Data Structures

:

92 3102

' 3102 ' .

mo.liamgnehmot.

1.0

, .

. .

1

1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 : . . . . . . . . . . . . . . 9

4.1 : . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1 " " : . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 02

1.2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 02

1.1.2 : . . . . . . . . . . . . . . . . . . . . 72

2.2 )TSB(: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

1.2.2 TSB: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03

2.2.2 TSB: . . . . . . . . . . . . . . . . . . . . . . . 03

3.2.2 TSB: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2.2 TSB: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2.2 TSB: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.2.2 TSB: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.2.2 TSB: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 lasrevarT TSB: . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

1.3.2 lasrevarTredrOnI: . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.3.2 lasrevarTredrOerP TSB: . . . . . . . . . . . . . . . . . . . . 83

3.3.2 lasrevarTredrOtsoP TSB: . . . . . . . . . . . . . . . . . . . . 93

4.2 LVA: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

1.4.2 LVA: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.2 LVA: . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.2 LVA : . . . . . . . . . . . . . 34

4.4.2 LVA: . . . . . . . . . . . . . . . . . . . . . 64

5.4.2 LVA : . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.4.2 LVA: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.4.2 tuC LVA: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8.4.2 LVA: . . . . . . . . . . . . . . . . . . . . . . . . . . . 05

9.4.2 LVA-t: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2

1.3 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 )snoitnuF hsaH(: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

1.4 ) (: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 06

1.1.4 gniniahC: . . . . . . . . . . . . . . . . . . . . . . . . . . . 06

2.1.4 )gnihsaH nepO(: . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

1.3.4 : . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.3.4 : . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 07

1.5 troskiuq : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 07

2.5 teleSkiuQ: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.2.5 teleSkiuQ: . . . . . . . . . . . . . . . . 67

2.2.5 9: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6 )paeH-xaM(: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 08

1.6 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 08

1.1.6 : . . . . . . . . . . . . . . . . . . . . . . 18

2.1.6 xaMtartxE: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.6 yeK_esaernI: . . . . . . . . . . . . . . . . . . . . 38

4.1.6 yeK_tresnI: . . . . . . . . . . . . . . . . . . . . . . 58

5.1.6 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.1.6 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.1.6 troSpaeH: . . . . . . . . . . . . . . . . . . . . . . . 98

8.1.6 11: . . . . . . . . . . . . 98

7 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.7 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.7 SFD: . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

1.2.7 tseroF tsriF htpeD : . . . . . . . . . . . . . . . . . . . 89

3.7 SFD: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 001

1.3.7 tseroF-tsriF-htpeD : . . . . . . . . . . . . . . . . . . . 101

2.3.7 : . . . . . . . . . . . . . . . . . . . . 701

3.3.7 : . . . . . . . 011

4.3.7 ) 21(: . . . . . . . . . . . . . . . . . . 311

4.7 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

1.4.7 ) 21(: . 611

3

2.4.7 ) 21(: . . . . . . . . . . . . . . . . . . . . 711

3.4.7 ) 21(: . . . . . . . . . . . . . . . . . . . . . . 811

4.4.7 GAD ) 21(: . . 911

5.7 SFB: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911

6.7 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

1.6.7 artskjiD: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

2.6.7 droF-namlleB ) (: . . . . . . . . . . . . . . . 621

8 dniF noinU: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

9 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

1.9 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

2.9 TSM: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.9 laksurK TSM: . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

4.9 mirP TSM: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

01 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731

1.01 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731

1.1.01 troS gnitnuoC: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731

2.1.01 troS xidaR: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931

3.1.01 troS noitresnI: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931

4.1.01 troStekuB: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 041

5.1.01 troSnoiteleS: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.1.01 : . . . . . . . . . . . . . . . 241

2.01 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

1.2.01 : . . . . . . . . . . . . . . . . . . . . . . . . . . . 741

4

1 :

1 :

1.1 :

)n( n . .

.

1.1 : R +N : g ,f :1. )g( O f 0 > C +N 0N 0N > n )n( g C )n( f.

2. )g( f 0 > C +N 0N 0N > n )n( g C )n( f.3. )g( o f 0 > +N 0N 0N > n |)n( g| < |)n( f|.

. 0 =6 )n( g " 0 n )n(g)n(f

4. )g( f 0 > +N 0N 0N > n |)n( g| > |)n( f|. . 0 =6 )n( g " n )n(g)n(f

5. )g( f )g( O f )g( f, 0 > 2C ,1C +N 0N 0N > n:

)n( g2C )n( f )n( g1C

2.1 :

1. )g( O , :

})n( g C )n( f : 0N > n t.s 0N,C | R +N : f{ = )g( O

2. )g( O = f )g( O f .

3. :

)g( O f f " g . )g( o f f " g.

)g( f f " g . )g( f f g.

)g( f f " g .

3.1 :

1. n = )n( f n = )n( g )g( O f 1 = C 1 = 0N 1 > n )n( g 1 = n3 n = )n( f. )f( O g 3 = g 1 = 0N 1 > n

)n( f 3 = n3 n3 = )n( g.

5

1 : 1.1 :

2. n = )n( f 2n = )n( g )g( O f 1 = C 1 = 0N 1 > n :)n( g = 2n n = )n( f

4.1 :

1. : )g( O f ) ,, ,o( )h( O g ) ,, ,o( )h( O f ) ,, ,o( .2. )g( O f " )f( g.

3. :

)( )g( f.)( )g( O f )f( O g)( )g( f )f( g.

)( )f( g.

4. )g( O f f h )g( O h.5. )g( f f h )g( h.6. )g( o f )g( O f )g( f )g( f.

7. )h + g( f )g( o h )g( f.8. )g( O f ) , ,o( +R c )g( O fc ) , ,o( .

:

1. :

)( )g( O f )h( O g, 0 > 2C ,1C +N 2N ,1N > n}2N ,1N{ xam :

)h( O f = )n( h2C1C )n( g1C )n( f

)( 0 > )g( o f )h( o g +N 2N ,1N }2N ,1N{ xam > n:

)h( o f = |)n( h| = |)n( h| < |)n( g| < |)n( f|

)( )g( f )h( g 0 > 2C ,1C N 2N ,1N }2N ,1N{ xam > n:

)h( f = )n( h2C1C )n( g1C )n( f

)( 0 > )g( f )h( g +N 2N ,1N }2N ,1N{ xam > n:

)h( f = |)n( h| = |)n( h| > |)n( g| > |)n( f|

6

1 :1.1 :

)( )g( f )h( g )g( O f )g( f )h( O g )h( g, )h( O f )h( f )h( f.

2. )g( O f 0 > C +N 0N 0N > n :

1 )n( g = )n( gC )n( fC)f( g = )n( f

)f( g 0 > C +N 0N 0N > n :

1 )n( f = )n( f C )n( gC)g( O f = )n( g

3. )f( O g )g( O f .4. 0 > C N N N > n :

)g( O h = )n( gC )n( f )n( h

5. 0 > C N N N > n :

)g( h = )n( gC )n( f )n( h

6. .

7. )h + g( f 0 > 2C ,1C +N 1N 1N > n :

))n( h + )n( g( 2C )n( f ))n( h + )n( g( 1C

)g( o h 0N 0N > n |)n( g| 21 < |)n( h|. )h( f( }1N ,0N{ xam > n :

1 12

) )n( f )n( g1C

(1 1

2

))n( g2C

)h( f }1N ,0N{ xam > n :8. )g( O f 0 > C +N 0N 0N > n )n( gC )n( f

)n( g cC )n( fc. .

5.1 : n4 = )n( f 2n = )n( g )g( O f )f( g 41 = C N n n = )n4( 41 2n.

7

1 :2.1 :

6.1 n :

:)A(troselbbuB

1-n ot 1=i rof

i-n ot 1=j rof

:neht ]1 + j[ A > ]j[ A fi

)]1 + j[ A , ]j[ A(paws

)n( T n, :

)n( T1n1=i

in + 1

1=j

5

1n =

1=i

5 )1 n( n5 + )1 n( = ))i n( 5 + 1(1n1=i

i

)1 n( n5 )1 n( n5 + )1 n( =2

1 n5.1 2n5.2 = )1 n( n5.2 + )1 n( =

. , 2n )1 +n5.1( . . )2n( T

)2n5.2( o )1 n5.1( )2n( 2n5.2. 7.1 :

. " n ) ... ,3n ,2n ,n(, n n. )2n( O, )ngol n( O, )n( O ) ,(.

)2n( O, )ngol n( O, )n( O.2.1 :

" :

1. : " .

2. : " .

3. : " "

.

:

8.1 : :

1. esaC-tsroW : \ . "

esaC-tsroW.

2. esaC-tseB : \ . "

esaC-tseB.

9.1 esaC-tseB esaC-tsroW.

8

1 :3.1 :

esaC-tsroW :

esaC-tsroW

. ))n( f( f :

1. ) (

))n( f( f .

2.

" )

( ))n( f( O f . ))n( f( ))n( f( . " )n( )ngol(

.

01.1 :

k n n )n( O k )n k( O ) k n )n( O k n )2n( O '(. k

)n k( .

3.1 :

"

. ,

, n. :

1. " )

n n( ) esaC-tsroWoireneS(.

2. "

) ".

:

11.1

)n( .

: A )n( / T, C N n n c < )n( T. 1 = C N 0n 0n < )n( T . 0n 0n . }0n ,... ,1{ j

j j j . A } 0,... ,1{

B A " Bnim =6 ]j[ B, A j

j ]j[ B 1 >

]j[ B B A

Bnim = ]j[

B B

B 1. j

B

B. A j ]j[

9

1 :3.1 :

B 1. A

B

j. ]j[

1 >B .

21.1

)n( .

: A " )n( / T. N 0n 0n < )n( T. 0n }0n ,... ,1{ j . 0n j "

" .

31.1

)n( .

: ", :

:)A(niMdniF

n ,... ,2 = i rof

]1[ A < ]i[ A fi

]1[ A htiw]i[ A paws

]1[ A nruter

: j j ]1[ A < ]j[ A ]j[ A ]1[ A. ]j[ A n i < j ]j[ A > ]i[ A

]1[ A .

: n )1( )n(.

01

1 :4.1 :

4.1 :

, " :

:)]n ,... ,1[ A(troselbbuB

1>n fi

)]n ,... ,1[ A( elbbub

)]1 n ,... ,1[ A(troselbbub elbbuB :

)]n ,... ,1[ A(elbbuB

1-n,...,0=j rof

neht ]1 + j[A > ]j[A fi

)1 + j[A ,]j[A(paws

) ( elbbuB n5 )1 , 1 3 ( ) ( n2. elbbuB )n( h )n( h n5 )n( h n2 " N n. troselbbuB )n( T "

, elbbuB )1-n(troselbbuB :

)n( + )1( = )n( TelbbuB

)1 n( T +)1 n(troselbbub

)n( + )1 n( T =

)1( = )1( T . :

41.1 :

1. )n( O )n( T : T )n( O n )n( T ) T ( )n( T )n( O. )n( O )n( T ) )n( O = )n( T( 0 > C,N N > n

nC )n( T. , .2. )1( )1( T : . T n n )" n ( 1 ) n(.

1 .

3. )n( O+ )1( O = )n( T ) )n( O+ )1( O )n( T( : )n( O+ )1( O, .

. 0 > 2C ,1C N N N > n n2C + 1C )n( T.

4. )n( O + )1 n( T = )n( T : )n( O + )1 n( T, ) )n( O+ f g )n( O+ f h + f )n( O h(. n 1 n n. 0 > C N N N > n nC + )1 n( T )n( T. n )n( T

)1 n( T .

11

1 :4.1 :

O , ' . , " ", "fo esubA

noitaton" , .

.

,

.

:

51.1 :

1. )2n( O = )2n( O+ )n( O )n( O f )2n( O g )2n( O g + f.2. )n( O = )ngol( O+ )n( O )n( O f )ngol( O g )n( O g + f.

.

k1=i

3. k kf ,... ,1f i )n( O if )n( O if

4. )n( O f )ngol( O g )ngol n( O g f.

:

1. g ,f 0 > 2C ,1C N N N > n :2n)2C + 1C( = 2n2C + 2n1C 2n2C +n1C )n( g + )n( f

O g + f 2C + 1C N .(2n)

2. g ,f 0 > 2C ,1C N N N > n :n)2C + 1C( = n2C +n1C ngol 2C +n1C )n( g + )n( f

n ngol N n. )n( O g + f .3. kf ,... ,1f , 0 > kC ,... ,1C N N N > n:

n)kC +... + 1C( == nkC +.... +n1C )n( kf +... + )n( 1f

)2n( O )2n( O.4. g ,f 0 > 2C ,1C N N N > n:

ngol n2C1C = ngol 2C n1C )n( g )n( f

)n( O n , )n( O?

)n( O nf ,... ,1f, :

= )n( T

n1=i

)n( ifn1=i

n= niC

n1=i

iC

21

1 : 4.1 :

)n( O T. n1=i

iC

n. :

n )n( Tn1=i

n iCn1=i

xamni1

xam n = iCni1

iC

n1=i

xam n= 1ni1

xam = n iCni1

niC2

)2n( O )n( O.1.4.1 :

troselbbuB:

)1 n( T + )n( = )n( T 0 > 2C ,1C N n :

)1 n( T +n1C )n( T )1 n( T +n2C 1C )1( T 2C.

61.1 " N n elbbuB )n( 1 = n.

" n ) 1 n, 2 n ( " , :

)2 n( T + )1 n( 1C +n1C )1 n( T +n1C )n( T

1C = 1 1C +... + )1 n( 1C +n1C ...n1=i

1C = i)1 +n( n

2

2)1+n(n 2C )n( T, N n : )n(2T

2C2+ 2n

2C2 )n( T n

)n(1T1C2+ 2n

1C2n

2T N n )n( 1T )n( T )n( 2T (2n)O 1T

(2n)

)2n( O T )2n( T, )2n( T. = )n( T )n( O if.

n1=i

)n( if

2.4.1 :

:

71.1

troselbbuB )n( O T.

81.1 " )2n( T )2n( T. )n( O T n1C

0 > 2C ,1C N N N > n n2C )n( T 2 " .

31

1 : 4.1 :

: : 1 = n )1( O )1( T ) (.: )1 n( O )1 n( T )n( O )n( T ) (.

: :

)n( T

esuba erom)n( T = )n( O+ )1 n( T =

esuba erom)n( O = )n( T = )n( O+ )1 n( O =

)n( O = )n( O + )1 n( O ) ( )n( O f )1 n( O g 0 > 2C ,1C N N N > n :

n)2C + 1C( 2C n)2C + 1C( = )1 n( 2C +n1C )n( )g + f( )n( O )1 n( O )n( O.

)n( O T, . +)1 n( O = )n( O+)1 n( T)n( O, "

.

91.1 O . . )n( O T

0 > C n nC )n( T. )1( O )1( T 0 > C C )1( T .

)n( O+ )1 n( T = )n( T C n nC + )1 n( T )n( T.

xam = C )n( O T .{C

C ,}

: C C )1( T. : )1 n( C )1 n( T. : C " n:

nC2 C nC2 = nC + )1 n( C nC + )1 n( C nC + )1 n( T )n( T

nC2 )n( T nC )n( T ) (. .

.

02.1

troSelbbuB )2n( T, .: 0 > C N n 2nC )n( T.

)1( )1( T 0 > C C )1( T. )n( + )1 n( T = )n( T C n nC + )1 n( T )n( T.

}C , C{xam = C. )2n( O T :41

1 : 4.1 :

21 C 21 C )1( T. 2)1 n( C )1 n( T :

2nC )1 +n1 2n( C = nC + 2)1 n( C nC + 2)1 n( C nC + )1 n( T )n( T C " 2nC )n( T

0 > D N n )n( T 2nD.

)1( )1( T D D )1( T. )n( + )1 n( T = )n( T D nD+ )1 n( T )n( T.

}D , D{nim 21 = D. )2n( )n( T : 21 D 21D )1( T.

2)1 n( D )1 n( T :

+ 2nD = nD+ 2)1 n( D nD+ )1 n( T )n( T( 0

2nD D+n)D2 D

)2n( O T )2n( T )2n( T.

12.1 :

:)]n ,... ,1[ A(gnihtemoSoD

2 > n fi

)A(troSelbbuB

3n q)]q ,... ,1[ A(gnihtemoSoD

)]n ,.... ,1 + q[A(gnihtemoSoD

" :

T = )n( Tn(3

)T +

(n2

3

)+

(2n)

)1( = )2( T = )1( T. , 0 > C n :

T )n( Tn(3

)T +

(n2

3

) 2nC +

(T

(1

3n 3

)T +

(2

3n 3

)C +

n(3

)2)

+

(T

(1

3

n2

3

)T +

(2

3n2 3

)C +

(n2

3

)2)T 4 2nC +

(n4

9

)C +

(+ 2n

2n5

9

)

Tk2 ... ((

2

3

k)n

)C +

(+ 2n

k1=i

(5

9

i)2n

)

51

1 :4.1 :

T. :((

23

k)n)2 1 =

k

k33 gol 3 < n

2 k n

C

(+ 2n

k1=i

(5

9

i)2n

)2nC

(+ 1

1=i

(5

9

)i)2nC =

(+ 1

5

4

)C =

2n

)2n( O T, 0 > D n :T )n( T

n(3

)T +

(n2

3

) 2nD+

(T

(1

3n 3

)T +

(2

3n 3

)D+

n(3

)2)

+

(T

(1

3

n2

3

)T +

(2

3n2 3

)D+

(n2

3

)2)T 2nD+

n(9

)+ 2nD+

2n

9

T ... n (k3

)D+ 2nD+

2n

9D+... +

n

k9

T k ":((

23

k)n) k n 31 gol 3 < k3n 1 =

nD+ 2nD+ 1 )n( T2

9D+... +

n

k92nD

)2n( T " )2n( T. " :

T = )n( T 0 > 2C ,1C N n :(n3

)T +

(n23

)+

(2n)

Tn(3

)T +

(n2

3

)n2C +

T )n( T 2n(3

)T +

(n2

3

)n1C +

2

)1( )3( T 0 > 2D ,1D 12D )3( T 22D. }1D ,1C{ xam9 = 1c }2D ,2C{ nim = 2c )2n( )n( T.

: 3 = n

D < 2c212c < 12D )3( T 2

k2c, n, :: n < k 2k1c )k( T 2

T )n( Tn(3

)T +

(n2

3

)n1C +

2

HI1c

n(3

2)1c +

(n2

3

2)+1c9= 2n

6

9n1c

2nc < 2

1C9 1c 91c 1C. :

T )n( Tn(3

)T +

(n2

3

)n2C +

2c 2n(3

2)2c +

(n2

3

2)n2c +

= 2

(5

91 +

)n2c

n2c > 22

2n1c )n( T 2n2c )2n( T .

61

1 : 5.1 " " :

5.1 " " :

22.1 troSegreM A n:

:)A(troSegreM

:od 1 > n fi

A(trosegreM[ 2n ,... ,1

])

A(trosegreM)]n ,... ,1 + 2n[

A(egreM[ 2n ,... ,1

]A,)n,]n ,... ,1 + 2n[

egreM :

:)]m+ k ,... ,1[ C , ]k ,... ,1[ B , ]m,... ,1[ A(egreM

.1=pC , 1=pB , 1=pA

:od m+ k pC elihW:)od 1 + k < pB dna 1 +m = pA ro ]pA[ .A < ]pB[ B( elihW

]pB[ B ]pC[ CpC tnemernI

pB tnemernI

:)od 1 +m < pA dna 1 + k = pB ro ]pA[ .B < ]pB[ A( elihW

]pA[ A ]pC[ CpC tnemernI

pA tnemernI

]m+ k ,... ,1[ C nruteR

:

1. .

2. n 2gol 1 . egreM

.

egreM: egreM m,k m+k.

m+ k C )m+ k(. m+ k.

troSegreM: n egreM )n( . trosegreM )n( T :

T2 = )n( Tn(2

))n(+

71

1 : 5.1 " " :

)1( = )1( T.

. 0 > 2C ,1C n :

T2n(2

)T2 )n( T n1C +

n(2

)n2C +

trosegrem . ]n ,... ,1[ :

[2n ,... ,1

],[nn ,... ,1 + 2

][

n... ,14],[n4n ,.... ,1 +

2],

[n22 +n3 ,... ,1 +

4],

[2 +n3

4n ,1 +

]

(nk

) k ) egrem(

1C.nT k

(nk

0 2C ,1C k kn 2C ) k k :

1Ck = n1Cn

kTk

n(k

)n2Ck

kn2C =

1 )n( 2gol :

n)n( 2gol 1C

)n( T)n(2gol1=k

Tkn(k

)n)n( 2gol 2C

)ngol n( T.

: " n" a .

(kn) bn.

:

T a = )n( Tn(b

)+

(kn)

32.1 )(:

Ta = )n( T )1( = )1( T 0 > k ,b ,a b ,a(nb

) +

(kn)

q = q :kb

.

1. 1 = q )ngol kn( T.2. 1 < q )kn( T.

3. 1 > q )a bgoln( T.C,C :

: )kn( +) bn( ta = )n( T 0 T a

n(b

)T a )n( T knC +

n(b

)C +

kn

m

((

nmb

)k) mbn ma

m :

Ca (kb

m)C ma = kn

n (mb

k)Tma

n (mb

) C ma

n (mb

k)C =

a (kb

m)kn

81

: " " 1.5: 1

: 1 logb (n)

C

logb(n)m=1

( abk

)mnk T (n)

logb(n)m=1

amt( nbm

)C

logb(n)m=1

( abk

)mnk:

: T (nk log (n)) abk

= 1 .1

logb(n)m=1

( abk

)m=

logb(n)m=1

1 = logb (n)

T (nklogb (n)) Clogb (n)nk T (n) C

logb (n)nk

: logb (n)nk (nk logn

)

logb (n)nk logb (n)nk =1

log (b)log (n)nk = logb (n)nk O

(nk logn

)

: T (nk) 0 < abk< 1 .2

Cnk =(Ca

bk

)nk =

{C

1m=1

( abk

)m}nk

C

logb(n)m=1

( abk

)mnk T (n)C

logb(n)m=1

( abk

)m nk {C

m=1

( abk

)m}nk =

(C

1 ( abk ))nk = Cnk

.C, C > 0

: T (nlogb a) abk> 1 .3

T (n) C

logb(n)m=1

( abk

)mnk = C((

abk

)logb(n)+1 1(abk

) 1)nk =

C (C(

abk

) 1)(( a

bk

)logb(n)+1 1)nk C

( abk

)logb(n)+1nk = C

alogb(n)+1

bklogb(n)+1nk C a

logb(n)+1

bk logb(n)+1nk = C

alogb(n) abk logb(n) bn

k

=

(Ca

b

)alogb(n)

blogb(nk)nk =

(Ca

b

)nlogb(a)

nknk =

(Ca

b

)nlogb(a) = Cnlogb(a)

.Cnlogb(a) T (n) C

19

2 :

2 :

1.2 epyT ataD tartsbA )TDA( :

, :

1. .

2. OFIL pop,hsup '.

3. OFIF eueuqed,eueuqne '.

:

)k(hraeS.

)k(tresnI. )k(eteleD.

. )rosse

uS( )rosseederP( .

.

.

1.2 :

2.2 : )E,V( = G V )" n = | V|( E )" m = |E|(. .

3.2 E V V )2v ,1v( )1v ,2v( ) "" ( .

4.2 )htaP(: )E,V( = G V }rv ,... ,1v{ }i r ,.. ,1{ i E )1iv ,iv(.

5.2 1v rv.

6.2 : )E,V( = G , V v E )w ,v( V w =6 v.

7.2 v ) v(.

8.2 : )E,V( = G }rv ,... ,1v{ 1 > r , rv = 1v rv = 1v ) (.

9.2 , 1 > r rv = 1v.

01.2 : )E,V( = G .

02

2 :1.2 :

11.2 : )E,V( = G V 2v ,1v 1v 2v.

21.2 : )E,V( = G , V C G .

31.2 C .

41.2 : )E,V( = G .

51.2 ) (:

)E,V( = G n m , :

1. 1 n < m G mn .2. G 1 n m.

3. G 3 n m .4. G 1 n = m ) 1 n (.

:

1. m, 0 = m 0 n n . G

1 m m. e G )}e{ \E,V( = 1 m 1+mn , G

:

e G G 1 +mn . e G G mn .

mn G, .2. G 1 mn 1 n m.

3. n = m . n. 3 = n = m "" . 1 n n n

. :

G 1: G

G 3 1 n 1 n G.

2 V v . 2

. v .

4. , 2 1 n m. 2 ,1 = n 1 ,0 . 3 n 3 n < m ) (. n < m 1 n 1 n = m,

.

12

2 :1.2 :

61.2

)E,V( = G " 1 | V| = |E|.

: = 1 | V| = |E| .= 1 | V| = |E| . " G

. G 2 | V| | V| | V|

1 | V| .

71.2 )eerT detooR(: )E,V( = G .

81.2 .

.

.

91.2 :

)E,V( = G )( .

: 0v V =6 0v. 0v . } ,rv ,... ,1v ,0v{ = m } ,su ,... ,1u ,0v{ = m. mm w , 0v = w w .

j ,i ju = iv = w, :

w = iv 7 1+iv 7 ... 7 1rv 7 7 ... 7 1+ju 7 ju = w

G .

02.2

.

12.2 : )E,V( = G , :

1. : .

2. : ) 1(.

3. : V v V w .4. : .

5. : ) 0(.

22

2 :1.2 :

22.2 : )E,V( = G :

1. : .

2. : )

(.

3. :

.

4. : 2 0 .

32.2 :

1. .

2. .

42.2 h )(:

h h2 .

: h:

: 0 = h 1 = 02.

: 1 h h.: h 1 h. 1h2.

h2 = 1h2 + 1h2, .

52.2 ) (:

T h, :

1. k k2 h2 .

2. 1 h2.3. 1 1+h2.

4. h2 h.

:

1. k:

: 0 = k 1 = 02.

: 1 k k.: 1 k 1k2 . k 1 k 1 k 2 . k

k2 = 1k2 2, .

32

2 :1.2 :

2. h:

: 0 = n , 0 = 1 02.: h < n 1 +n.

: 1 +n n n. n 1 n2. n n2

. 1 1+n2 = 1 n2 2 = n2 + 1 n2.3. 1 1+h2 = 1 h2 + h2.

4. 1 d d2

) T(thgieh2. h2 h, .

62.2 )(:

h " h2.

: ) T( l T.

= h h2.= T h h2 , h .

: T 0 = h .

: 1 h h.: T h h2 = ) T( l. :

1v. " T 1 h. l.(T

)l 1 h 1h2

(T

)T h2 =

T

rv ,lv rT ,lT 1 h. )lT( l + )rT( l = ) T( l = h2 1h2 )rT( L , )lT( l.

2 < )rT( L, : 1h

2 = )lT( l1h2 = )1 2( 1h2 = 1h2 h2 > )rT( L h

2 = )lT( l. 1h2 )lT( l 1h2 = )rT( l, 1h

rT ,lT 1h2 . 1 h ) lT )lT(htped2 1h2 1 h )lT( htped(. rT ,lT

1 h 1h2 T .

72.2 : d d .

82.2 2.

42

2 : 1.2 :

92.2 )(:

1. d h hd .

2. d n n dgol.

:

1. h:

: 0 = h 1 = 0d.

: d 1 h.: d h d 1 h.

1hd. hd = 1hd d, .2. .

k kd ) (. ) h( hd , hd n

)n( dgol h, .

03.2 : )E,V( = T .

13.2 6:

) T( D T, :

1. pi |pi| = ) T( D pi.2. .

:

1. , :

)1(

)2( )3(

)4( )5(

)6( )8( )9( )01(

01 8 5 3 4 6 9 .2. ) ( :

52

2 :1.2 :

:)x(maiD

:llun=x fi

)1 ,1( nruter)tfel.x( maiD )l_thgieh,r_maid()thgir.x( maiD )r_thgieh,r_maid(1+}r_thgieh,l_thgieh{ xam = a}2 +r_thgieh+l_thgieh,r_maid,l_maid{ xam = b)b ,a( nruter

:

23.2

T x, :

1. 1 +}))thgir.x( T( h , ))tfel.x( T( h{ xam = ) T( h.2. }2 + ))thgir.x( T( h + ))tfel.x( T( h , ))thgir.x( T( D , ))tfel.x( T( D{ xam = ) T( D.

: 1.

1. : 1 = n 0 :

0 = 1 +}1,1{ xam = ) T( h

: n < k n. T n x. . "

.

1 +))thgir.x( T( h = ) T( h .

2. : 1 = n 0 :

0 = }2 + )1( + )1( ,1,1{ xam = ) T( D

: n < k n. T n x, :

" ".

)thgir.T( D = ) T( D .

thgir.x, )thgir.x ,x(, )tfel.x ,x( tfel.x. ))thgir.x( T( H

))tfel.x( T( H :

2 + ))thgir.x( T( h + ))tfel.x( T( h = ) T( D

62

2 : 1.2 :

n :

: 0 = n )1,1( . n < k n.

:

. .

1

. ,

.

: )1( )n( ) n (

1.1.2 :

)n( , )ngol n( .

33.2 : A X )" ( ]j[ X < ]i[ X si

j ,i . :

)jX < iX(on

sey

)kX < jX( )....(...

...

" X. X ) (

.

43.2

.

53.2

))n( gol n( .

:

1.

. h )ngol n( h.2.

n !n . n !n.

72

2 :1.2 :

: L , !n L h2 L. h2 !n, h )!n( 2gol. :

= )!n( 2gol hn1=i

> )i( 2gol

n 2n=i

)i( 2gol

n 2n=i

2gol

n(2

)

=n

22gol

n(2

)=n

2))n( gol n( )1 )n( 2gol(

2n i n )i( 2gol2gol. )1 )n( 2gol( 2n h N n .

(n2

)

63.2 )ngol n( )!n( gol, , :

= )!n( goln1=i

< )i( goln1=i

)n( gol n = )n( gol

.

73.2

)ngol( .

83.2 "", " " " ".

:

.

y : x )x ( :

y < x y < x.

y > x y > x.

y = x y = x.

. n K n K n . )n( h n, "

".

h2 )n(h3. )n(h3 n )n( 3gol )n( h N n ))n( 3gol( h. )ngol( )n( 3gol ))n( gol( h, .

82

2 : 2.2 )TSB(:

93.2

)n( gol

:

n )n( h . " )

( .

n , )n(h2 , )n(h2 n )n( 2gol )n( h N n. )ngol( )n( 2gol ))n( gol( h, .

04.2 " )n( .

2.2 )TSB(:

14.2 : )E,V( = T toor. T V i :

tnerap.i ) (. tfel.i thgir.i ) \ (.

)i( yek . :

1. V y V x )x( yek )y( yek.2. V y V x )x( yek < )y( yek .

24.2 .

34.2 : )E,V( = T TSB T x :1. )x( T x.

2. )tfel.x( T x.

3. )thgir.x( T x.

44.2 T T. T x x .

54.2 :

)71(

)003( )1( )053( )21( )21(

)22(

92

2 : 2.2 )TSB(:

21 71. :

)71(

)003( )1( )053( )02( )21(

)22(

1.2.2 TSB:

k x:

:)k,x(hraeSeerT

"dnuof ton" nruter llun = x fi

x nruter k = )x( yek fi esle

esle

)x( yek < k fi

)k,tfel.x(hraeSeerT nruter

esle

)k,thgir.x(hraeSeerT nruter

: esaC tsroW )h( h

. , )h(:

)k,x(hraeSeerTevitaretI

k =6 )x( yek dna llun =6 x elihw)x(yek < k fi

tfel.x xesle

thgir.xx.x nruter

2.2.2 TSB:

x :

:)x(niMeerT

llun =6 tfel.x elihwtfel.xx

x nruter

03

2 :2.2 )TSB(:

:)x(xaMeerT

llun =6 thgir.x elihwthgir.xx

x nruter

: )h( x \ " .

3.2.2 TSB:

:

64.2 :

x )x( s ) ( .

1. x .

2. x )x( s x.

3. )x( s )( x x x.

74.2 V y V x y x .

: )2(: )x( T x. )x( T x x ) (, ))thgir.x( T( nim = w.

)x( s = w )x( T / z )w( yek < )z( yek < )x( yek. 84.2 :

T TSB T x T z ))x( T( xam )z( yek ))x( T( nim " )x( T z.

: )x( T / z ) toor.T =6 x( ))x( T( xam )z( yek ))x( T( nim. z toor. T x toor. T ) (, y. x =6 y )x( T / z x = y )x( T z. y

x :

1. )y( yek < )x( yek x y )x( T y )y( yek. y z

y y :

)z( yek )y( yek < ))x( T ni gnihtyna( yek ))x( T( xam > )z( yek .

2. )x( yek < )y( yek x y )x( T y )y( yek. y z

y y :

))x( T ni gnihtyna( yek < )y( yek )z( yek ))x( T( nim < )z( yek .

13

2 :2.2 )TSB(:

)x( T z ))x( T( xam )z( yek ))x( T( nim.

z )w( yek < )z( yek < )x( yek )x( T w :))x( T( xam )w( yek < )z( yek < )x( yek ))x( T( nim

)x( T z )z( yek < )x( yek )thgir.x( T z < )z( yek)w( yek ))thgir.x( T( nim = w, z ".

)3(: w, x w . x w w ) )w( yek < )x( yek( ) ( x x . w w x w,x ) w( "

.

:

:)x(

uS

llun =6 thgir.x fi)thgir.x(niMeerT nruter

esle

tnerap.x=tnerap

)x = thgir.tnerap dna llun =6 tnerap( elihwtnerap xtnerap.x tnerap

tnerap nruter

:

.

)h( = )h2(.

4.2.2 TSB:

:

:)x(derP

llun =6 tfel.x fi)tfel.x(xaMeerT nruter

esle

tnerap.x=tnerap

)x = tfel.tnerap dna llun =6 tnerap( elihwtnerapx

23

2 :2.2 )TSB(:

tnerap.xtneraptnerap nruter

":

94.2 ::

T TSB T x T z ))x( T( xam )z( yek ))x( T( nim " )x( T z. 05.2 :

x )x( p ) ( .

1. x .

2. x )x( p x.

3. )x( p ) ( x x x.

: )2(: )x( T x. )x( T x x, )tfel.x( xam = w. )x( p = w )x( T / z )x( yek < )z( yek < )w( yek "

z )x( yek < )z( yek < )w( yek )x( T w :

))x( T( xam )x( yek < )z( yek < )w( yek ))x( T( nim

)x( T z )x( yek < )z( yek )tfel.x( T z )z( yek < )w( yek ))x( tfel( xam = w, z ".

)3(: w, x w . x w w ) )x( yek < )w( yek( ) ( x x . w w x w,x ) w(

" .

:

.

)h( = )h2(.

5.2.2 TSB:

)x,T(eteled x T, :

1. x .

2. x x tnerap.x. TSB x x

" TSB.

3. x .

33

2 :2.2 )TSB(:

15.2

T T x x )x( s .

: x )x( s )thgir.x( T. ) ( .

25.2 :

35.2

T T x x )x( p .

: x )x( p )tfel.x( T. ) ( .

x :

)x(s 2 ) (. x .

" TSB :

)x(s: )x( s . x: x x )x( s. )x( s

x x )x( s.

: x )x( s )h(, .

6.2.2 TSB:

T k :

:)k,T(tresnIeerT

:lluN=toor.T fi

k=yek.toor.T

: esle

toor.T = x

:yek.x < k fi

:lluN=tfel.x fi

k=yek.tfel.x

:esle

)k ,tfel.x( tresnIeerT

:esle

43

2 : 2.2 )TSB(:

:lluN=thgir.x fi

k=yek.thgir.x

:esle

)k ,thgir.x( tresnIeerT

k x :

1. yek.x < k tfel.x .

2. yek.x > k thgir.x .

TSB k k x.

: h. esaC tsroW k . eerTtresnI tresnI 3 )

(. esaC tsroW )h(.

45.2 : TSB , :

)2(

)4( )1(

)3(

2 1 :

)2(

)4( )1(

)3(

=

)3(

= )4( )1(

)3(

)4(

1 2 :

)2(

)4( )1(

)3(

=

)2(

)4(

)3(

=

)4(

)3(

:

. x y z x y x ) x y ( z y.

53

2 : 2.2 )TSB(:

: " x y . TSB y ) (

TSB. TSB y < x < z :

y fo dlih yreve < x < z fo dlih yreve

)z( T )y( T ))y( T( nim )z( T ))y( T( nim .

: :

y ,x )1(. y )h( ) y (.

z " )1(.

" " )h(.

:

. ,

z.

.

7.2.2 TSB:

A tresnIeerT :

:)]n ,... ,1[ A(TSBdliuB

]1[ A toor.T: n ,... ,2 = i rof

)]i[ A,T(tresnIeerT

tresnIeerT. A ]i[ A ]1 +i[ A tresnIeerT

i )i( 0 > C :

= )n( T

n1=i

C = iC)1 +n( n

2C =

(2n

2+n

2

))2n( O

)2n( O T )2n( T " )2n( T . 55.2

TSB n )ngol n( .

: A TSB n. B n, A TSB )(redrOnI . )(redrOnI B , B . )ngol n(

63

2 : 3.2 lasrevarT TSB:

" B A B " . AT A BT B )ngol n( .

)(redrOnI )n( :

)n( AT + )n( = )n( BT

)ngol n( = )n( BT

0 > C ngol nC )n( BT n . )ngol n( / AT N 0n 0ngol 0nC > )0n( AT, 0n 0ngol 0nC = )n( AT > )0n( BT

)ngol n( BT C .

3.2 lasrevarT TSB:

lasrevarT TSB ,

lasrevarT:

1. .lasrevarTredrOnI .

2. .lasrevarTredrOerP ,

.

3. .lasrevarTredrOtsoP ,

.

1.3.2 lasrevarTredrOnI:

lasrevarTredrOnI :

:)x(redrOnI

lluN=6 tfel.x fi)tfel.x(redrOnI

yek.x tnirp

llun=6 )x(thgir fi)thgir.x(redrOnI

: . 0 = n . n < k n . n x n " . x x n " . x

x " .

: )n( )n ( ) ( )1( .

lasrevarTredrOnI:

:)x(redrOnI

73

2 :3.2 lasrevarT TSB:

)x( niMeerT xllun =6 x elihw

)yek.x(tnirp

)x(

uS x

: .

x x .

: n )h( O n )n( O )2n( O )

(. n )n( .

65.2

" )n(.

: )n( )n( O. , )1( O

. )v ,u( ) u v( :

1. v u )yek.u < yek.v(: u ) v( )v ,u( . u u .

u u )v ,u(.

2. v u )yek.u > yek.v(: u )u(

uS = )thgir.u( niMeerT)v( niMeerT )v ,u( . )v( xaMeerT = m, m )m(

uS u m u ) "( )v ,u( . u

)v ,u(.

)n( O, .

75.2 TSB: lasrevarTredrOnI

. lasrevarTredrOnI

. )2n( O . n lasrevarTredrOnI )n( O

)2n( O.2.3.2 lasrevarTredrOerP TSB:

lasrevarTredrOterP:

:)x(redrOerP

yek.x tnirp


83

2 :4.2 LVA:


: redrOnI x, ) (

) ( .

: redrOnI )n(.

3.3.2 lasrevarTredrOtsoP TSB:

lasrevarTredrOtsoP:

:)x(redrOtsoP



yek.x tnirp

: redrOnI

, x.

: redrOnI )n(.

4.2 LVA:

: TSB ),, '( )h( O. ""

.

85.2 : TSB "" n ))n( gol( O h.

95.2 LVA: TSB LVA x :

1 |))thgir.x( T( h ))tfel.x( T( h|

))x( T( h ) x(

06.2 LVA:

)21(

)61( )8( )41( )01( )4(

)6( )2(

93

2 :4.2 LVA:

16.2 x h.x .

26.2 LVA TSB :

LVA .

: kn LVA k. )kngol( O k LVA n kn n )ngol( O k ngol kngol .

36.2

kn k 2kn + 1kn + 1 = kn.

: LVA T k kn . 1 = k 2 = kn 1 > k ) (. lT rT . 1 k 1kn 1kn 1k 1kn T

. :

1kn > eertbus rehto ni sedon#+ 1kn +1 = kn

" lT 1 k T LVA rT 1 k 2 k ) 1(. T rT 2 k 2kn 2kn,

:

2kn + 1kn + 1 = kn

kn:

2kn + 1kn + 1 = kn

2kn>1kn 2k 2kn 2k 2 > ... > )4kn2( 2 > 2kn2 >

2k :

k2 k2=

{neve si k 0

ddo si k 1

:

2k= 2k 2kn 2

{2

kneve si k 0n 2

2kddo si k 1n 2

=

{2

k 2

neve si k

2k1+ 2

ddo si k

2gol > )kn( 2gol

(2

k1+ 2

) 1 +2k =

k

2k

2)kngol( O k = )kn( 2gol 2 < k = )kn( 2gol < 1 +

04

2 : 4.2 LVA:

46.2

T LVA n k )ngol( O k.

: kn kn n kngol ngol )kngol( O k )ngol( O k.

56.2 n n 2gol )ngol( k " LVA n )ngol(.

kn :

66.2 ' " 0 = 0F, 1 = 1F 2nF + 1nF = nF 1 > n.

76.2

1 = .5

= ) ( 2+15

= nF 2nn

5 N n

(: , 1 +x = 2x 2nx + 1nx = nx :2nx + 1nx = nx = 0 = )1 x 2x( 2nx = 0 = )1 x 2x

, ':

2n + 1n = n

2n + 1n = n

b ,a :

a = nUnb + n

:

a = nU2nU + 1nU = 2nb + 2na + 1nb + 1n

{ b ,a :0 = b +a

1 = b +a

nF = nU " :

a = 0U0 = 0b + 0

a = 1U1 = 1b + 1

:

1 = a = 1 = )a( +a=

1+15

1 25

2

=151 = b =

5

b ,a nF :

= nF15+ n

(1

5

)= n

n n5

14

2 :4.2 LVA:

' LVA. kn LVA k 1 + 2kn + 1kn = kn 0 = 0n 1 = 1n.

1 + kn = km :

2km+ 1km = 1 + 2kn + 1 + 1kn = 1 + 1 + 2kn + 1kn = 1 + kn = km

3F = 1 = 0m 4F = 2 = 1m, 3+kF = km k :

= 1 3+kF = kn3+k 3+k

5

kn :

= kn3+k 3+k

5>3+k

5gol > kn gol =

(3+k

5

)gol )3 + k( =

(5)

+ kn gol < k =(gol

(5)3

)

C=

= C + kn gol < k =kngolgol

C + kngol 44.1 C +

)kngol( O k, .

1.4.2 LVA:

T .

2.4.2 LVA:

: LVA TSB . LVA "

. "" .

: TSB .

LVA.

86.2 :

T x :

1. thgir.x tfel.x x .

2. )x( T x.

3. Rx Lx x .

24

2 : 4.2 LVA:

96.2 LVA: a LVA a )L( a :

)a(

)b( \La/ \Rb/ \Lb/

)L(a=

)b(

\Rb/ )a(

\Lb/ \La/

a )R( a :

)a(

\Ra/ )b(

\Rb/ \Lb/

)R(a=

)b(

)a( \Lb/

\Ra/ \Rb/

07.2 " )1(.

17.2 .

3.4.2 LVA :

27.2 LL LVA: LL a T a a L)tfel.a(

) tfel.a = b(, " :

)a(

)Ra( )b(

1h)Rb( h)Lb(

h LB :

1. a )b( T LVA .

2. )b( T LVA Lb h Rb 1 +h ,1 h ,h. a Lb Rb 1 + h h ) Rb(. Rb

1 h )b( T 1 +h.3. )b( T = La 1 + h Ra 1 + h ,1 h ,h )a( T LVA 1. Ra 1+h " h

Ra 1 h.

34

2 : 4.2 LVA:

" LL :

2+h)a(

1h h\Ra/ 1+h)b(

1h\Rb/ h\Lb/

37.2 LL:

LL a )R( a .

: a:

2+h)a(

1h\Ra/ 1+h)b(

1h\Rb/ h\Lb/

)R(a=

1+h)b(

h)a( h\Lb/

1h\Ra/ 1h\Rb/ , TSB " :

1. b a b a a b .

2. Lb b .

3. Rb b a .

4. Ra a .

5. Ra a b b.

LL.

47.2 RR LVA : RR a a a R)thgir.a( )

thgir.a = b(, " :

)a(

)b( \La/

h\Rb/ 1h\Lb/

h Rb Lb La 1 h . 57.2 RR:

RR a )L( a .

44

2 : 4.2 LVA:

: a:

2+h)a(

1+h)B( 1h\La/

h\Rb/ 1h\Lb/

)L(a=

1+h)b(

h\Rb/ h)a(

1h\Lb/ 1h\La/ LVA TSB :

1. b a b a a b .

2. Rb b .

3. Lb b a .

4. La a .

5. La a b b.

67.2 RL LVA:

RL a a a R)tfel.a( ) tfel.a = b(, " :

)a(

h\Ra/ 2+h)b(

1+h\Rb/ h\Lb/ a Rb.

77.2 RL :

RL A )L( B )R( A .

: Rb " . "

Rb. )L( b:

3+h)a(

h\Ra/ 2+h)b(

1+h)c( h\Lb/

h\Rc/ 1h\Lc/

)L(b=

3+h)A(

h\Ra/ 2+h)c(

h\Rc/ 1+h)b(

1h\Lc/ h\Lb/

54

2 : 4.2 LVA:

)R( a :

3+h)A(

h\Ra/ 2+h)c(

h\Rc/ 1+h)b(

1h\Lc/ h\Lb/

)R(a=

2+h)c(

1+h)a( 1+h)b(

h\Ra/ h\Rc/ 1h\Lc/ h\Lb/

, TSB " :

1. c a b a < c < b a c b c .

2. Ra a c Ra ni gnihtyreve < a < c

3. Rc c a .

4. Lc c b .

5. Lb c b c < b < Lb ni gnihtyreve.

87.2 LR LVA:

LR a a a L)thgir.a( ) thgir.a = b(, " :

h)a(

2+h)b( h\La/

h\Rb/ 1+h\Lb/ A Lb.

97.2 Lb Lb " .

08.2 LR :

LR A )R( b )L( a .

: .

4.4.2 LVA:

18.2 RR :

RR a )L( a .

64

2 : 4.2 LVA:

: La ) a( 1+h , " La h . LVA Ra 1 + h 2 + h h. 1 +h h Ra 2 +h. RR thgir.a = b Rb 1 +h LVA Lb h 1 + h )2 + h Ra(

)L( a :

3+h)a(

2+h)b( h\La/

1+h\Rb/ 1+h/h\Lb/

)L(a=

2+h)b(

1+h\Rb/ 2+h/1+h)a(

1+h/h\Lb/ h\La/ Rb 1 + h, Lb h 1 + h La h

A 1 +h 2 +h " .

28.2 RR :

38.2 LR,RL,LL

.

5.4.2 LVA :

n LVA. LVA .

)ngol n( :

48.2

n LVA )ngol n(.

: 2n LVA 2n n. .

. :

: 0 = n 1 = n .

: n 1 +n . n . 1 +n ,

:

1. LVA

.

2. LVA RR

. " ) (

.

)ngol( " )ngol n(.

n2

" )ngol n(.

74

2 :4.2 LVA:

.

. :

:)]dne ,... ,trats[ A(LVAdliuB

1+dne-trats NLIN nruter 0 < N fi

21dne + 1 r)]r[ A( edoNwen eert:1 > N fi

)]1-r,...,trats[ A( LVAdliuB tfel. ]r[ A)]dne...,1+r[ A( LVAdliuB thgir. ]r[ A

eert nruter

:

1. LIN

2. r .

3. ".

4. , .

5.

.

: TSB

. LVA,

n.

: 1 = n LVA .

: n < k n. n LVA. n 21n LVA . n . H rH ,lH

n 2

n 1 2

. rH lH 1 + rH = H. 1 + lH rH LVA.

58.2 7 . .

: ) N, r, edoN ( " )n( n .

84

2 :4.2 LVA:

6.4.2 LVA:

LVA 2T ,1T . LVA T . .

n " )ngol n( ngol LVA n )ngol n(.

: in = |iT| }in{ xam = m.

1. lasrevarTredrOnI

2A ,1A " )m(.

2. )m( .

3. LVA m2 " )m( .

" )m( " .

7.4.2 tuC LVA:

LVA . )k ,T( tuC TSB . , k ,

k :

1. 1T k.

2. 2T k.

3. k.

tuC LVA LVA: LVA, tuC

LVA. n , :

k , LVA n " )ngol(. .

A lasrevarTredrOnI, " )n(.

k )ngol( i. LVAdliuB ]1 i ,... ,1[ A 1T )n(. LVAdliuB ]n ,... ,1 +i[ A 2T )n(.

k ,2T ,1T .

" .

tuC TSB : LVA,

tuC LVA. n , :

94

2 :4.2 LVA:

k )n gol(, . k , n )ngol( )1(

)ngol( . TSB k ) x( .

)tfel.x( T, )thgir.x( T x.

" )ngol(.

8.4.2 LVA:

k :

LVA n , k .

1: )ngol( O ) ( 1 k . k 1 k . )n( O

n )n( O. 2: lasrevarTredrOnI k .

redrOnI .

3: )ngol( O : " . LVA

x mun.x x ) x(. 1 +mun.thgir.x +mun.tfel.x = mun.x " l 0 = mun.thgir.l = mun.tfel.l 1 = mun.l. . mun

\ . n )ngol( O.

k :

:)k,x(tsellamSKdniF

:k == mun.thgir.x-mun.x fi

x nruter

:k mun.tfel.x fi)k ,tfel.x(tsellamSKdniF nruter

:esle

)1 mun.tfel.x k ,thgir.x(tsellamSKdniF nruter

k " .

: )ngol( O LVA.

: .

: 1 = n 1 = n k 1 1 = k r k = 1 = 0 1 = mun.thgir.r mun.r .

: n n :

05

2 : 4.2 LVA:

mun.thgir.r mun.r = k 1 mun.thgir.r mun.r x x. x r

k .

k mun.tfel.r n k " k )r

( .

k < mun.tfel.r ) n ( y "1 mun.tfel.r k" ". mun.tfel.r y r y 1 mun.tfel.r k

k .

9.4.2 LVA-t:

68.2 LVA-t: TSB LVA-t x :

t |))thgir.x( T( h ))tfel.x( T( h|

))x( T( h ) x(

78.2

T LVA-t k n , :

1. k,tn LVA-t k k,tn k.

2. t ))k,tn( gol( O k3. t )ngol( O k.

:

1. x T LVA-t k k,tn ) 0 = k 1 = 0,tn 1 = k 2 = 1,tn(. T k 1 k T " 1k,tn. T LVA-t )1 +t( k ) ))thgir.x( T( h ))tfel.x( T( h t " 1 k )1 +t( k(. T )1 +t( k "

)1+t(k,tn. :

1k,tn > 1 + )1+t(k,tn + 1k,tn = k,tn

1 .

2. k ,t :

2 > )1+t(k,tn2 > )1+t(k,tn + 1k,tn > k,tn()1+t(2k,tn + 1)1+t(k,tn

)2 > ... > )1+t(2k,tn4 >

k k k,tn 1+t

)1+t( 1+t

:

k k1 +t

= )1 +t( {neve si k 0

ddo si k 1

15

2 :4.2 LVA:

1 = 0,tn 2 = 1,tn :

2k

k k,tn 1+t= )1+t( 1+t

{2

kneve si k 0,tn 1+t

2k

ddo si k 1,tn 1+t=

{2

k 1+t

neve si k

2k

1+ 1+tddo si k

2 > k,tn k 1+ 1+t

k

1 +tk

1 +t))k,tn( gol( O )k,tn( 2gol )1 +t( < k = )k,tn( 2gol < 1 +

3. T LVA-t k n , k,tn n k,tn < )k,tn( golngol. ))k,tn( gol( O k )ngol( O k, .

25

3 :

3 :

1.3 : X X )X( P X2 :

}X A| A{ = )X( P

)X( P X. 2.3 :

X |X|2 = |)X( P|.

3.3 X , X "

.

: :

4.3

X : X}1 ,0{ )X( P, X}1 ,0{ }1 ,0{ X : f.

: " )X( P X}1 ,0{. )X( P B, }1 ,0{ X : BX B X 1 = )a( BX B a 0 = )a( BX . X a

}1 ,0{ )a( BX X}1 ,0{ BX. A}1 ,0{ )X( P : BX = )B( . ": )X( P 2B ,1B 2B =6 1B " 1B 1b 2B / 1b :

)2B( =6 )1B( = )1b( )2B( = )1b( 2BX = 0 =6 1 = )1b( 1BX = )1b( )1B(

: X}1 ,0{ f, X 0X 0X a 1 = )a( f, 0XX = f f = )0X( , .

:

= |)X( P||X|2 = |X||}1 ,0{| = X}1 ,0{

: :

: X X 12 = )X( P.: X 1 n X n . :

= )}nx ,... ,1x{( P C

}}1nx ,... ,1x{ P A| }nx{ A{ )}1nx ,... ,1x{( P , )}nx ,... ,1x{( P B :

1. B / nx )}1nx ,... ,1x{( P B.

35

3 :

B = B.2. B nx }nx{ \B = B }1nx ,... ,1x{ P B }nx{

C )}nx ,... ,1x{( P, C A C }nx ,... ,1x{ )}nx ,... ,1x{( P A . " 1n2 = |)}nx ,... ,1x{( P| }}1nx ,... ,1x{ P A| }nx{ A{ 1n2

n2 = 1n2 + 1n2 = |)}nx ,... ,1x{( P|, .

5.3 : )P, F,( :

1. .

2. F F )( P.3. P R F : P :

= )( P.

1 = )( P

F A ]1 ,0[ )A( P. F B,A = B A )B( P + )A( P = )B A( P.

6.3 , )P,( .

= )A( P .

A2 = F A )( P

= )A( P.A

7.3 : F A )( P

8.3 F .

9.3 : F B,A = B A, 0 = )B A( P.

01.3 B,A )B( P + )A( P = )B A( P.

11.3 : )P, F,( F A , A A\ = cA ) (.

21.3 : )P, F,( F B,A 0 =6 )B( P, A B )B( P)BA( P = )B|A( P.

31.3 ) (:

)P, F,( F B,A , :

1. : )A( P 1 = )cA( P.

2. : )B A( P )B( P + )A( P = )B A( P.

45

3 :1.3 :

3. : )A( P )A|B( P = )B( P )B|A( P = )B A( P.4. F B,A B A )B( P )A( P.

41.3 ) (:

)P, F,( F A . }nB ,... ,2B ,1B{ ) ( :

= )A( Pn1=i

= )iB A( Pn1=i

)iB( P )iB|A( P

51.3 : )P, F,( ||1 = )( P.

1.3 :

61.3 : )P, F,( R : X. 71.3 " ) R (

.

81.3 :: X )P, F,( X :

= ] X[ E = A

)( P)( X

91.3 " :

X )P, F,( :

= ] X[ E

)X(mIx]x = X[ P x

.

: :

)X(mIx

= ]x = X[ P x

)X(mIx= )}x = )( X| {( P x

)X(mIx

x )x(1X

)( P

=

)X(mIx

)x(1X

)( P x1=

)X(mIx

)x(1X

)( P )( X2=

)X( E = )}{( P )( X

:

1. " )x( 1X x = )( X .

55

3 :1.3 :

.

)X(mIx

)x(1X

}{2. =

02.3 1 0005001 0005 0000001 0000001 0005, }0005001 ,... ,1{ = }0000001 ,0005{ : X

0005 = X 0005 0000001 000001, :

= )0005 = X( P0005 nrae taht sdlohesuoh fo rebmun

= ||0000001

0005001

= )0000001 = X( P0000001 nrae taht sdlohesuoh fo rebmun

= ||0005

0005001

:

0000001 0005 = ] X[ E0005001

0005 0000001 +0005001

000 ,01

12.3 ) (:

)P, F,( R : Y,X , :1. R b ,a b + ] X[ Ea = ]b + Xa[ E )(.2. R c c = ]c[ E ) (.

3. ] Y[ E + ] X[ E = ] Y + X[ E.

22.3 ) (:

X )P, F,( }nB ,...2B ,1B{ :

= ] X[ E

n1=i

)iA( P ]iA| X[ E

]A| X[ E X F A:

= ]A| X[ EA

= )( P)( X

)A|X(mIx)x = X( Px

32.3

.

42.3 :

X 0 > a :

] X[ E )a X( Pa

=

a

65

3 :1.3 :

1 > :

1 ) X( P

: :

= ] X[ E =

a

4 )snoitnuF hsaH(:

4 )snoitnuF hsaH(:

)TDA( hraeS,eteleD,tresnI.

.

.

)1( O.

1.4 N ) 8 (. )1( O 801 . TSB ))N( gol( O.

)1( O .

"" ) (

)1( O .

2.4 ) '( U " " .

3.4 : h U .

4.4 " N U m N . U N .

N .

5.4 :

1. : k m domk = )k( h m . .

2. : k )kA kA( m = )k( h A m .

.

:

1. h ) )1( O(.2. )N( O m.

3. .

6.4 )1( O U . "

.

7.4 : h U y ,x )y( h = )x( h.

85

4 )snoitnuF hsaH(:

8.4

U N m m N |U|. U .

U N h }1 m,... ,0{ U : h U

: h U ". U iU i = )k( h )}1 m,... ,0{ i h (. i

N < |iU| :

= |U|1m0=i

< |iU|1m0=i

m N= N

U i N |iU|. N iU h i h.

) (:

. U }1 m,... ,0{ U : 2h ,1h .

U . 9.4

U N m 2m N |U|. U .

U N 1h 2h }1 m,... ,0{ U : 2h ,1h U

: }1 m,... ,0{ U : 2h ,1h. mN = N m N = 2mN |U|. 1h

U U N mN = N = U . " 2h.

U N 2h .

mN UU " 1h 2h " .

01.4 : }1 m,... ,0{ U : h U j }1 m,... ,0{ i m1 = ]i = )j( h[ P.

11.4 m1 .

21.4 : }1 m,... ,0{ U : h N m h mN = .

31.4 " .

41.4 :

}1 m,... ,0{ U : h . }1 m,... ,0{ i N i mN = .

95

4 )snoitnuF hsaH(:1.4 ) (:

: j,i1 " j i" ) N j 1

= iW :

N1=j

m i 1(. i j,i1

E = ]iW[ E

N 1=j

j,i1

N =

1=j

= ] j,i1[ E

m1=j

= )1 = j,i1( P

N1=j

1

m=N

m

51.4 h .

1.4 ) (:

1.1.4 gniniahC:

}1 m,... ,0{ U : h , .

, gniniahC.

)1( O )1( O .

61.4 N gniniahC N )1( O ) h( " )N( O.

71.4 gniniahC :

h gniniahC ) + 1(.

: k )k( h, " )k(hn. )1( :

1. k :

E + )1( = ]emiT hraeS[ E[)k(hn

]+ )1( =

N

m

(+ 1

N

m

)E .

[)k(hn

] =

2. )k(hn . k

:

)k(hn1=i

= i ]i noitaol ni si k[ P)k(hn1=i

1

)k(hn1 = i

)k(hn

)k(hn(1 )k(hn

)2

=1 )k(hn

2

:

E + )1( = ]emiT hraeS[ E

[1 )k(hn

2

]+ )1( =

1

2E[)k(hn

1 ]2

+ )1( =1

21

2) + 1(

06

4 )snoitnuF hsaH(:1.4 ) (:

81.4 h .

2.1.4 )gnihsaH nepO(:

1 . .

h }1 m,... ,0{ }1 m,... ,0{ U : h )i ,k( h 0=i1m})i ,k( h{ }1 m,... ,0{ k.

: k )i ,k( h .

: k )0 ,k( h " )1 ,k( h , .

91.4 gniborP raeniL: h " gniborp raenil, h m domi + )k( h = )1 ,k( h.

r " m1+r ) 1 + r h "(. .

02.4 gniborP itardauQ: gniborP itardauQ,

i2c +i1c +)k( h = )i ,k( h 2 h m dom

m,2c ,1c . sretsul h .

12.4 gnihsaH elbuoD: , :

m dom )k( 2h i + )k( 1h = )i ,k( h

2h ,1h ) (. 2h )k( 2h m k, m 2h m < )k( 2h k m 2 2h

2 dom1 )k( 2h k.

22.4 m m m }1 m,... ,0{.

.

32.4 :

U k }1 m,... ,0{ U : 2h ,1h . d = ))k( 2h,m( dg :

m dom )k( 2h i + )k( 1h = )i ,k( h

)k( 1h.m k d

16

4 )snoitnuF hsaH(: 2.4 :

: d )k( 2h })k( 2h ,.. ,1{ l l d = )k( 2h :

h(,km

d

)=(+ )k( 1h

m

d)k( 2h

)m+ )k( 1h =

ld

dlm+ )k( 1h =

)k( 1h m domm k d

m d

, .

42.4 gniniahC

N }Nk ,... ,1k{ )N( O. )i ,k( h i N i 2 i ik

:

N1=i

= i)1 + N( N

2=

1

2

(N+ 2N

)2N( O )

2.4 :

.

.

52.4 : }}1 m,... ,0{ U : h{ =: H U y =6 x :

= ])y( h = )x( h[ HhP|})y( h = )x( h| H h{|

|H|1

m

H h .

62.4 " U y =6 x m|H| H = )x( h)y( h.

72.4 :

H U m. U U N = | U|, U U x H) |H|1 (. U x x mN +1.

82.4 U " i mN.

: U y ,x y,x1 ")y( h = )x( h" ) h = xW

Uy

(. U x y,x1

26

4 )snoitnuF hsaH(:2.4 :

:

E = ]xW[ E

Uy

y,x1

=

Uy= ]y,x1[ E

Uy

]1 = y,x1[ P

1=+]1 = x,x1[ P

Uy=6 x

1

m+ 1 =

1 Nm

N + 1 m

92.4 m H U. U H " .

y =6 x :

1 ])y( h = )x( h[ HhP = ]1 = y,x1[ Pm

03.4 :

H U m. U U N = | U|, U U x H.

U k )mN + 1( . 13.4 gniniahC.

: k h )k( h, )k( h )k(hn.

)1( :

1. k :

E + )1( = ]emiT hraeS[ E[)k(hn

]+ )1( =

N

m

(+ 1

N

m

)E .

[)k(hn

] =

2. )k(hn . k

:

)k(hn1=i

= i ]i noitaol ni si k[ P)k(hn1=i

1

)k(hn1 = i

)k(hn

)k(hn(1 )k(hn

)2

=1 )k(hn

2

:

E + )1( = ]emiT hraeS[ E

[1 )k(hn

2

]+ )1( =

1

2E[)k(hn

1 ]2

+ )1( =1

21

2) + 1(

36


23.4 k N N )1(.

33.4 :

)2Z( lrM =: H l2 |U| r2 m l < r. H M U k k b]k[ l. b]k[ M r

H k 01 " 2. 43.4

H " . : U y =6 x , y x = z, l 1 = jz ) y =6 x( :

]|0 = zM[ HMP = ]yM= xM[ HMP = ])y( h = )x( h[ HhP

z l )2Z( lrM M

= zM i 0 = z

l1=i

0 = zM. iM i zM iziM

" :

= zM

1=iz | i+ jM= iM

1=iz | i=6 j

iM

:

P = ]0 = zM[ P

+ jM

1=iz | i=6 jiM

0 =

P =

= jM

1=iz | i=6 j

iM

] = jM[ P =

r r21 ) " r2(. :

1 = ]|0 = zM[ HMP = ])y( h = )x( h[ HhPr2

=1

m

53.4 :

U m p p |U|. b ,a :

m dom)p domb + ka( = )k( b,ah

:

}pZ b , pZ a | b,ah{ = H . U 2k =6 1k p |U| pZ 2k ,1k. pZ a pZ b )b + ka( = ir. 2k =6 1k

2r =6 1r pZ :p dom0 6 )2k 1k( a )b + 2ka( )b + 1ka( 2r 1r

46


pZ 0 )2k 1k( a 0 a 0 )2k 1k( " . )1 p( p pZ a pZ b 2r ,1r 2k =6 1k. H b,ah U 2k =6 1k m dom2r 1r. 1r b ,a m dom2r 1r. pZ m dom ) p < m(

m, pZ 1r mp pZ 2r m dom2r 1r :

= |pZ||}m dom2r 1r | PZ 2r{| = ]m dom2r 1r[ Pmp p

pm

p=

1

m

2k =6 1k H b,ah m1 ])2k( b,ah = )1k( b,ah[ P, .

3.4 :

""

) (. "

)

(.

" ", .

"

.

1.3.4 :

2N m 2N2, :

1. H h ) " )1( O(.2. )N( O m )N( O m .

3. .

63.4 ]2N2 ,2N[ m m 2 ) " 2( m

2 .

73.4 :

H h 21 < ]dab si h[ P h "" y =6 x )y( h = )x( h.

83.4 21 > ]doog si h[ P.

: N ) 2N( . H y ,x m|H| h .

( m|H| ) 2N( H . :N

2

)|H| m

=)1 N( N

2

|H|m

C NC < ]doog si1h| )N( T[ E. 1h ,

:

1 + NC ])N( T[ E2]dab si1h| )N( T[ E

1h H 2h , :1

21 = ]dab si1h| )N( T[ E

2)]dab si2h[ P]dab si 2h| )N( T[ E + ]doog si2h[ P]doog si 2h| )N( T[ E(

1 21 + ]doog si2h| )N( T[ E

41 ]dab si2h| )N( T[ E

2+ NC

1

4]dab si2h| )N( T[ E

" :

1 + NC ])N( T[ E2+ NC

1

4]dab si2h| )N( T[ E

N M :

+ NC ])N( T[ E1M1=k

1

k2+ NC

1

M2]dab si mh| )N( T[ E

M :1

M20 M ]dab si mh| )N( T[ E

:

NC + NC ])N( T[ E1=k

1

k2)N( O N C3 = NC2 + NC = )1( O+

04.4

.

66


2.3.4 :

)N( m :

1. H h H ) " )1( O(.

2. 1m j 0 jn j ) j = )k( h(.n2. jh

23. 1 m j 0 j2n jm j

".

4. .

14.4 :

1.

" . |H| N "" )N( m 2

(2N) h )N( m

) (

.

2. m jn j2n jm ) (, .

.

24.4 :

N.

: h

E. j jm j2n21m +m

0=j

jm

1m +m, )N( O

0=j

jm

:

E

1m +m

0=j

jm

=

m=E+ ]m[ E

1m0=j

jm

E +m

1m0=j

j2n2

E2 +m =

1m0=j

j2n

E )N( O m :1m0=j

j2n

)N( O

E

1m +m

0=j

jm

E2 +m

1m0=j

j2n

)N( O =

y,x1 )y( h = )x( h, :

x

y

= edillo taht sriap deredro fo rebmun#= y,x11m0=j

lle ht'j eht ni edillo taht sriap deredro fo rebmun#

=

1m0=j

= )yek emas eht fo sriap gnidulni( lle ht'j eht ni sriap fo rebmun#

n1=j

j2n

76


:

E

1m0=j

j2n

E =

[x

y

y,x1

]E =

x

y=6 x

y,x1

E +

[x

y=x

y,x1

]

x

y=6 x

+ ]y,x1[ Ex

1== ]x,x1[ E

x

y=6 x

N+ ]1 = y,x1[ P

=x

y=6 x

N+ ])y( h = )x( h[ P

+ N

x

y=6 x

1

m+ N =

1

m

x

y=6 x

+ N = 1)1 N( N

m

NmN2

h .

E .

1m0=j

j2n

)N( O

:

1. h N m > N2.

h , j

1m0=j

E2 > j2n

1m0=j

j2n

2.

j2n jm.

.

1m0=j

E N 4 > j2n1m0=j

j2n

3.

:

P = ]dab si h[ P

1m0=j

E2 > j2n

1m0=j

j2n

E

1m0=j

j2n

E2

1m0=j

j2n

=1

2

" " :

P = ]dab si h[ P

1m0=j

N4 > j2n

E

1m0=j

j2n

N4

N2

N4=

1

2

h

E. 21 h

1m0=j

j2n

N2

.

E.

1m0=j

j2n

N2 =

86


34.4

H . m < jm )

( H m. 44.4 :

N.

: :

n 1mi0

21. h N4 i

m , in h }1 m,... ,0{ i N .

, :

h , h )1( O N )N( O.

) gniniah( hsah, )1( O )N( O ".

, " N )N( O.

)1( O N4 )1( O " )N( O.

2. )n( t ) h ( )n( T.

" , :

E = ])N( T[ E

[1m0=i

)in( t

]=

1m0=i

])in( t[ E

)n( O 0 > C 1 m i 0 inC ])in( t[ E :

E = ])N( T[ E

[1m0=i

)in( t

]=

1m0=i

])in( t[ E1m0=i

C = inC

1m0=i

NC = in

)N( O ] T[ E .3. 21 h . X < p1 = ] X[ E.

112

" 1 p < 21 2 = )1( O.

" )N( O .

96

5 :

5 :

1.5 troskiuq :

troskiuq n. . :

1. tovip .

2. tovip

tovip )" noititrap(.

3. noititrap .

" :

:)]r ,l[ A(troskiuQ

:r < l fi

)]r ,l[ A(noititrap m)]1 m,l[ A(troskiuQ)]r ,1 +m[ A(troskiuQ

thgiR,tfeL :

:)]r ,l[ A(noititraP

])r ,l( modnar[ A tovip.l r ezis fo yarra ytpme na etaer // B tini: r ,... ,l = j rof

neht tovip < ]j[ A fi

1 + l l , ]j[ A ]l[ Bneht tovip > ]j[ A fi esle

1 r r , ]j[ A ]r[ B]tovip[ A ]l[ BB Al nruter

:

1. noititraP ]r ,l[ A ) (.

2. toviP " .

3. B l r tovip tovip .

toviP toviP.

4. toviP.

5. noititraP toviP .

07

5 : 1.5 troskiuq :

1.5 toviP

. B p toviP tovip = ]p[ B.

: noititraP toviP ) (

.

)]1 m,l[ A(troskiuQ )]r ,1 +m[ A(troskiuQ noititraP .

: noititraP )n( ) ( ) (. m toviP

)n m( troskiuq :)1 +mn( T + )1 m( T + )n( = )n( T

:

]| )1 +mn( T + )1 m( T[ E + ])n( [ E = ])1 +mn( T + )1 m( T + )n( [ E = ])n( T[ E

+ )n( =

n1=k

)]k = m[ P ]k = m| )1 +mn( T + )1 m( T[ E(

+ )n( =

n1=k

1

n= ])1 + k n( T + )1 k( T[ E

n1=k

1

n)])1 + k n( T[ E + ])1 k( T[ E(

}n ,... ,1{ k n1 = ]k = m[ P k. :

n1=k

1

n= )])1 + k n( T[ E + ])1 k( T[ E(

n1=k

1

n+ ])1 k( T[ E

n1=k

1

n])1 + k n( T[ E

=1n0=k

1

n+ ])k( T[ E

1n0=k

1

n2 = ])k( T[ E

1n0=k

1

n])k( T[ E

, :

2 = ])n( T[ E

1n0=k

1

n)n( + ])k( T[ E

2C ,1C )n( 2C )n( n1C N n, N n:

2

1n0=m

1

n2 ])n( T[ E n1C + ])m( T[ E

1n0=m

1

nn2C + ])m( T[ E

:

2 = )n( CU1n0=m

1

nnC + ))m( CU(

N n :)n( 1CU ])n( T[ E )n( 2CU

17

5 : 1.5 troskiuq :

)n( CU , n :

2 = )n( CUn

1n0=m

nC + )m( CU2

)1 +n( CU 1 +n :

2 = )1 +n( CU)1 +n(

n0=m

)1 +n( C + )m( CU2

:

C +nC2 + )n( CU2 = )n( CUn )1 +n( CU)1 +n(

C +nC2 + )n( CU)2 +n( = )1 +n( CU)1 +n(

= )1 +n( CU

2 +n

1 +n+ )n( CU

C +nC2

1 +n

= C)1 +n( C

1 +n=C +nC

1 +nC +nC2

1 +nC2 +nC2

1 +n=

)1 +n( C2

1 +nC2 =

:

2 +n )1 +n( CU1 +n

C2 + )n( CU

" :

2 +n )1 +n( CU1 +n

2 +n C2 + )n( CU1 +n

(1 +n

nC2 + )1 n( CU

)C2 +

2 +n 1 +n

1 +n

n

((n

C2 + )2 n( CU1 n)C2 +

)C2 +

...

2 +n 1

C2+)0( CU 0=

2 +n +...

n+ C2

2 +n

1 +nC2 )2 +n( = C2 + C2

2+n1=j

1

j

) ( ,

:

= )n( H

n1=j

1

j1 =

0S

+1

2+

1

31S

+1

4+

1

5+

1

6+

1

72S

+...+1

n

1 k2 , k, 11+k2

1k2

k :

1

2k2 =

1

1+k2

seulav k2sahkS1

k2+

1

1 + k2+... +

1

1 1+k21

k2+... +

1

k2k2 =

1

k21 =

27

5 :1.5 troskiuq :

2 = n )1 +n( 2gol = k : 1 k

1

2= )1 +n( 2gol

1

2 k

)n(Hk1=i

)1 +n( 2gol = k kS

:

)ngol( = ))1 +n( 2gol( )n( H

2.5 :

)n( gol

2 gol= )1 +n( 2gol )n( 2gol =

)1 +n( gol

2 gol)n2( gol

2 gol=

ngol + 2 gol

2 gol+ 1 =

)n( gol

2 gol

":

1

2 gol+ 1 )1 +n( 2gol )n( gol

)n( gol

2 gol

" :

C2 )2 +n( )1 +n( CU2+n1=j

1

j)n + 2( H C2 )2 +n( =

))n( gol n( = ))2 +n( gol )2 +n((

3.5 )ngol( )n( H 0 > 2c ,1c :

ngol 2c )n + 2( H ngol 1c

)2 +n( gol )2 +n( )2c C2( )n + 2( H C2 )2 +n( )2 +n( gol )2 +n( )1c C2(

))2 +n( gol )2 +n(( )n + 2( H C2 )2 +n(

N n :

))n( gol n( 2 gol 2 = )n( gol + 2 gol n2 = )n2( gol n2 )2 +n( gol )2 +n( ngol n

)ngol n( )2 +n( gol )2 +n( )ngol n( )1 +n( CU.

:

)n( 1CU ])n( T[ E )n( 2CU

))n( gol n( )n( 2CU , )n( 1CU ))n( gol n( ] T[ E.

37

5 : 2.5 teleSkiuQ:

4.5 )ngol( nH x1 = )x( f N n :

= )n( gol

n

1

1

x xd

)n(Hn1=j

1

j+ )1( f

n

1

1

x)n( gol + 1 = xd

N n )n( gol + 1 )n( H )n( gol ))n( gol( H.

5.5 esaC-tsroW troskiuQ :

toviP

1 ) (.

n )2n( O.1=i

= i)1 +n( n

2

troskiuQ: noititraP toviP

]thgir[ A snoititraP . troskiuQ

(2n)

noititraP :

:)]r ,l[ A(noititraP

]r[ A tovip1 l i:od 1 r ,... ,l = j rof

neht tovip ]j[ A fi]j[ A ]i[ Aegnahxe1 +i i

]r[ A ]1 +i[ Aegnahxe1 +i nruter

.(2n) )n( troskiuQ

2.5 teleSkiuQ:

A n n k 1 k ) 1 k (. 1 = k n = k . ) ( k ,

" )ngol n( troskiuQ.

teleSkiuQ:

noititraP noititraPR , teleSkiuQ

:

:)k , ]n ,.... ,1[ A(teleSkiuQ

esa siht ni 1 = k // ]1[ A nruter 1 = n fi

:esle

47

5 : 2.5 teleSkiuQ:

)]n ,... ,1[ A( noititraPR r:k = r fi

]k[ A nruter

:r < k fi

)k , ]1 r ,... ,1[ A(teleSkiuQ nruter:esle

)1 + r k , ]n ,1 + r[ A(teleSkiuQ nruter

: . , n < k n, :

1. k = r r noititraPR noititraPR 1 r ,... ,1 ]r[ A n ,... ,1+r ]r[ A ]r[ A

k .

2. r < k k ]1 r ,... ,1[ A, n )k , ]1 r ,... ,1[ A(teleSkiuQ

k " k .

3. k > r ]n ,... ,1 + r[ A ) ( ]r ,... ,1[ A k 1+rk ]n ,1 + r[ A

)1 + r k , ]n ,1 + r[ A(teleSkiuQ.

: ":

toviP noititraPR toviP 43 . n n 4 ) ( tovip " 4n

. 2n ,n34

tovip :

stovip doog#

stovip elbissop fo rebmun#=

n2

n=

1

2

)n( T . 21 tovip3 n4

1 tovip . 2

1 2

n43 . tovip 1 2

n. )n( t noititraPR )n( tovip " :

])n( t[ E + ]toviP dab[ P ]tovip dab | )n( T[ E + ]toviP dooG[ P ]tovip doog | )n( T[ E = ])n( T[ E

=1

21 + ]tovip doog | )n( T[ E

21 )n( t + ]tovip dab | )n( T[ E

2E

[T

(3

4n

])+

1

2)n( t + ])n( T[ E

1 2E

[T

(3

4n

])+

1

2nC + ])n( T[ E

)n( t C ".

57

5 :2.5 teleSkiuQ:

:

1 ])n( T[ E2E

[T

(3

4n

])+

1

21 = nC + ])n( T[ E

21 ])n( T[ E

2E

[T

(3

4n

])nC +

E ])n( T[ E =[T

(3

4n

])E nC2 +

[T

((3

4

2)n

])3 C2 +

4nC2 +n

E ... [T

((3

4

k)n

])nC2 +

1k0=j

(3

4

j)E

[T

((3

4

k)n

])nC2 +

0=j

(3

4

j)

E =

[T

((3

4

k)n

])C2 +

1

43 1E =

[T

((3

4

k)n

])nC8 +

k " :(34

k) n k 1 = n

)n( O nC8 +tsnoC = C8 + ])1( T[ E ])n( T[ E

" .

1.2.5 teleSkiuQ:

teleSkiuQ

noititraP . .

, )ngol n( )n( 2gol toviP ". teleSkiuQ noititraP )n(

n 2gol )n( )ngol n(.

6.5 A A i ) ( . ) ( .

.

teleSkiuQ :

1. A n 5n 5 n 5 5 domn.

n5

2. troSnoitresnI

) (,

5nm,... ,1m.

q.[ 5nm,... ,1m

]3.

4. noititraP q.

5. m ) ]q ,... ,1[ A(. m k k ]q ,... ,1[ A m > k

)m k( ]n ,... ,q[ A.

: .

: )n(.

67

5 : 2.5 teleSkiuQ:

7.5

A q 6 01n3 .

nA ,... ,1A 5: n 5 5n 5

) [5nm,... ,1m

] 5nm,... ,1m . q

q (. 1 5n 21 q " 3 q. q 3 q. n 5

(11 5n 2

) 1 01n3 = 2 +

21 5 q ) n5

2 5 domn q( 3 x "

:

3

(1

2

n5

2

)=

3

2

n5

3 6

2

n

5n3 = 6

016

q. " 1 3 x . "" ""

6 01n3 q .

)n( T n, :

5n )n( O . 5 )1( O )

.n5

(. )n( O = )1( OT.(

n5

)

n5

noititrap-deidom ) 5n( O )n( O. tovip k tovip kn . 6 + 01n7 = )6 01n3 ( n

T.(n76 + 01

)" n :

T )n( Tn(

5

)T +

(n7

016 +

))n( O+

)n( O T:: 1 = n .

: n < k, n.

: 0 > 1C :

T )n( Tn(

5

)T +

(n7

016 +

)n1C +

77

5 : 2.5 teleSkiuQ:

0 > C :n5

n < 6 + 01n7 n 2

n 2

". " 2n

.

2n teleS .

)n( O )n( O " )n( O.

2n )1( O , " )n( O.

6: A n ]1 ,0[ ) A x ]1 ,0[ )b ,a( a b = )]b ,a[ x( P(. " )n( O : " }n ,... ,1{ k k A"

)n( O. : A :

n = m xm = )x( h. ) k )mk , m1k[ m k 1.

)mk , m1k[ x )k ,1 k[ mx k = mx = )x( h(. )k( tnuo )m k 1( ".

87

5 :2.5 teleSkiuQ:

n )n( O. : A n 0 1 ]1 ,0[. A A )A( mX ,... )A( 1X iX A h k ) A )mi , m1i [(, m k 1

:

= ])A( kX[ E

n1=i

= }]) mk , m1k[)i(A[{In1=i

P

[ )i( A

[1 km

,k

m

])tsiD mrofinU

=

n1=i

(k

m1 k

m

)

n =

(k

m1 k

m

)=

n

m=

n

nnn=n

( O. )n

l A l , :

nim = )l( j

{| }m,... ,1{ j

1j1=i

l )i( tnuo1j1=i

l > )i( tnuo

}

l )l( j . teles )l( j . l

= m )l( j. teles

( l

1j1=i

)i( tnuo

)

)n( O )l( j )n( O. )l( j )n( O n = m.

" )n( O, .

97

6 )paeH-xaM(:

6 )paeH-xaM(:

1.6 :

TDA " " :

1. )A( xaM .

2. )A( xaMtartxE .

3. )x ,A( tresnI x.

4. )yek ,x ,A( esaernI x yek.

TSB TDA " " .

paeH-xaM TDA " .

1.6 )paeH-xaM(: paeH-xaM "

", . "

" "

.

2.6 :

1. )1(.

2. " "

.

3.6 :

. .

.

4.6 ) (:

1. .

2. h h2 1 1+h2 .3. n n 2gol.

4. .

5. .

:

T. x

1. T T x T

. x T T T ) (.

T T )

T .

T (

08

6 )paeH-xaM(:1.6 :

2. 1h ) 1h ( 1 h 1 h2, h h " h2 . , h2

1 1+h2 = h2 + 1 h2 .3. h :

1+h2 < 1 1+h2 n h2

1 +h < n 2gol h n 2gol h < 1 n 2gol h n 2gol = h.H x

4. x H, . 1

H H

H . "

" .

5. . 0 = h . h 1 + h. 1 +h a .

h. a a

, .

1.1.6 :

. .

,

:

= A[] )5(lav )4(lav )3(lav )2(lav )1(lav

) (:

)1(

)3( )2( )7( )6( )5( )4(

i :

1. i i2 = tfel.i.

2. i 1 +i2 = thgir.i.

3. i 2i = tnerap.i.4. )A( ezispaeH.

18

6 )paeH-xaM(:1.6 :

5.6 13 i.

"

.

. , A ]i[ A )A i( i .

6.6 , .

7.6 :

.

yfipaeH-xaM yfipaeH-niM

yfipaeH-xaM.

2.1.6 xaMtartxE:

)A( xaMtartxE . )A( ezispaeH .

) (. yfipaeH-xaM:

:)i ,A(yfipaeHxaM

i2 = tfel.i L1 +i2 = thgir.i Ri tsegraL:neht ]tsegraL[ A > ]L[ A dna )A(ezispaeh L fi

L tsegraL:neht ]tsegraL[ A > ]R[ A dna )A(ezispaeh R fi

R tsegraL:neht i =6 tsegraL fi

]tsegral[ A ]i[ A paws.)tsegraL,A(yfipaeH_xaM

:

8.6 yfipaeHxaM:

)i ,A(yfipaeHxaM i ) 1+i2 ,i2 ( i .

: h = i .

: 0 = h .

: h 1 +h.

: 1 +h, :

1. i i yfipaeHxaM .

28

6 )paeH-xaM(:1.6 :

2. i i =6 tsegraL, " L = tsegral. yfipaeHxaM ]i[ A ]L[ A )L,A(yfipaeHxaM ) L , yfipaeHxaM i(. L h " . R . i )

yfipaeHxaM (, :

]L[ A R R " .

]L[ A ]i[ A > ]L[ A )L,A(yfipaeHxaM i .

.

yfipaeHxaM: )ngol( ) )A(ezispaeH= n( " )n( 2gol.

xaMtartxE :

:)A(xaMtartxE

]1[ A m])A( ezispaeH[ A ]1[ A)A(ezispaeH tnemered

)1 ,A(yfipaeHxaM

.m nruter

: A )A(xaMtartxE . .

yfipaeHxaM

8.6 yfipaeHxaM

: )ngol( ) )A(ezispaeH= n( xaMtartxE yfipaeHxaM .

3.1.6 yeK_esaernI:

:

: )yek,i ,A(yeKesaernI

"rorre" nruter ]i[ A i elihw

]tnerap.i[ A ]i[ A paws)i( tnerap i

38

6 )paeH-xaM(:1.6 :

,

.

:

i. " i. A

. :

,

.

" . .

]i[ A :

1. 1 = i ) ( .

2. ]tnerap.i[ A ]i[ A, .

" .

9.6

yeKesaernI .

: }h ,... ,0{ d .: 0 = d

.

: d 1 +d.

: )yek ,i ,A(yeKesaernI i 1 +d, :

1. ]tnerap.i[ A ]i[ A: )i( tnerap tnerap.i. i ]i[ A . i yeKesaernI i . A

.

2. ]tnerap.i[ A > ]i[ A, ]tnerap.i[ A ]i[ A. i ]tnerap.i[ A ]i[ A . )yek ,tnerap.i ,A(yeKesarnI " )i( tnerap yek ) ]i[ A ]tnerap.i[ A( )yek ,i ,A( yeKesaernI. )i( tnerap d )yek ,tnerap.i ,A(yeKesarnI "

)yek ,i ,A(yeKesarnI .

48

6 )paeH-xaM(: 1.6 :

yeKesaernI .

: )ngol(.

01.6 yeKesaereD :

) (. yfipaeHxaM .

yfipaeHxaM "

.

4.1.6 yeK_tresnI:

:

)yek,A(yeKtresnI

)A(ezispaeH tnemernI

])A( ezispaeH[ A)yek , )A( ezispaeH,A(yeKesaernI

: )

( )

(,

. yeKesaernI " .

yeKesaernI tresnI

yeKesaernI.

: " yeKesaernI )ngol(

5.1.6 :

6.1.6 :

n:

1. " ))n( gol n(.

2. tresnI yeKtresnI.

11.6

)ngol n(.

58

6 )paeH-xaM(:1.6 :

: yeKtresnI n n )ngol( O )ngol n( O. ]n ,... ,1[. i )1 i( 2gol. "

)i gol( . 2n :

)n( Tn

2n=iC > i gol C

n 2n=i

gol(n2)C =

n2

gol

(n2)

ngol n )ngol n( .

:

paeHdliuB n :

:)A(paeHdliuB

)A( htgneL n:1otnwod n = i rof

)i ,A(yfipaeHxaM

.A nruteR

: paeHdliuB .

Data Structures

Documents