Corrigé

n
{a1, . . . , an}1 = a1< a2. . . < an
S S
Syst n a1, .., an
1 = a1< a2< ... < an
S
k1, ..., knPn
i=1 ki∗ai=SPn i=1 ki
an
an−1
{1,20,30}40 {1,5,6}
10
Effg(S, Syst)Syst S
a1= 1, a2=d, a3=d2, . . . , ai=di−1, . . . , an=dn−1d d > 1
•
g=g1, ..., gn
o=o1, ..., on
g=o g j gj6=oj
gj> oj
i=n
X
i=1
gi∗ai=
i=n
X
i=1
oi∗ai
gi=oii j
i=n
X
i=j+1
gi∗ai=
i=n
X
i=j+1
oi∗ai
i=j
X
i=1
gi∗ai=
i=j
X
i=1
oi∗ai

i=j−1
X
i=1
oi∗ai=
i=j−1
X
i=1
gi∗ai+ (gj−oj)∗aj≥(gj−oj)∗aj≥aj
oi1≤i≤j−1d
i=j−1
X
i=1
oi∗ai≤
i=j−1
X
i=1
(d−1) ∗ai≤(d−1) ∗
i=j−1
X
i=1
ai
Pi=j−1
i=1 ai=Pi=j−1 i=1 di−1=Pi=j−2 i=0 di= (dj−1−1)/(d−1) Pi=j−1 i=1 oi∗ai≤dj−1−1< aj
j= 1
i1≤i≤j−1oid o0
o0
l=oll i i + 1 o0
i−1=oi−1o0 i=oi−d, o0 i+1 =oi+1 + 1
o0o o
g=o g
diR+k∗di
•
d−1aii < n
d d
on=s%d, on−1= (s div d)%d, ..., ..., o1=s div dn−1
•d
O(S∗n)
Eff(S, Syst)S
S
•ap
S < apS a1, ..., ap−1
S≥apap
Eff(S, Syst, k)S k
Syst
Eff(S, Syst, p) = Eff(S, Syst, p −1) S < ap
Eff(S, Syst, p) = min(Ef f(S, Syst, p −1),1 + Eff(S−ap, Syst, p)) S≥ap
Eff(0, Syst, p) = 0
Eff(S, Syst, 1) = S a1= 1
0S
1n
•ap
ap0S/ap
Eff(S, Syst, p) = min0≤k≤s/ap(k+Ef f(S−k∗ap, Syst, p −1)).
Eff(S, Syst, p) = min(Ef f(S, Syst, p −1), S/ap+Eff(S%ap, p))
{1,3,4}S= 10 4 0 2

•
k1p ak≤S
Eff(S, Syst) = 1 + min1≤i≤kEf f(S−ai, Syst)
Eff(S, Syst, p) = 1 + min1≤i≤kEf f(S−ai, Syst, i)
min(Eff(S, Syst, p −
1),1 + Eff(S−val(p), Syst, p) 1 + Eff(S−val(p), Syst, p)
log S +Plog ai+log n O(S∗n)
X
Effmoy(X, Syst)Eff(S, Syst)S1X
Effmoyg(X, Syst)Effg(S, Syst)S1X
O(X)Effmoyg(X, Syst)
O(X)
Effmoyg(s, Syst)s0X
Effmoyg(s, Syst)
O(X∗n)Effmoy(X, Syst)

X99