1. Ìàòåìàòè÷åñêèå îñíîâû Computer Science
×àñòü 2: Âåðîÿòíîñòíûé ìåòîä. Ëåêöèÿ 8.
Ìåòîä âòîðîãî ìîìåíòà
Äìèòðèé Èöûêñîí
ÏÎÌÈ ÐÀÍ
6 äåêàáðÿ 2009
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2. Ñîäåðæàíèå ëåêöèè
1 Äèñïåðñèÿ
2 Íåðàâåíñòâî ×åáûøåâà
3 Ïîðîã äëÿ 4-êëèêè
4 Ñåïàðàòîð
Ëèòåðàòóðà
1 Í. Àëîí, Äæ. Ñïåíñåð. Âåðîÿòíîñòíûé ìåòîä.
2 S. Jukna. Extremal combinatorics.
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3. Äèñïåðñèÿ
Îïðåäåëåíèå. Äèñïåðñèåé ñëó÷àéíîé âåëè÷èíû ξ íàçûâàåòñÿ
âåëè÷èíà D ξ = E[(ξ − E ξ)2].
D ξ = E[ξ − E[ξ]]2 = E[ξ 2 − 2ξ E[ξ] + (E[ξ])2 ] = E[ξ 2 ] − (E[ξ])2 ≥ 0.
Ëåììà. (Íåðàâåíñòâî ×åáûøåâà). D[ξ] = σ 2 , òîãäà
1
Pr[|ξ − E[ξ]| ≥ kσ] ≤ .
k2
Äîêàçàòåëüñòâî. η = (ξ − E[ξ])2, E[η] = σ2.
(íåð-âî Ìàðêîâà) 1
2 2
Pr[|ξ − E ξ| ≥ kσ} = Pr{η ≥ k σ ] ≤ k2
.
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4. Äèñïåðñèÿ ñóììû
• Cov [ξ1 , ξ2 ] = E[ξ1 ξ2 ] − E[ξ1 ] E[ξ2 ]
• D[ n ξ1 ] = n D[ξi ] + 2 i<j Cov [ξi , ξj ]
i=1 i=1
• D[ ξi ] = E[( i ξi )2 ] − (E[ i ξi ])2
i
• E[( 2 2 2
i ξi ) ] = E[ i ξi +2 i<j ξi ξj ] = i E[ξi ]+2 i<j E[ξi ξj ]
• (E[ i ξi ])2 = ( i E[ξi ])2 = i (E[ξi ])2 + 2 i<j E[ξi ] E[ξj ]
• D[ i ξi ] = i E[ξi2 ] − i (E[ξi ])2 + 2 i<j E[ξi ξj ] −
2 i<j E[ξi ] E[ξj ] = i D[ξi ] + 2 i<j Cov [ξi , ξj ]
• Åñëè ïîïàðíî íåçàâèñèìû, òî
ξ1 , ξ2 , . . . , ξn
D[ξ1 + ξ2 + · · · + ξn ] = D[ξ1 ] + D[ξ2 ] + · · · + D[ξn ].
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5. Ïîðîã ñóùåñòâîâàíèÿ 4-êëèêè
• G (n, p) ñëó÷àéíûé ãðàô íà n âåðøèíàõ, êàæäîå ðåáðî
äîáàâëÿåòñÿ íåçàâèñèìî ñ âåðîÿòíîñòüþ p.
• Çàäà÷à: ïðè êàêèõ p â ãðàôå G (n, p) ñ áîëüøîé
âåðîÿòíîñòüþ åñòü êëèêà íà 4 âåðøèíàõ?
• Òåîðåìà. Åñëè p = o(n−2/3), òî â ãðàôå G (n, p) åñòü
4-êëèêà ñ âåðîÿòíîñòüþ o(1). Åñëè p = ω(n−2/3), òî â
ãðàôå G (n, p) åñòü 4-êëèêà ñ âåðîÿòíîñòüþ 1 − o(1).
Äîêàçàòåëüñòâî.
• S ⊆ V , |S| = 4, AS : S îáðàçóåò êëèêó.
1, åñëè S îáðàçóåò êëèêó G
• XS =
0, èíà÷å
• E[XS ] = Pr[XS = 1] = p 6
• X = XS
4 6
• Pr[X ≥ 1] ≤ E[X ] = Cn p 6 ≤ n24 = o(1),
4 p
åñëè p = o(n−2/3 )
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6. Ïîðîã ñóùåñòâîâàíèÿ 4-êëèêè
D[X ]
• Pr[X = 0] ≤ E[X ]2
• D[X ] = S D[XS ] + S=T Cov (XS , XT )
• D[XS ] = 2
E[XS ] − E[XS ] 2 ≤ E[X 2 ] = E[X ] = p6
S S
• S D[XS ] = O(n4 p 6 )
• Åñëè AS è AT íåçàâèñèìû, òî Cov (XS , XT ) = 0.
• Åñëè AS è AT çàâèñèìû, åñëè |S ∩ T | = 2 èëè |S ∩ T | = 3.
• Åñëè |S ∩ T | = 2, òàêèõ ïàð O(n6),
Cov [XS , XT ] ≤ E[XS XT ] = p 11
• Åñëè |S ∩ T | = 3, òàêèõ ïàð O(n5),
Cov [XS , XT ] ≤ E[XS XT ] = p 9
• D[X ] = O(n4 p 6 + n6 p 11 + n5 p 9 ) = o(n8 p 12 ) = o((E[X ])2 )
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7. Ñåïàðàòîðû
Òåîðåìà. X = {1, 2, . . . , n}, F ⊆ . 1) ∀A ∈ F, |A| ≤ r ; 2)
2X
dx = |{A ∈ F | x ∈ A}|; 3) d = ;
x∈X x d
Òîãäà ñóùåñòâóåò ïàðà ìíîæåñòâ S, T ⊆ X , ÷òî S ∩ T = ∅ è
n
äëÿ ëþáîãî A ∈ F âûïîëíÿåòñÿ ëèáî A ∩ S = ∅, ëèáî
A ∩ T = ∅, êðîìå òîãî |S|, |T | ≥ (1 − δ)2−d n, ãäå δ = dr 2 .
d+1
Äîêàçàòåëüñòâî.
n
• Ïîêðàñèì âñå ìíîæåñòâà èç F ñëó÷àéíûì îáðàçîì â ñèíèé
è êðàñíûå öâåòà.
• S = {x ∈ X | ∀A ∈ F, x ∈ A =⇒ A ñèíåå}
• T = {x ∈ X | ∀A ∈ F, x ∈ A =⇒ A êðàñíîå}
• S ∩ T = ∅, ∀A ∈ F ëèáî A ∩ S = ∅, ëèáî A ∩ T = ∅.
• Öåëü: ïîêàçàòü, ÷òî ñ ïîëîæèòåëüíîé âåðîÿòíîñòüþ
|S|, |T | ≥ (1 − δ)2−d n.
• Zx =
1, x ∈ S
0, èíà÷å
, E[Zx ] = Pr[Zx = 1] = 2−d
x
• Z = x Zx , Z = |S|
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8. Ñåïàðàòîðû
èíà÷å ,
1, x ∈ S
• Zx = E[Zx ] = Pr[Zx = 1] = 2−dx
0,
• Z = x Zx Z = |S|,
• E[Z ] = x 2−dx ≥ n2− x dx /n = n2−d
• D[Z ] = x D[Zx ] + x=y Cov [Zx , Zy ]
• Åñëè íåò A∈F, ÷òî , òî
x, y ∈ A è
Zx Z y íåçàâèñèìû,
ïîýòîìó .
Cov (Zx , Zy ) = 0
• Äëÿ x ∈Xñóùåñòâóåò íå áîëåå (r − 1)dx ýëåìåíòîâ , ÷òî
y
x, yëåæàò â îäíîì A∈F
• Cov (Zx , Zy ) ≤ E[Zx Zy ] ≤ E[Zx ] ≤ 2−dx
−dx ≤
• y =x Cov (Zx , Zy ) ≤ (r − 1) x dx 2
r −1 −dx )(
n ( x2 x dx ) = d(r − 1)E [Z ]
• D[Z ] ≤ (d(r − 1) + 1) E[Z ] ≤ dr E[Z ]
• Pr[Z (1 − δ) E[Z ]] δ2 (E[Z]])2 ≤ δ2 dr ] = dr 2d+1 E[Z ] =
D[Z
E[Z
drn
n
2d+1 E[Z ]
≤ n
2d+1 2−d n
= 1
2 .
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