• Aucun résultat trouvé

f (l, a 1 , ..., a k ) = g(l, a 1 , ..., a k , f (h(l), i 1 (a 1 )..., i k (a k )))

N/A
N/A
Protected

Academic year: 2022

Partager "f (l, a 1 , ..., a k ) = g(l, a 1 , ..., a k , f (h(l), i 1 (a 1 )..., i k (a k )))"

Copied!
3
0
0

Texte intégral

(1)

2436587:9;<>=@?ACBDBEGFH=I.JA=KELNM

OQP

RTSVU

*W

U

YX

Z\[G]G^G_`acbd[ e^gfihcjekhcae`je

lm>npoqprsqnGt:u> WKv)wxzy y@{)

U

|)y~})W,} C*C*€}‚|W

U

} (*„ƒ

f(l) = g(l, f (h(l)))

U

}>…†

U

} €y*ˆ‡

U

)W

U

k + 1

Š.}wz„ƒ

f (l, a 1 , ..., a k ) = g(l, a 1 , ..., a k , f (h(l), i 1 (a 1 )..., i k (a k )))

‹ U

} WKvQ0Œp})‡

U

)W

U

$W„wŽy‘}.p.…)y

U

) (T’“*Œp}Q

U

$W

U

U

)y).ƒ

”

‹•–0—:˜~–™›š~œ0˜Yc—–D•ž—~–.Ÿ

 

œ0—Q™¢¡£•~ž—~–¥¤Y¢—p¡

¦T§©¨0ª

—˜p«–

¬

œ®­p­¯•

¨0ª¥°±

—®²¯š~œ.˜Q¢—~–:•³>´p´

+

‹•–0—:˜~–™

  –  ~µ

˜–Dœ:•¶Ÿ

 

œ0—Q™¢¡£• ¤Y¢—p¡

¦T§ ¨0ª¶·

œp•~ž.–

¬ µ

­­&¸

¨0ª

œ.Ÿ

µ

¬¬

  –  ~µ

˜~–:œD¸´´

¹

‹•–0—:˜~–™Dž

±

  

–º~

 

š~œc˜£•~pž¢—~–Ÿ

 

œ0—Q™¢¡£•~ž—~–¥¤Y¢—p¡

¦T§©¨0ª¶»

¬D¦

œ

§Q¨0ª

œ

¬

œ®­­

µ

­p­T¼

¨ª

œ¶½¾ž

±

  

–º~

 

š~œc˜:¼

¿ÀpÁÁ*Â0ÄÄ@ŐÀÇƛÈ

É¢Ê

g(l, b) = ¬b h(a :: l) = l f ([]) = true

ËÊ

g(a, b :: l, c) = (a = b) ∨ c i 1 (a) = a

h(a :: l) = l f (a, []) = f alse

ÌpÊ

g(a :: b :: l, s) = a + s

h(a :: b :: l) = l

f ([]) = 0

f (a :: []) = a

(2)

lm>npoqprsqn£u>† )W† y~}" †Q ˆ‡}) Xˆ*V }0„ƒ† } C C})

l

…

W U

wŽwz)W*€†Q VW

U

) } €*y}z

U

})¯T*

l p

l i

…Œp}QW

U

)pV *† *WwzpV*8wzp

y ’)y~W*†Q €(wŽ†Q (y

l

R

W*

U † K

U *(†† *W&†Q X ‹



R

†Q ,‡s})

U

YX

l p

l i

… †}

U

G‡ })

U

) *,y}G*, **

l triee p

l i triee

…

U

Qp

)

l triee

‹

|}††

U

y~†

U

€y ’})wŽ†pK

U

g W„wŽ\y*ˆ‡

U

)W

U

)€y†Q K

U C‡s})

U

ž–0š~œ.˜~–.˜­ œD•ž—~–

¨ª

²œ:•~ž—~–œ:•~pž¢—~–)³

·

«)ž

±0°

­ ²œD•ž—~– œ:•~pž¢—~–)³

¨0ª

œD•ž—~–

” ‹

8W W„wŽ>‘‡

U

)W U

—˜Q0º

·

«Qž

±.°

­ œ:•~ž—~–

¨ª

œ:•ž¢—–‹

+

‹"!

{Q ŽŒp})

—˜Q0º

·

«Qž

±.° >’††Q pz†Q0z‰|T’“)w

‚yz‡

U

)W

U

)Žy@{)K

* **Wd

KWv)wx

f (l) = g(l, f (h(l))

…,† U † U ‘} WKv)wx †}),Š) KT…8Œp}V)W}‘y*,‡

U

)W

U

)

W U wŽwz

—p˜Y0º

·

«)žÇ

±0°

RU }

—p˜Y0º0˜œ0šY$#ǖ

X ‹

¹ ‹ S

W} (,W

U

wŽ†~ y

—p˜Y0º

·

«)žÇ

±0° d wz*Cy~}N

U w

yW

U

wŽ†Q K

U

)„…)K0WKvQp€Œ})

U w

$y‚W

U

wŽ†Q K

U

) )*W**K *z†

U

} Ž‡})

U

) y} * yN

U

Š.})}

n

*d}

wŽ}

n

…)}N†

2n − 1

…)€Œ}) T’

U † K

U

|y†Q K

U

‚) ‡ C†Q0€yW

U

wŽ†Q K

U

)

‹

¿ÀpÁÁ*Â0ÄÄ@ŐÀÇƛÈ

É¢Ê

%'&$(*)+&,-().0/01$23.5456*%87:9;$(<,=>%:?.('=

@BADC0EF@BA

GH@

;

AIC0E

%

G / C0E

%'&$(>%JLK%'M87D30&5N;$)+&$)H%D.6O1$23.5406QPR().0/$1$23.5406*%<J$STPU().0/$1023.54068%'M<S

VV

ËÊXWZY\[^][^_a`bc["de$f+g+bh[ji5k+k[lbLmnaolf+mpRqcrs[aot<i5k+k["i5f\pRotnluviwpRqhuxkbc[¢ÊzyXY{k[lf+_|r}i5qhmR[$~

f (l) = g(l, f(h 1 (l)), f (h 2 (l)))

 [o^q€[p_pRf‚ƒpRi0Y'_„ke0fm

().0/$1$23.5406 [l_

().0/0)+;0N.†…&

ÊL‡ˆbhfp‰$nlY+nami0bc[aux[aY'_a`

f (l) = g(l, f(h 1 (l)), ..., f (h n (l)))

e$f€`iŠ$[aoXf+Y\pRot+nauvi‹f+YqhŒ'f+[$~

f (l) = g(l, map f liste



des



h i (l))

ÌpʏŽe0qc_

n = longueur(l) = 2 k

Ê"‡[lY<d+i5Y'_bc[oli0bholf+b‘d[ ().0/$1023.5406

l

`Le0Yr’fpRqce$Y+Y+[

2 k 1

ki0qcm[ap“d[xbcq”pU_R[ap d+["bce$Y+‰0f+[af+m

1

`

2 k 2

k<i5qhmR[p•d+["bcq”p_[ap|d["bhe0Y‰0f+[af+m

2

`)ÊÊÊh`

1

ki5qhm[“d["bhq”p_[ap|d["bhe0Y+‰$f+[lfm

2 k 1

Ê

–

e0Y<o5`+bh["Y+e0uwg+m[—_e5_i0b€d[jo^e$uki5mi5q”pUe$Ypz[apU_—i5f{uxqh[lf+˜~

P k 1

i =0 2 k 1 i 2 i = k2 k 1

i0f™k+qcm[0~

P k 1

i =0 2 k 1 i (2 i +1 − 1) = k2 k − P k 1

i =0 2 k 1 i = k2 k − 2 k + 1 = (k − 1)2 k + 1

di5Yp„bh[apd+[lf˜šoai0pa`bhiole0uk+bh[^˜q›_n"[apU_

O(n × log 2 (n))

lm>npoqprsqnœ¾u’p K*W

U y y}g*K) †

U

)

l 1

l 2

*})zzŒ}8W

U

pH*

wzpŒ}††Q Kp‰‘,‡

U

(y))

l 1

(y))

l 2

‹

U

)W

U

x}~ ˆ}V†

U

} †

U

Š. KwŽwz T’ *W

U

x*

  –  Çµ

˜~–®­ œ

¨ª

œ:•ž¢—

¨0ª

µ

±±

• R

~ WW* ” X ‹

(3)

” ‹

8W Ž ‡ )W W„wŽ

 —–0˜Qž.–~™¢—Q ­ œ:•~ž— œ:•~pž¢— œD•ž—

… Œ}® „

T’“*Œ}Q

U

f (l 1 , l 2 ) = g(l 1 , l 2 , f (h(l 1 ), i 1 (l 2 )))

g(a :: l 1 , l 2 , r) =

a :: r

  –  ~µ ˜~–

(a, l 2 )

r

U

h(a :: l) = l i 1 (l) = l

U

} 4W„W} T’ *W

U y y} * **„…,>’“* †Q0|)*W**KK y ’}K  *

y ’††Q Q)W*

‹

+ ‹

8W ,‡

U

)W U

 °

—–0˜)ž–~™¢—Q

±.°

­ ²œD•ž—QœD•ž—Y³

¨ª

œ:•~pž¢—

} p

—˜Q.º

·

«Qž

±0° C>’} p(†Q0

  –  ~µ

˜–‹

¹ ‹ SVU

wŽ†Q ‘*W

U

wŽ†~*‘y

 °

—~–0˜Qž.–™c—Q

±0°

 °

—~–.˜)ž–~™c—Y

±.°

…\G wz*‘y~}G U w

dy

W U

wŽ†Q K

U

)C*(y ’“Šp

R

Œ}>’

U

‚) y~Š.}) †Q0 X ‹

¿ÀpÁÁ*Â0ÄÄ@ŐÀÇƛÈ

É¢Ê

%'&$(*)+&,*.6(+&0)30&,†(.54068%H79;0(,=>%?L.('=

@BAIC5E @BA

G

; C0E

%'&0(O)+&+37D.6(&$)30&,(.5406 .6

.†1 PP9&9)+&:;<S ('=&06>;B)+&3

&%+30&:)+&+3

VV

ËÊ

%'&$(*)+&,*.6(+&0)30&,†(.5406M%H7>%'&$( %0(7'().0/$1$23.54068%*;06…0(7'().5/$1$23.†406 .6

9;$(,= %0(ˆK$(:?L.($=

@BA

K C5EZ@BA

G

ZK

@BA<C0EZ@BA

G

;ˆK .

C0E

.†1>;$7*('=&06>;.6(+&$)<30&,†(.†406M .&%+35&

.1>;O($=&06I.6(+&0)30&,†(.5406MFP .S-&%+30&

.6(+&$)<30&,†(.†406MIP;S8.

VV

ÌpʏŽe0qh[lY'_

n = longueur(l 1 )

`

m = longueur(l 2 )

Ê .6(+&$)30&+,†(.5406

l 1 l 2

ole0uke0mR_R[

n

i0k+k<[abhp“d[ 9&9)+&

a l 2

Ê  ti$Œ'f+["i5k+k[lb ole0f_[ji5f™uqc[l˜

1

`i5f™k+qhmR[

m

Ê – e0Yob”ivo^e$uxkbc[l˜qc_Rn"d[

.a6(+&$)35&,†(.5456

[apU_—i5f{uqc[af˜

O(n)

`<i0fškqcm[

O(n × m

Ê

Y+[\r’e$qhpƒ_Rmqcn bh[apƒbcq”p_[apa` .6(+&$)30&+,†(.5406+M [^˜[aolf_R[ i5f:uqc[af˜

min(m, n)

_[apU_pa`„i0f8k+qhmR[

2min(m, n)

_[apU_pa`<de$Yo"b”ivo^e$uxkbc[l˜qc_Rnjd+[ .6(+&$)30&+,†(.5406+M [apU_

O(n × log 2 (n) + m × log 2 (m)) = O(max(n, m) × log 2 (max(n, m)))

lm>npoqprsqn ®

U

} €H†

U † U

U

|}0.ƒ

U

}

U

} 

l intersection l l = l

‹

¿ÀpÁÁ*Â0ÄÄ@ŐÀÇÆ È

br}i0f_k+me0f+Š$[lm•k+bcf<p|‰0naY+nlmi5bh[lu[lY'_|Œ'f+[0`+pRq

l

[apU_„fY+[jpUe$fp bhqhpU_R[d[

l

i5bhe0mp .a6(+&$)35&,†(.5456

l l 0

!

l

"

r’e0mux[abcbh[lu[aY$_`

l

pUe$fp bhqhpU_R[d[

l 0

p$#naolmRqc_

∃f : {0, .., longueur(l) − 1} → {0, .., longueur(l 0 ) − 1}

_R[abcbh[jŒ'f+[

∀ 0 ≤ i ≤ longuer(l) − 1 elt(l, i) = elt(l 0 , f (i))

e%

elt(a :: j, 0) = a

`

elt(a :: j, n + 1) = elt(j, n)

 [l_—nlY+e$Yo^nwpU[‹d+[lue0Y'_Rm[r}i0olqcbh[lu[aY$_Xki5mqhYdf<o _Rqhe0Y `iŠ$[aob”ivmR[auvi5mŒ$f[Œ'f+[wpUq

l = a :: j

[^_

f

[lYŠ$e0q

l

pRf+m

l 0

i0f\pU[aYp|d[bhivd+n'&Y+qc_Rqhe0Y\olq( Bd[appRfpl`i0bce$mp

n 7→ f (n + 1)

[aYŠ0e0q

j

pRf+m

l 0

Ê

Références

Documents relatifs

Capes externe de math´ ematiques, session 1997, 1` ere composition.. Danc ce probl` eme, α d´ esigne un nombre r´ eel

Calculer les aires des parallélogrammes EFGH, IJKL et MNOP en effectuant les mesures

Dans chaque cas, tracer la section du solide par le plan (IJK)

Dans cette série d’exercices, on cherchera à déterminer la section du solide par un plan quelconque (donc pas nécessairement parallèle à une face) défini par des points situés

&amp; SAMPSON, G., On weak restricted estimates and end point problems for con- volutions with oscillating kernels (I), submitted for publication.. Thesis, Washington

All algebras are supposed to have identity element (denoted e ) b u t are not assumed associ- ative or finite-dimensional in general.. The term normed algebra is

Weiter gibt es ja in jedem alge- braisehen KSrper unendlieh viele Primideale ersten Grades... die Kongruenzen (10)haben keine LSsungen. ])as Ergebnis l~sst sich

unit time. I t is easy to give an interpretation of Fishers likelihood- function in terms of information theory. As an example we are going to treat the case