data structures
DESCRIPTION
Lecture notesTRANSCRIPT
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:
92 3102
' 3102 ' .
mo.liamgnehmot.
1.0
, .
. .
1
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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
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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
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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
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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
lluN=6 tfel.x fi)tfel.x(redrOnI
83
-
2 :4.2 LVA:
llun=6 )x(thgir fi)thgir.x(redrOnI
: redrOnI x, ) (
) ( .
: redrOnI )n(.
3.3.2 lasrevarTredrOtsoP TSB:
lasrevarTredrOtsoP:
:)x(redrOtsoP
lluN=6 tfel.x fi)tfel.x(redrOnI
llun=6 )x(thgir fi)thgir.x(redrOnI
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
-
4 )snoitnuF hsaH(: 2.4 :
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
-
4 )snoitnuF hsaH(: 3.4 :
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
-
4 )snoitnuF hsaH(: 3.4 :
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
-
4 )snoitnuF hsaH(:3.4 :
:
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
-
4 )snoitnuF hsaH(:3.4 :
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 .