Âû÷èñëèòåëüíî òðóäíûå çàäà÷è è äåðàíäîìèçàöèÿ Ëåêöèÿ 6

Âû÷èñëèòåëüíî òðóäíûå çàäà÷è è
äåðàíäîìèçàöèÿ
Ëåêöèÿ 6: Ïîâûøåíèå òðóäíîñòè ôóíêöèè.
XOR -ëåììà
ßî.
Äìèòðèé Èöûêñîí
ÏÎÌÈ ÐÀÍ
12 àïðåëÿ 2009
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Ïëàí
ρ
Havg
(f ) = max {S | ∀C , |C | ≤ S =⇒ Prx←Un [C (x)= f (x)] < ρ}
,
1
+1
1
2
S
Hwrs (f ) = Havg
Havg (f ) = max S | Havg
(f ) ≥ S
•
1−δ (f 0 ) ñòðîèì f 00 ñ áîëüøîé H + (f 00 ).
Ïî f 0 ñ áîëüøîé Havg
avg
1
2
• XOR
•
-ëåììà ßî
1−δ (f 0 ).
Ïî f ñ áîëüøîé Hwrs (f ) ñòðîèì f 0 ñ áîëüøîé Havg
•
Êîäû, èñïðàâëÿþùèå îøèáêè
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Ëåììà Èìïàëüÿööî
•
•
1−δ (f ) ≥ S
Ïóñòü Havg
Êàæäàÿ ñõåìà ðàçìåðà ≤ S îøèáàåòñÿ õîòÿ áû íà δ2n
âõîäàõ
1
2
Äëÿ êàæäîé ñõåìû ýòè òðóäíûå âõîäû ñâîè
Òðóäíûå âõîäû îáùèå äëÿ âñåõ ñõåì
Ëåììà. (Ëåììà Èìïàëüÿööî î òðóäíîì ðàñïðåäåëåíèè) Ïóñòü
1−δ (f ) ≥ S , òîãäà ñóùåñòâóåò òàêîå ðàñïðåäåëåíèå H íà
Havg
{0, 1}, ÷òî
1
1 ∀x ∈ {0, 1}n , H(x) ≤ δ 2−n (ïëîòíîñòü H ðàâíà δ )
S
1
2 ∀ ñõåìû C ðàçìåðà ≤ 100n : Prx←H [C (x) = f (x)] ≤ 2 + 2
3/9
MIN-MAX òåîðåìà
. . a1m • Pm pi = 1, Pn qi = 1
i=1
. . a2m • a i=1
ij âûèãðûø ñòðî÷íîãî
.
. . . . .
èãðîêà P
.
. . . . .
• qAp = i,j pi qj aij qn an1 an2 . . anm
ìàòîæèäàíèå âûèãðûøà
p1 p2 . . pm
Òåîðåìà. Ñóùåñòâóþò òàêèå ñòðàòåãèè p∗, q∗, ÷òî äëÿ âñåõ
ñòðàòåãèé p, q âûïîëíÿþòñÿ íåðàâåíñòâà:
q1
q2
a11
a21
a12
a22
qAp ∗ ≤ q ∗ Ap ∗ ≤ q ∗ Ap
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Äîêàçàòåëüñòâî ëåììû Èìïàëüÿööî
Ëåììà. (Ëåììà Èìïàëüÿööî î òðóäíîì ðàñïðåäåëåíèè) Ïóñòü
1−δ (f ) ≥ S , òîãäà ñóùåñòâóåò òàêîå ðàñïðåäåëåíèå H íà
Havg
{0, 1}, ÷òî
1
1 ∀x ∈ {0, 1}n , H(x) ≤ δ 2−n (ïëîòíîñòü H ðàâíà δ )
S
1
2 ∀ ñõåìû C ðàçìåðà ≤ 100n : Prx←H [C (x) = f (x)] ≤ 2 + Äîêàçàòåëüñòâî.
S
• Äèìà âûáèðàåò ñõåìó C ðàçìåðà ≤ S 0 = 100n
, Ýäèê
âûáèðàåò ðàñïðåäåëåíèå H ïëîòíîñòè δ.
• Ýäèê ïëàòèò Äèìå Prx←H [C (x) = f (x)] ðóáëåé.
• Ðàñïðåäåëåíèå íà ðàñïðåäåëåíèÿõ ïëîòíîñòè δ ýòî
ðàñïðåäåëåíèå ïëîòíîñòè δ
2
2
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Ïðîäîëæåíèå äîêàçàòåëüñòâà
•
•
•
•
•
S
Äèìà âûáèðàåò ñõåìó C ðàçìåðà ≤ S 0 = 100n
, Ýäèê
âûáèðàåò ðàñïðåäåëåíèå H ïëîòíîñòè δ.
Ýäèê ïëàòèò Äèìå Prx←H [C (x) = f (x)] ðóáëåé.
Ðàñïðåäåëåíèå íà ðàñïðåäåëåíèÿõ ïëîòíîñòè δ ýòî
ðàñïðåäåëåíèå ïëîòíîñòè δ
Ïóñòü C, H ∗ ýòî ðåøåíèå èãðû.
∀H ïëîòíîñòè δ , PrC ←C,x←H [C (x) = f (x)] ≥
2
PrC ←C,x←H ∗ [C (x) = f (x)] ≥
1
2
+
ïëîõîé, åñëè PrC ←C [C (x) = f (x)] ≤ 12 + 2
Ïëîõèõ x ìåíüøå δ2n
C1 , C2 , . . . , Ct ← C , t = 50n
, C = majority (C1, C2, . . . , Cn ).
• x ∈ {0, 1}n
•
•
2
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Ïðîäîëæåíèå äîêàçàòåëüñòâà
ïëîòíîñòè δ, PrC ←C,x←H [C (x) = f (x)] ≥ 12 + x ∈ {0, 1}n ïëîõîé, åñëè PrC ←C [C (x) = f (x)] ≤ 12 + 2
Ïëîõèõ x ìåíüøå δ2n
C1 , C2 , . . . , Ct ← C , t = 50n
, C = majority (C1, C2, . . . , Cn ).
(Îöåíêè ×åðíîâà â àääèòèâíîé ôîðìå) Xi ∈ {0, 1} íåçàâèñèìûå îäèíàêîâî ðàñïðåäåëåííûå
ñëó÷àéíûå
hP
i
X
âåëè÷èíû. E Xi = µ. Òîãäà Pr | n − µ| > ε ≤ 2e −2ε n
Äëÿ õîðîøåãî x Pr[C (x) = f (x)] ≥ 1 − 2e −50n
Õîðîøèõ x íå áîëåå 2n , ïîýòîìó íàéäåòñÿ ñõåìà C ,
êîòîðàÿ âû÷èñëÿåò f íà âñåõ õîðîøèõ x .
Çíà÷èò, Prx←U [C (x) = f (x)] ≥ 1 − δ
|C | ≤ tS 0 + cn = S2 + cn < S , ïðîòèâîðå÷èå!!!
• ∀H
•
•
•
•
2
n
i=1
•
•
•
•
i
2
n
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Òåîðåìà. (ßî, 1982). f
f ⊕k (x
XOR-ëåììà
,
: {0, 1}n f ⊕k : {0, 1}nk → {0, 1}
1 , x2 , . . . , xk ) = f (x1 ) ⊕ f (x2 ) ⊕ · · · ⊕ f (xk )
. Òîãäà
:
1−δ (f ).
âûïîëíÿåòñÿ Havg+(f ⊕k ) ≥ 400n
Havg
Äîêàçàòåëüñòâî. Ïóñòü íàéäåòñÿ ñõåìà C ðàçìåðà S 0 = 400n
S,
÷òî
∀δ > 0, ∀ > 2(1 − δ)k
1
2
2
2
"
Pr
(x1 ,x2 ,...,xk )←Unk
•
C (x1 , x2 , . . . , xk ) =
k
M
i=1
#
f (xk ) ≥
1
+
2
Ïóñòü H òðóäíîå ðàñïðåäåëåíèå èç ëåììû Èìïàëüÿööî
äëÿ 2 . H(x) ≤ 1δ 2−n .
• Un = (1 − δ)G + δH
• Un2 = (1 − δ)2 G 2 + (1 − δ)δGH + δ(1 − δ)HG + δ 2 H 2
• Unk = (1 − δ)k G k + (1 − δ)k−1 δG k−1 H + · · · + δ k H k
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Ïðîäîëæåíèå äîêàçàòåëüñòâà
• Un = (1 − δ)G + δH
• Unk = (1 − δ)k G k + (1 − δ)k−1 δG k−1 H + · · · + δ k H k
• D
ðàñïðåäåëåíèå
íà {0, 1}nk .
h
PD = Prx←D C (x1 , x2 , . . . , xk ) =
•
•
1
2
1
2
Lk
i
i=1 f (xk )
+ ≤ PUnk = (1 − δ)k PG k + (1 − δ)k−1 δPG k−1 H + · · · + δ k PH k
+
2
≤ (1 − δ)k−1 δPG k−1 H + · · · + δ k PH k
+
h 2
i
Lk
• Prx←G (i) H (k−i) C (x1 , x2 , . . . , xk ) =
f
(x
)
> 12 + 2
k
i=1
h
i
L
• Prx` ←H C (x1 , x2 , . . . , xk ) =
f
(x
)
⊕
f
(x
)
> 12 + 2
i
l
i6=`
• PG (i) H (k−i) >
•
1
2
Ïðîòèâîðå÷èå ñ ëåììîé
Èìïàëüÿööî äëÿ ñõåìû
L
C (x1 , x2 , . . . , xk ) ⊕ i6=` f (xi ).
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