c„/ö
4
VU X
Ì Wqé<È@è}é<ϟëÓëÍπÆ<Ç/ÈwÉçí;ÑÉ»Ï
Â:ÃÄaÅÆ<Ç/ÈwÉ
ZY
ëÓÇ £Ã»ÄóÒÕÔõÖÈwÇ×ÉwÆ<ÅØ
ðj>c8dÞ
`<‘ÁÙk§W¹t¢%«€·
xیÜ#Ým_„>jkrgs7dÞ'‡d±r s}öùnpfódfgc8df7hó_a`bMsf
`„>i<npb.j>c.ŠeGd±_7†8ds7hn “ _ np„>…¯f7npeEjÕ`„/”
“ df
sf
däÝdeE_„kd”
[ ]\
_^
a`g¿}u£¢¦¥%¢][£v'` ž¼cbfˆ‡npf7†.jkhdFb8dO„kd)_b8npá„GFeGdŒhŸö j>eE_b8d†8†.jknpfÁ^rgsj>„>j>áb8^dŒhŸö sf
_a`b3`ib3`_Š7dY†.sb[sf7dJj>eE_b.j>eq`fgc8d”e}
dJc8däݯc8dJhŸöùdf c.b8^dqd†.c[sf7dE†8^r s7df7‡dEhd eGnpc8†)hdJ„knpfis7dsb8†
àeGd†.sb8^d†wdf‡
`b3`p‡c
F b8d†
”
b f†8npsŠa`j>c8dj>eE_b.j>eGdb£‡d;_a`b3`ib3`_Š7dònhd#lo`pønpfü"^rgsj>„>j>áb8^däý
`1 , ` 2 , . . . , ` n
†8sb;sfG‡db.c3`j>fEf7npeáb8dãhd{„>j>if7d†r sj7‡npfgc.jkdff7df c»sfEeq`Ýmj>eseWhd
‡
`b3`p‡c
F b8d†‡8Ša`p‡sf7d”1 f jmsf7d
Hvt £¥ ¶¼ž¾¡ k¢%¶'¹£vž x
x¢¦t«
„>j>if7d˜hnpff7^d˜‡npfgc.jkdfgc„kd†ðeGnpc8†òhd
F
i
÷
à
M
j i≤j
à dcòrgs}öùnpf\„Õ`jk†8†8ddäÝ`p‡c8deGdf còsf2d†._a`p‡ddfgc.b8d[hdsmÝ
gD7fQhd„Õ`Œ„>j>if7d)‡npb.b8d†._
eGnpc8†à„kd[f7npeáb8dOhd‡
`b3`p‡c b8d†hŸöùd†._a`p‡deGdf c†.s__„k^eGdf c3`j>b8d†˜÷Œ„Õ`
sij = M − j + i −
f j£„kd†eGnpc8†
‡
`p†
sij
j
X
`h .
h=i
÷
f7dOc.jkdff7dfgc[_a`p†˜†.sb„Õ`G„>j>if7dà
BI‡npf ß<df7npf7†[_„>schpc˜r s7dh7`f7†[‡d
†8db3`Ef7^ig`c.j,l
i j
sij
”bfÁ†3npsŠa`j>c8dJeEj>fj>eEjk†8db„Õ`\†8npeEeGdàŸ†.sb[c8npsc8d†[„kd†[„>j>if7d†
à
=∞
F
npf7h:÷
hd†‡
`b.b8^†)hd†[f7npeáb8d†
ND7f¾hdO‡8Ša`pr s7d)„>j>if7d”(i#`b[däÝmdeE_„kdà†.j}„kd)c8däÝmc8d
hdO‡
`b3`p‡c b8d†hŸöùd†8_a`p‡deGdfgc†.s__„k^eGdfgc3`j>b8d†÷G„Õ`
c.jkdf cȠ.sb
3
F
„>j>if7d†dcr s7d{„kd†;f7npeáb8d†ðhd‡
`b3`p‡c b8d†»hŸöùd†._a`p‡deGdf cð†.s__„k^eGdf c3`j>b8d†»_ npsb‡8Ša`p‡sf7d
"!#%$%
'&)(*,+
-.$/%01234 5#67128:9;&<$>=@?8-?
hd†˜„>j>if7d†[†8npf c
E
à
hd)hmj>eGdf7†.jknpf
dc
’
.D
àa„Õ`q†8npeEeGdJhd†‡
`b.b8^†[†8db3`
2 4
1
hnpf cƒsf7dOdfgc.b8^d
n×n
E[j, k]
F
”€][^ 7fjk†8†8npf7†˜sf¾c3`á„kd
`s
22 + 42 + 12 = 21
b8d_b8^†8df c8d„Õ`E_„>s7†ƒ_ dc.j>c8d)†8npeEeGdOhd†ƒ‡
`b.b8^†˜hd†
f7npeOáb8d†)hdJ‡
`b3`p‡c b8d†hŸöùd†._a`p‡deGdfgcO†.s__„k^eGdfgc3`j>b8d†_ npsb[j>eE_b.j>eGdb„kd†
k
„>j>if7d†”
á‘ÁÙ¦­E¹t€¢¦«€·Û€][npff7dÞsf7d»lonpb.es„kd{_ npsb;‡
`„k‡s„kdb
sc.j>„>jk†3dÞ
sij
E[j, k]
”‘
kblQb8dc.b8npsß<d
‡‘ÁÙ¦­ˆ¹t¢%«€·Û
j
_b8deEjkdb8†[eGnpc8††.sb
j
BI
÷_a`b.c.j>b;hŸö•`sc.b8d†‡
`p†8d†#hmsJc3`á„kd
`s}”a‹ "f7hmjk‡d
“ c “ npf:„Õ`Jߏ`„kdsbãhd„Õ`J†8np„>sc.jknpfþ
ha‘ÁÙ¦­ˆ¹t¢%«€·Û#]˜npff7dބkdc8deE_7†ƒhŸöùdäÝ^‡sc.jknpf:hdß<npc.b8d`„>i<npb.j>c.ŠeGdOdfMf7npc3`c.jknpfˆ`p†.…¯eE_c8npc.jkr s7d”
m
d‘ÁÙ¦­Q¹t€¢%«€·Û ònpeE_„k^c8dÞ)„kd˜c3`á„kd
`sM_ npsb{„kd[_€dc.j>c{c8däÝmc8d˜†8sj>ßp`f c
à7`ß<d‡
M = 10
‡3Ša`s7h¯ý
o]p?q1rs't
{}|#~ t
u
v
w
x
y
z
u
v
w
x
y
I<üZnf2á€npf2á
np„Ÿhd[b.j>Þ
z
zx
uz
u1∞
uw ∞ ∞ ∞ ∞
€
z1y ∞ ∞ ∞ ∞
∞ ∞ ∞
w1v
∞
∞ ∞
uz
∞
∞
∞
∞
‚çÌαÐ/è}È@égÇoÆ<ê»ëìÄaÅ\êÄÃégÇ"ÅÆ<ÇZY£Ã»ÄaÅMÄƾπÖðÖ»é<ÈïÇ×ëìπÆ<Ç)ƒ.űÒÕÔÍÖÈ@Ç×ÉwÆ<ÅØ
Â:ÃÄaÅÆ<Ç/ÈwÉ
`<‘ÁÙ¦­:¹»t¢%«€·Û£Ü;ݯ_„>jkr s7dÞ)„/ö•`ߏ`fgc3`i<dhŸö sf7dŠ7dsb.jk†.c.jkr s7dhd[b8d‡8Š7db8‡8Š7dJ`ß<d‡[c3`á nps7†_a`bb3`__ npb.c÷Osf7d
Š7dsb.jk†8c.jkrgs7d)hŸö•`eG^„>jknpb3`c.jknpfˆ„kn¯‡
`„kd)hd[áa`p†8d”
á‘ÁÙ¦­
¹t€¢%«€·Û)]˜npff7dÞ¾„Õ`±ig`b3`f c.jkd¾†.sbY„Õ`±ß`„kdsb
`__b8n
ݯj>eq`c.j,l
c
GF
V
hŸö sf7d¾†3np„>sc.jknpfºb8dc8npsb.f7^d¾_a`bqsfì`„>i<npb.j>c.ŠeGd
“ `á7†8np„>sõ_ npsbEsfÀ_b8npᄠeGd:hd2eq`Ýmj>eEjk†3`c.jknpfº^c3`f cYhnpff7^Mr s7d2„Õ`Áߏ`„kdsbGhŸö sf7d
†3np„>sc.jknpf:np_c.j>eq`„kdd†.c
V
?”
Qiwnpsb\„kdQ_b8npá„GFeGdÁhmsÍß<n
…g`i<dsbMhd¾‡npeEeGdb8‡d¾eG^c.b.jkr s7dàðf7nps7†2`ß<npf7†\߯sÍdf쇄Õ`p†8†8d
hdsmÝ'`„>i<npb.j>c.ŠeGd†O`__b8n
ݯj>eq`c.j,lo†„I „kdŒ_b8deEjkdb^c3`f c
s7d„>„kdGd†.c
“ b8d„Õ`c.j,lãdc„kdE†8d‡npf7hŸà 1 “ b8d„Õ`c.j,l.”…
‡‘ÁÙ¦­
¹t€¢%«€·Û
1
„Õ`Ghmj,â ^b8df7‡dOd†8†8dfgc.jkd„>„kddfgc.b8d)„kd†ƒhdsmÝMr sjw`Œ_ db.eEjk†ƒhŸö•`eG^„>jknpb8db[`j>f7†.j}„Õ`Œig`b32`fgc.jkdŒhd)r sa`„>j>c8^)hd
„/ö•`__b8n
ݯj>eq`c.jknpfþ
[ ]\
†[t6‡%·
žˆ9f s__ n†8npf7†{r s}öùnpf\†8dhnpff7dƒsf2df7†8deOá„kd˜hd npá1ˆ.dc8†
à¯n lq„Õ`c3`j>„>„kd
n
FeGdGnpá1ˆ.dc†3`c.jk† l/`j>c
”ebf¼†8npsŠa`j>c8dGb3`fi<dbc8nps7†)„kd†npá1ˆ dc8†)h7`f7†sf'eEj>fj>ese
ti
i
0 < ti < 1
hdá n2‰kc8d†ðhdƒc3`j>„>„kdƒsfj>c3`j>b8d”dð
Ša`prgs7dá n2‰kc8dã_ dscð‡npfgc8dfj>b{f}ö j>eE_ npb.c8dr s7d„†3nps7† “ df7†8deá„kd[hŸöùnpá1ˆ.dc8†
P
hnpf c)„Õ`\c3`j>„>„kdJc8npc3`„kdEf}öùdä݇:
F hdE_a`p† ”…f¯npj>c
j>„d†.c˜lo`p‡j>„kdEhdJb8^
`„>jk†3dbrgs7dŒ„kdJf7npeOáb8d
1
T = ni=1 ti Š
eEj>fj>eOse
hd[á n2k‰ c8d†f7^‡d†8† `j>b8d†
à
a
à
d
.
†
ƒ
c
`
M
s
G
e
p
n
>
j
7
f
†
”
N?
dT e
ö•`„>i<npb.j>c.ŠeGd‹kŒ-5Ž{
»
_b8df7h¾‡8Ša`pr s7dOnpá1.ˆ dc˜„/ö sfˆ`_bF †„/ö•`sc.b8dOdc„kd)_„Õ`p‡dŒh7`f7†ƒ„Õ`G_b8deEjGF b8dOá n2k‰ c8d
r sjŸ_ dscã„/ö•`p‡‡s7dj>„>„>j>b
I †8npj>c
„kd[f7npeáb8dOhd[á€n2k‰ c8d†ƒsc.j>„>jk†8^d†_a`bã„/ö•`„>i<npb.j>c.ŠeGd”
N
ha‘ÁÙ¦­¼¹»t¢%«€·ÛXò
npf7†.c.b.sjk†3dތsf±däÝdeE_„Õ`j>b8dEhmsÁ_b8npá„GF eGdJ_€npsb[„kdr s7d„9Œ-5Žƒ
sc.j>„>jk†8dE`sˆeGnpj>f7†hdsmÝ
á n2k‰ c8d†ãhdc.b8np_}”
Hvt £¥ ¶¼žY¡£ž2¶¼¢¦x
ž2ž «
hms
"!#%$%
'&)(*,+
-.$/%01234 5#67128:9;&<$>=@?8-?
<
„IŸ‚ƒb.iseGdf c8dÞYhŸö•`á
–
‘ eGd)hŸö sf:`„>i<npb.j>c.ŠeGdhŸö•`__b8n
ݯj>eq`c.jknpf “ b8d„Õ`c.j,l.”@‹"j f “
‹Œ-5Ž„Õ`jk†8†8dG`s±_„>s7†sf7dGá€n2‰kc8dJb8deE_„>jkdq÷\eGnpj>c.1jk^qnps±eGnpj>f7†
”
][^eGnpf c.b8dÞãdf7†.sj>c8d{rgs7dò„kdðf7npeáb8dhdðá€n2k‰ c8d†£sc.j>„>jk†8^d†#_a`bkŒ’-.Ž“Ÿ
f}öùd†.c.ˆ `eq`jk†;†8s_€^b.jkdsb;÷
”‘
d‘ÁÙk§Á¹»t¢%«€·
xÛ ¾npf c.b8dÞrgs}ö j>„}†ö•`ij>cc8npscãhd[e
hmjk‡d†
npb8h'rgs7d
d2T e
XW
… )”
G• )•
6–
—U
Â:ÃÄaÅÆ<Ç/ÈwÉæÔçÌ qé<È ðÐ ëÍÄúíÄñÐ8î Ð aëìÄaÉwÆ:ëìÏ È}égÇoƏϟÇ/égÄºÒ ÓÖÈ@Ç×ÉwÆ<ÅØ
f¯npj>c
di T
sfMc3`á„kd
`s:hd
lonpjk†
” £`bdäÝmdeE_„kdà
2
n
'ˆ
d†8ceq`
6n
˜‘P™: Œ.]) ]Œ")h7`f7†
df c.jkdb8†
” f2^„k^eGdf cd†.chmj>c
npb.j>c3`j>b8dOh7`f7†
T : [2, 8, 5, 2, 2]
”
T
'‰
†
ö j>„ …M`__a`b3` kcã_„>s7†hd
n
2
š!#›5œ".Œ.N]6?-.œ")_€npsbãh^‡jkhdb˜†.j ‡npfgc.jkdf cƒsf:^„k^eGdf cƒeq`'ˆ.n “
b.j>c3`j>b8dOdc„kd[b8dc8npsb.fa`fgc
à„kd‡
`p†ƒ^‡3Š7^
`fgc
”9ò
dc˜`„>i<npb.j>c.ŠeGdhnpj>cã_b8df7hmTb8dsf:c8deE_7†ƒh7`f7†
”
Θ(n)
á‘ÁÙk§ç¹t¢%«€·
xÛ[][^‡b.j>ß<dÞQsfº`„>i<npb.j>c.ŠeGdŒž,>ž#>.??-.œ¾
" _ npsbGh^‡jkhdbq†.j
‡npfgc.jkdf cqsfõ^„k^eGdfgcqeq` “
T
ˆ.npb.j>c3`j>b8dqdcO„kdGb8dc8npsb.fa`f c
à@„kdY‡
`p†^‡8Š7^
`f c
”kò
dcO`„>i<npb.j>c.ŠeGd2hnpj>c)_b8df7hmb8dqsf²c8deE_7†h7`f7†
”
Θ(n)
][npff7dÞdckˆs7†.c.jTDadÞ)† `Œ_b8npáa`áj>„>j>c8^hd)†.s7‡‡:
F †”
`<‘ÁÙk§'¹t¢%«€·
xÛ#][^‡b.j>ß<dÞ)sf¾`„>i<npb.j>c.ŠeGd
@d˜i;b8nl%d†3†8dsb/I
Ÿ˜j>„>„kd†Ciwd†3`f cŒ‹
# ’– A
Žmàc8d„/” ’– Ž “/–
AAOdäݯc
” – A – <‘