Âû÷èñëèòåëüíî òðóäíûå çàäà÷è è äåðàíäîìèçàöèÿ Ëåêöèÿ 6: Ïîâûøåíèå òðóäíîñòè ôóíêöèè. XOR -ëåììà ßî. Äìèòðèé Èöûêñîí ÏÎÌÈ ÐÀÍ 12 àïðåëÿ 2009 1/9 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 2/9 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 4/9 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 5/9 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 6/9 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 7/9 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 8/9 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 ). 9/9