# Untitled

Untitled

Ω Θ

Ω Θ

Ω Θ

Ω Θ

CO2

CO2

CO2

NO2

NO2 40 µg/m3

200 µg/m3

100 µg/m 3 0−50 µg/m 3

CO2

• CO2

CO HC NOx

NOx

8, 5◦ 8, 5◦ 8, 5◦ ≥ 8, 5◦

NOx SO2 CO HC CO2

CO2

CO2

CO2

Ω Θ

Ω Θ

f g

C k

|f(x)| ≤ C|g(x)|

f(x) O(g(x)) x > k C k f(x)

O(g(x)

f O(g(x)) g

g f x

g f f

f(n) f(n) n

f(n) = 1 + 2 + ... + n ≤ n + n + ... + n

n

f(n) n n n f(n)

= n 2

n n n 2

k = 1 C = 1

f(n) O(n 2 )

x, y ∈

|x| + |y| ≥ |x + y|

x ≥ 0 y ≥ 0 |x| + |y| = x + y = |x + y|

x ≥ 0 y < 0 |x| + |y| = x + (−y) > x + y = |x + y| y

−y > y

x < 0 y ≥ 0

|x| + |y| = (−x) + y > x + y = |x + y|

x < 0 y < 0 |x| + |y| = (−x) + (−y) = −(x + y) = |x + y|

f(x) = an·x n +an−1·x n−1 +...+a1·x+a0 a0, a1, ..., an−1, an

f(x) O(x n ) O(x n )

f

k = 1 x > 1

|f(x)| = |anx n + an−1x n−1 + ... + a1x + a0|

≤ |an|x n + |an−1|x n−1 + ... + |a1|x + |a0|

= x n

|an| + |an−1|

x

|a1|x |a0|

+ ... + +

xn−1 xn

≤ x n (|an| + |an−1| + ... + |a1| + |a0|)

= Cx n

x

x x

x n

k = 1

C = |an| + |an−1| + ... + |a1|n + |a0|

f(x) = 2 · x 7 − 5 · x 5 + 3 · x 4 − 3 · x + 5

f(x) O(x 7 ) f(x)

f1(x) O(g1(x)) f2(x) O(g2(x)) (f1+f2)(x) O(max(|g1(x)|, |g2(x)|))

f1(x) O(g1(x)) f2(x) O(g2(x)) (f1 · f2)(x) O(g1(x) · g2(x))

Ω Θ

Ω Θ

Ω Θ

f g

C k

|f(x)| ≥ C|g(x)|

f(x) Ω(g(x)) x > k

g C k g f x k

Θ

f g

f(x) O(g(x)) f(x) Ω(g(x)) f(x) Θ(g(x))

f(x) g(x)

f(x) O(g(x)) Ω(g(x))

C1 C2 k1 k2 k1 k2

f(x) Θ(g(x)) f(x) O(g(x)) f(x) Ω(g(x))

Θ

Θ

x > k Θ

Ω Θ

b = 2

n

10−9

2(n) 7 · 10−9

n 10 −7

n · 2(n)) n(n) 7 · 10 −7

n b 10 −5

b n 4 · 10 13

n! 10 100

10 9

10−9

n

b 10000

n ≈ ∞

n < 10

10100

∩ = ∅

n

c1, ..., c4

n+1 i = n i

n

T (n) = 1 · c1 + (n + 1) · c2 + n · c3 + n · c4

= (c2 + c3 + c4) · n + c1 + c2

≤ c · n

O(n)

n · n

n

n · n

T (n) = n · c1 + (n · n) · c2 + (n · n) · c3

= n · c1 + (n 2 )(c2 + c3)

≤ c · n 2

O(n 2 )

G = (V, E) V

E

V = {v1, v2, . . . , vn}

E = {e1, e2, . . . , em}

(vi, vj) vi vj

a

b

a b

a b

v deg(v)

G

v v v deg(v)

v

v

v v

L(vi) vi

vi w(vi, vj) vi vj

G

n

a z

i := 1 n

L(vi) := ∞

L(a) := 0

S := ∅

w(vi, vj) = ∞ w(vi, vi) = 0

z /∈ S

u := S L(u)

S := S ∪ u

v S

L(u) + w(u, v) < L(v)

L(v) := L(u) + w(u, v)

S

S

O(n 2 ) n

G

n n

n − 1

n · (n − 1) + n = n 2 − n + n = n 2

O(n 2 )

w(vi, vj) a h a

L(vi)

a h

a

a a b c

b = 4 c = 3

c

c d e

d 3 + 2 = 5 e 3 + 6 = 9

b b

d d b d

e d

h

h

(a, c, e, g, h)

h

G

n

k v

v ∈ S

k v

S v /∈ S

k = 0 : S = ∅

∞.

k v

S k (k + 1)

v u

v

k + 1

u S (k + 1)

u S v v

u Lk(u) v

Lk(v) + w(u, v) k + 1

Lk+1(u) Lk(u) Lk(v) + w(u, v)

vi vj vh

G n

∞ G

h := 1 n

i := 1 n

j := 1 n

w(vi, vh) + w(vh, vj) < w(vi, vj)

w(vi, vj) := w(vi, vh) + w(vh, vj)

k vi vj i ≤

k ∧ j ≤ k

k = 0 :

vi vj

k w(vi, vj) vi vj

(k + 1) n

k + 1 vk+1

vi vj

vi vk+1 vk+1 vj

w(vi, vj) w(vi, vj)

w(vi, vk+1) + w(vk+1, vj)

w(v1, vj)

k + 1

O(n 3 ) n

G

n

n · n · n = n 3

O(n 3 )

∞ w(vi, vj)

i j i j

0 2 ∞

4

⎢ 2

⎣ ∞

0

2

2

0

5 ⎥

3 ⎦

4 5 3 0

0 2 4

4

⎢ 2

⎣ 4

0

2

2

0

5 ⎥

3 ⎦

4 5 3 0

O(n3 )

O(n2 )

(n2 − n)/2

O(n2 ) · O(n2 ) = O(n2 · n2 ) = O(n4 )

(162 −16)/2

O(n 2 ) · O(120) =

O(120 · n 2 ) n

G = (V, E) v0, v1, . . . , vn−1, vn,

vi = vj 0 ≤ i < j ≤ n

G

G

G = (V, E)

n > 1

v0, v1, . . . , vn−1, vn, v0

v0, v1, . . . , vn−1, vn

1

T SP NP T SP NP

HK NP

HK NP NP

HK T SP NP

T SP T SP NP

HK T SP

HK T SP

C w(C) ≤ B

B

G = (V, E) v1, v2, . . . , vp

HK

G ′ = (V, E ′ )

G

G ′

1 (vi, vj) ∈ G

w(vivj) =

2

(vi, vj)

vi vj w(vi, vj) B = p p

G G ′

p

2

n = p r = 2

n!

r!(n − r)!

p! p · (p − 1)

= =

2!(p − 2)! 2

p2 − p

2

G G ′ O(p 2 )

G ′ G

C w(C) ≤ B = p G

G ′

G w(C) ≤ B = p ⇔ G ′ w(C ′ ) ≤ B = p

⇒:

C G C

v1, . . . , vp, v1 G ′

p C ′ v1, . . . , vp, v1 G ′

p

⇐:

C ′ G ′ B = p p

1 1

C ′ 1 G

C G p

n

n − 1

O(e · log v)

O(e · log(e)) e v

v v · (v − 1)/2

G

S G

K

S = V

(u, v) G u S u S

K

G n V

O(e·log v) e

v

e O(log(e))

G

v · (v − 1)/2

v · (v − 1)

log

= log(v · (v − 1)) − log(2) = log(v) + log(v − 1) − log(2)

2

O(log(v)) v

O(v · log(v))

K

G n

G

K G T

K

S V \S K S

V \S G

S

(u, v) S V \S

(u, v) K

(u, v) T

(u, v) /∈ T (u, v)

T (u, v) T (u, v)

S V \S S V \S e

T e (u, v) T ′ (u, v) e

(T ) = (e) +

(T ′ ) = ((u, v)) +

(u, v) S V \S

((u, v)) ≤ (e) (T ′ ) ≤ (T )

T T ′

k K

k = 0 : K

K k S K

K (u, v)

k + 1

O(n 2 )

n

G T

(u, v) ∈ T u v

u ∈ T ′ P = (x1, x2, ..., xn) x1 = xn = u

deg(vny) =

2 · deg(vgammel)

u v t u t u v

v t

T G T

T

P

P

T

(T ) < ( ) ⇔ 2 · (T ) < 2 · ( )

T

() = 2 · (T )

() < 2 · ( )

( ) < 2 · ( )

T

T

P a

P = (a, b, c, d, c, b, a)

P = (a, b, c, d)

c a 28

P = NP K

v

K P = NP

K

G = (V, E)

G ′ = (V, E ′ ) G ′

w(e) =

1 e ∈ G

v · p + 1

e G ′ p G ′

O(p 2 )

• G G ′

p

v · p

• G G ′

v·p+1 ≥ p−1+v·p+1 > v·p

G ′

> v · p

P = NP

P = NP

P = NP

P = NP

P = NP K

v

K P = NP

K

G = (V, E)

G ′ = (V, E ′ ) G ′

w(e) =

1 e ∈ G

v · p + 1

e G ′ p G ′

O(p 2 )

• G G ′

p

v · p

• G G ′

v·p+1 ≥ p−1+v·p+1 > v·p

G ′

> v · p

P = NP

P = NP

P = NP

P = NP

1/2

1/2

1/2

≤ 16

O(n2 )

(m2 − m)/2 m

O(n2 · m2 ) m2 m = 7 n = 2000

n

O(m!) m

n

m

O(n) O(n2 )

O(n2 )

O(m)

b a

a b

m2−m 2 O(m2 ·d)

d

for

O(n2 )

while

O(n) for

n

while n

O(n2 ) while

for n

O(n2 )

for

for n for

for

for if

n for

for

for

n

for n

while O(n3 )

O(n)

O(n3 )

O(m2 )

O(m2 · n3 )

n

for

m − 2

(m − 2)!

O(m!)

O(n · a) a

a ≤ n O(n2 )

O(m2 · n3 + m!)

n m

O(m2 · n3 + m!)

for

m − 2

(m − 2)!

O(m!)

O(n · a) a

a ≤ n O(n2 )

O(m2 · n3 + m!)

n m

O(m2 · n3 + m!)

(,, /) =

/ + 2 · 10 10 · · (0, 07 − (/) 2 ) 5

= 50 = 0 200

0, 25

= 0

0 0, 13

0, 39

0 0, 25

> 0, 25 < 0

a c

a =

· (0, 07 − ( )2 ) 5

b =

c = · (0, 07 − ( ) 2 ) 5

d =

+1

e = 100 +

f =

)+

(

100

g =

− ( ) 1,5

a c

a

a c

a c

a c

f

0, 012 − 0, 016 0, 23

e f

/ 0 0, 39

e 100 f

/

d g /

/

a c

1/2

1054, 09+1135, 07+1204, 45−2·679, 01 = 2035, 59

100%

1054, 09+1135, 07+1204, 45−2·679, 01 = 2035, 59

100%

CO2

O(n2 ) O(n3 )

Ω Θ

T SP P = NP

T SP P = NP

n n · (n − 1)/2

n = 1

n = k

k k · (k − 1)/2

k k

k · (k − 1)

2

+ k =

k · (k − 1)

2

k · (k − 1)

2

k2 − k + 2 · k

=

2

(k + 1) · k

n = k + 1

+ 2 · k

+ k

2

= k · (k − 1) + 2 · k

2

2

n n − 1

n = 1

n = k k

k − 1

k + 1

k n = k + 1

n

(n 2 − n)/2

n = 1

n = k

k 2 − k

2

k

k2 − k

+ k

2

k2 − k

2 + k = k2 − k + 2 · k

2

= k2 + k

2

k2 + k

2

= (k + 1)2 − (k + 1)

2

n = k + 1

=

=

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