# Эффективные алгоритмы, осень 2007: Вероятностные алгоритмы

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Computer Science http://logic.pdmi.ras.ru/infclub/

. (CS ) 6. 1 / 32

http://logic.pdmi.ras.ru/~infclub/

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. (CS ) 6. 2 / 32

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. (CS ) 6. 2 / 32

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. (CS ) 6. 2 / 32

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. (CS ) 6. 2 / 32

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. (CS ) 6. 3 / 32

• n n A, B , C . A B = C . A B C . O(n3), O(n2.376). , O(n2) .

. (CS ) 6. 4 / 32

• n n A, B , C . A B = C .

A B C . O(n3), O(n2.376). , O(n2) .

. (CS ) 6. 4 / 32

• n n A, B , C . A B = C . A B C .

O(n3), O(n2.376). , O(n2) .

. (CS ) 6. 4 / 32

• n n A, B , C . A B = C . A B C . O(n3), O(n2.376).

, O(n2) .

. (CS ) 6. 4 / 32

• n n A, B , C . A B = C . A B C . O(n3), O(n2.376). , O(n2) .

. (CS ) 6. 4 / 32

• Rand-Matrix-Equiv(A, B , C )

x {0, 1}n

y = B x z = A y t = C x , z = t

A (B x) = C x . .

. (CS ) 6. 5 / 32

• Rand-Matrix-Equiv(A, B , C )

x {0, 1}n

y = B x z = A y t = C x , z = t

A (B x) = C x . .

. (CS ) 6. 5 / 32

• Rand-Matrix-Equiv(A, B , C )

x {0, 1}n

y = B x

z = A y t = C x , z = t

A (B x) = C x . .

. (CS ) 6. 5 / 32

• Rand-Matrix-Equiv(A, B , C )

x {0, 1}n

y = B x z = A y

t = C x , z = t

A (B x) = C x . .

. (CS ) 6. 5 / 32

• Rand-Matrix-Equiv(A, B , C )

x {0, 1}n

y = B x z = A y t = C x

, z = t

A (B x) = C x . .

. (CS ) 6. 5 / 32

• Rand-Matrix-Equiv(A, B , C )

x {0, 1}n

y = B x z = A y t = C x , z = t

A (B x) = C x . .

. (CS ) 6. 5 / 32

• Rand-Matrix-Equiv(A, B , C )

x {0, 1}n

y = B x z = A y t = C x , z = t

A (B x) = C x . .

. (CS ) 6. 5 / 32

• Rand-Matrix-Equiv(A, B , C )

x {0, 1}n

y = B x z = A y t = C x , z = t

A (B x) = C x . .

. (CS ) 6. 5 / 32

• Rand-Matrix-Equiv(A, B , C )

x {0, 1}n

y = B x z = A y t = C x , z = t

A (B x) = C x .

.

. (CS ) 6. 5 / 32

• Rand-Matrix-Equiv(A, B , C )

x {0, 1}n

y = B x z = A y t = C x , z = t

A (B x) = C x . .

. (CS ) 6. 5 / 32

• Rand-Matrix-Equiv O(n2) 1/2. , A B = C .

, A B = C . , , , (A B C ) x = 0. d = (d1, . . . , dn) A B C ( , A B = C )., d1 = 0.

. (CS ) 6. 6 / 32

• Rand-Matrix-Equiv O(n2) 1/2. , A B = C .

, A B = C .

, , , (A B C ) x = 0. d = (d1, . . . , dn) A B C ( , A B = C )., d1 = 0.

. (CS ) 6. 6 / 32

• Rand-Matrix-Equiv O(n2) 1/2. , A B = C .

, A B = C . , , , (A B C ) x = 0.

d = (d1, . . . , dn) A B C ( , A B = C )., d1 = 0.

. (CS ) 6. 6 / 32

• Rand-Matrix-Equiv O(n2) 1/2. , A B = C .

, A B = C . , , , (A B C ) x = 0. d = (d1, . . . , dn) A B C ( , A B = C ).

, d1 = 0.

. (CS ) 6. 6 / 32

• Rand-Matrix-Equiv O(n2) 1/2. , A B = C .

, A B = C . , , , (A B C ) x = 0. d = (d1, . . . , dn) A B C ( , A B = C )., d1 = 0.

. (CS ) 6. 6 / 32

• ()

x n

i=1 dixi . 0 , x1 = h(x2, . . . , xn),

h(x2, . . . , xn) = n

i=2 dixid1

.

, h , 0 1. 1/2 x1 = h(x2, . . . , xn).

. (CS ) 6. 7 / 32

• ()

x

ni=1 dixi .

0 , x1 = h(x2, . . . , xn),

h(x2, . . . , xn) = n

i=2 dixid1

.

, h , 0 1. 1/2 x1 = h(x2, . . . , xn).

. (CS ) 6. 7 / 32

• ()

x

ni=1 dixi .

0 , x1 = h(x2, . . . , xn),

h(x2, . . . , xn) = n

i=2 dixid1

.

, h , 0 1. 1/2 x1 = h(x2, . . . , xn).

. (CS ) 6. 7 / 32

• ()

x

ni=1 dixi .

0 , x1 = h(x2, . . . , xn),

h(x2, . . . , xn) = n

i=2 dixid1

.

, h , 0 1.

1/2 x1 = h(x2, . . . , xn).

. (CS ) 6. 7 / 32

• ()

x

ni=1 dixi .

0 , x1 = h(x2, . . . , xn),

h(x2, . . . , xn) = n

i=2 dixid1

.

, h , 0 1. 1/2 x1 = h(x2, . . . , xn).

. (CS ) 6. 7 / 32

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. (CS ) 6. 8 / 32

• a b, , ( , ). : , . a mod p b mod p. 2 log2 p .

. (CS ) 6. 9 / 32

• a b, , ( , ).

: , . a mod p b mod p. 2 log2 p .

. (CS ) 6. 9 / 32

• a b, , ( , ). : , .

a mod p b mod p. 2 log2 p .

. (CS ) 6. 9 / 32

• a b, , ( , ). : , . a mod p b mod p.

2 log2 p .

. (CS ) 6. 9 / 32

• a b, , ( , ). : , . a mod p b mod p. 2 log2 p .

. (CS ) 6. 9 / 32

• (prime distribution function)(n) , n.

:

limn

(n)n/ ln n

= 1.

. (CS ) 6. 10 / 32

• (prime distribution function)(n) , n.

:

limn

(n)n/ ln n

= 1.

. (CS ) 6. 10 / 32

• Rand-String-Equiv(a, b)

|a| = |b| = n, N = n2 log2 n2. p [2..N]. , a mod p = b mod p. .

Rand-String-Equiv O(1/n) O(log n) .

. (CS ) 6. 11 / 32

• Rand-String-Equiv(a, b)

|a| = |b| = n, N = n2 log2 n2.

p [2..N]. , a mod p = b mod p. .

Rand-String-Equiv O(1/n) O(log n) .

. (CS ) 6. 11 / 32

• Rand-String-Equiv(a, b)

|a| = |b| = n, N = n2 log2 n2. p [2..N].

, a mod p = b mod p. .

Rand-String-Equiv O(1/n) O(log n) .

. (CS ) 6. 11 / 32

• Rand-String-Equiv(a, b)

|a| = |b| = n, N = n2 log2 n2. p [2..N]. , a mod p = b mod p.

.

Rand-String-Equiv O(1/n) O(log n) .

. (CS ) 6. 11 / 32

• Rand-String-Equiv(a, b)

|a| = |b| = n, N = n2 log2 n2. p [2..N]. , a mod p = b mod p. .

Rand-String-Equiv O(1/n) O(log n) .

. (CS ) 6. 11 / 32

• Rand-String-Equiv(a, b)

|a| = |b| = n, N = n2 log2 n2. p [2..N]. , a mod p = b mod p. .

Rand-String-Equiv O(1/n) O(log n) .

. (CS ) 6. 11 / 32

• :2 log2 p 2 log2 N = 2 log2(n2 log2 n2) = O(log2 n). , a = b, p (a b) 0 mod p. p

|{p P : (a b) 0 mod p}| n.

,

O(

nN/ lnN

)= O

(n(log n2 + log log n2)

n2 log n2

)= O

(1n

).

. (CS ) 6. 12 / 32

• :2 log2 p 2 log2 N = 2 log2(n2 log2 n2) = O(log2 n).

, a = b, p (a b) 0 mod p. p

|{p P : (a b) 0 mod p}| n.

,

O(

nN/ lnN

)= O

(n(log n2 + log log n2)

n2 log n2

)= O

(1n

).

. (CS ) 6. 12 / 32

• :2 log2 p 2 log2 N = 2 log2(n2 log2 n2) = O(log2 n). , a = b, p (a b) 0 mod p.

p

|{p P : (a b) 0 mod p}| n.

,

O(

nN/ lnN

)= O

(n(log n2 + log log n2)

n2 log n2

)= O

(1n

).

. (CS ) 6. 12 / 32

• :2 log2 p 2 log2 N = 2 log2(n2 log2 n2) = O(log2 n). , a = b, p (a b) 0 mod p. p

|{p P : (a b) 0 mod p}| n.

,

O(

nN/ lnN

)= O

(n(log n2 + log log n2)

n2 log n2

)= O

(1n

).

. (CS ) 6. 12 / 32

• :2 log2 p 2 log2 N = 2 log2(n2 log2 n2) = O(log2 n). , a = b, p (a b) 0 mod p. p

|{p P : (a b) 0 mod p}| n.

,

O(

nN/ lnN

)= O

(n(log n2 + log log n2)

n2 log n2

)= O

(1n

).

. (CS ) 6. 12 / 32

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. (CS ) 6. 13 / 32

• f (x1, . . . , xn), f = 0 (f 0 ). , , , , , 0 . . : . 0, , 0. 0, .

. (CS ) 6. 14 / 32

• f (x1, . . . , xn), f = 0 (f 0 ).

, , , , , 0 . . : . 0, , 0. 0, .

. (CS ) 6. 14 / 32

• f (x1, . . . , xn), f = 0 (f 0 ). , , , , , 0 . .

: . 0, , 0. 0, .

. (CS ) 6. 14 / 32

• f (x1, . . . , xn), f = 0 (f 0 ). , , , , , 0 . . : .

0, , 0. 0, .

. (CS ) 6. 14 / 32

• f (x1, . . . , xn), f = 0 (f 0 ). , , , , , 0 . . : . 0, , 0.

0, .

. (CS ) 6. 14 / 32

• f (x1, . . . , xn), f = 0 (f 0 ). , , , , , 0 . . : . 0, , 0. 0, .

. (CS ) 6. 14 / 32

• F , f (x1, . . . , xn) n . f