30
1:
1.5:
T(n)=aT(n-b)+f(n)
T(n)=T(n/a)+T(n/b)+f(n)T(n)=T(n/a)+T(n/b)+f(n)
! " # $ "
!" #$ %&
' ( ( ) ( ' * (+(
, ! # %-
' ( ( ) ( ' * (+(
!" # !$ #% & ' " ($ #%
. ! /" % /$ %-
' ( ( ) 0 *
, ' ( 1) (
. ' ( ( 1 . 2
3 4
. %&" ! ' ! "
!& ! ' ! " () :
%)%(
! &! ! 3.6 * ! (%)+' ! " T(n)=T(n/a)+T(n/b)+f(n)
" (% + , " * ! (%)+' ! " T(n)=T(n-1)+f(n)
%)%( -
!" # !$ #% & ' " ($ #%
%)%( -
" * ! (%)+' ! " T(n)=T(n/a)+T(n/b)+f(n)
" (% + , " * ! (%)+' ! " T(n)=aT(n-b)+c
%)%(
" '% + * $ . * !/ * ! (%)+' ! "
T(n)=T(n/a)+T(n/b)+f(n)
B. ( + *) (/
T(n)=aT(n-b)+f(n)
)!" # !$ #% & ' " ($ #%
H (%)+' ! " " " T(n)=aT(n-b)+f(n) *) (! ( !
(% + , .
(+(! ' ( $ ( " %( %!0 ( " '! " ! " ":
f(n)=c, %&!( % $%!( T(n)=aT(n-b)+c % !() !
+ ! " (% + , " % ' () ( ! % * $ (
..
a=1, % $%!( T(n)=T(n-b)+f(n) % ' +$ (! (
($ + ( %( & ! &% % ' % !(+() % ++ * ! " ( & '
(% + , ".
*( . ! " " * ( a, b, f(n) () ( !&" $+ ".

-. ( + *) (/
1. T(n)=aT(n-b)+c
!" # !$ #% & ' " ($ #%
T(n)=aT(n-b)+c +$ (! ( ! (% + , "
1. 1 ' ( 3 (. * " ! " " " ( .! ' ( !
. )
2. !) ! " ( " % ' % $%!( (! %& k (% + ,( " ( "
( ) ....3...)( +−⋅= bnTnT
( )+−⋅=*() & " )
3. % + *)2 ' ( %&!( ! ! ( ( !/ &% ' n0
' !( ! $ ! " " +$ ' ( /" % " k). . . n0=0
!&!( n/b
4. ! ! ! ! ' k ! & !$% ! ' 3 ! " 2.
5. % + * &" ! ' ) ! " % ' % ',(.
( ) .......)( +−⋅= kbnTnT
0nkbn =−
B. ( + *) (/
1. T(n)=aT(n-b)+c
1. %)+' ( ! (% + , " (- 1: 3 (. * " ! ' & )
*!" # !$ #% & ' " ($ #%
! 1 3 (. &2 ' ( ! & & 3 . " ' ( ! "
% 4( " % ' % $%! ' .
:
+$ (!( ! :
:
( )
=
>+−
=
0,1
0,225
)(
n
nnT
nT
αν
αν
:
( )
( )[ ] ( )
( )[ ] ( ) 22525652252655
2254522455
225)(
232
2
+⋅+⋅+−=+⋅++−=
+⋅+−=++−=
+−=
nTnT
nTnT
nTnT
B. ( + *) (/
1. T(n)=aT(n-b)+c
1. %)+' ( ! (% + , " (- 2: !) ! 3 k)
+!" # !$ #% & ' " ($ #%
! 2 3 ( ! ( ! ( % ' % $,( (! %& k (% + ,( "
( " *() & "
(… ' ( …)
( ) .......)( +−⋅= kbnTnT
( )
...
2252565)( 23
==
=+⋅+⋅+−= nTnT
( ) 22525...2525
...
21
+⋅+⋅++⋅+−=
==
−kk
knT
B. ( + *) (/
1. T(n)=aT(n-b)+c
1. %)+' ( ! (% + , " (- 3: % + * &" ! ' k)
,!" # !$ #% & ' " ($ #%
% + *)2 ' ( %&!( ! ! ( ( !/ &% ' n0
' !( ! $ ! " " +$ ' ( /" % " k).
(… ' ( …)
0nkbn =−
(… ' ( …)
! ! &!
2/
2
02
nk
kn
kn
=
=
=−

B. ( + *) (/
1. T(n)=aT(n-b)+c
1. %)+' ( ! (% + , " (- 4: ! ! ! ! ' k)
-!" # !$ #% & ' " ($ #%
! ! $ ( ! k % ' 3 ( ! % ! % ' % ',( ! 3
2. % %( % +( . () &" & " ( ! '
!( ! $ ! " ".
(… ' ( …)
! ! " k=n/2 ! (n) ' (:
( )
22525...255
22525...2505)(
212/2/
212/2/
+⋅+⋅++⋅+=
+⋅+⋅++⋅+=
−
−
nn
nn
TnT
B. ( + *) (/
1. T(n)=aT(n-b)+c
1. %)+' ( ! (% + , " (- 5: % + * &" ) ! ")
!" # !$ #% & ' " ($ #%
% ' % $%!( '% + *)2(! ( ! * / !& !$% ! '
'% + * $ ) ! " & / *(/ (! " % & ':
(… ' ( …)
=
+
−
−
=
n
i
n
i
x
x
x
0
1
1
1
! ! " k=n/2 ! (n) ' (:
5,055,1
)15(5,05
15
15
25
525
]5...551[25
]25...25252[5
22525...255)(
2/
2/2/
112/
2/
12/
0
2/
12/22/
12/22/
212/2/
−⋅=
=−+=
=
−
−
+=
=+=
=+++++=
=⋅++⋅+⋅++=
=+⋅+⋅++⋅+=
+−
−
=
−
−
−
n
nn
n
n
n
i
in
nn
nn
nn
nT
-. ( + *) (/
2. T(n)=T(n-b)+f(n)
!" # !$ #% & ' " ($ #%
T(n)=T(n-1)+f(n) +$ (! ( ! (% + , "
1. . ' ( &+ '" ! '" $" & '" T(n), T(n-1),… !
%( )%!/ ! " "
2. ! ' ( ! " (4 0 ( " ! +
3. % + *)2 ' ( ! % ' % $%!(
B. ( + *) (/
2. T(n)=aT(n-b)+c
1. %)+' ( ! (% + , " (- 1: , !/ & / )
!" # !$ #% & ' " ($ #%
! 1 3 * . ' ( &+ '" ! '" $" & '" %& ! T(n)
! & T(n0) &% ' n0 () %( )%!/ ! " "
:
+$ (!( ! :
:
( )1 3 , 0
( )
1, 0
T n n n
T n
n
αν
αν
− + >
=
=
:
1)0(
13)0()1(
23)1()2(
...
)2(3)3()2(
)1(3)2()1(
3)1()(
=
⋅+=
⋅+=
−+−=−
−+−=−
+−=
T
TT
TT
nnTnT
nnTnT
nnTnT

B. ( + *) (/
2. T(n)=aT(n-b)+c
1. %)+' ( ! (% + , " (- 2: & ( ! + )
!" # !$ #% & ' " ($ #%
! 2 3 % ! ' ( ! + ! " ( " % ' ' ( * ,( !
% * $ ( 3 :
3)1()( +−= nnTnT
1)0(
13)0()1(
23)1()2(
...
)2(3)3()2(
)1(3)2()1(
3)1()(
=
⋅+=
⋅+=
−+−=−
−+−=−
+−=
T
TT
TT
nnTnT
nnTnT
nnTnT
)(+
11323...)2(3)1(33)( +⋅+⋅++−+−+= nnnnT
B. ( + *) (/
2. T(n)=aT(n-b)+c
1. %)+' ( ! (% + , " (- 3: % + * &" ! ' ) ! ")
)!" # !$ #% & ' " ($ #%
! 3 3 '% + *)2 ' ( ! % ' % $%!( % ' ' /"
() ! % & ". 5 . $ ! (4 " ) ! :
=
+
=
n
i
nn
i
1 2
)1(
=
++
=
n
i
nnn
i
1
2
6
)12)(1(
=
+
Θ=
n
i
kk
ni
1
1
)(
( & ( '%& ( 4 )
( ' ( …)
15,15,1
2
)1(
3131
])1()2(...21[31
3)1(3)2(3...23131
11323...)2(3)1(33)(
2
1
++=
=
+
+=+=
+−+−++++=
+−+−++⋅+⋅+=
+⋅+⋅++−+−+=
=
nn
nn
i
nnn
nnn
nnnnT
n
i
B. ( + *) (/
T(n)=T(n/a)+T(n/b)+f(n)
!" # !$ #% & ' " ($ #%
H (%)+' ! " " " T(n)=T(n/a)+T(n/b)+f(n) *) (! :
( (. * ! " ! &! ! " 3.6 ( )
( ! " ( )
1
11
≤+
ba
1
11
>+
ba
( ! " ( )
% ( " '% + * $ . * !/ ! % ) (
(. &2 ' ( % ! , % & " ! 2 ! ( ! !
1>+
ba
-. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
*!" # !$ #% & ' " ($ #%
T(n)=T(n/a)+ (n/b)+f(n) +$ (! ( ! ! '
' "
1. %!'4 ! ' ' 0 1+ (/ ( ! 2 (%)%(
2. ( ( & 3 ( 0 ' ( %& (" % 4( " *) ! ( %& ! f(n))2. ( ( & 3 ( 0 ' ( %& (" % 4( " *) ! ( %& ! f(n))
3. % + * &" % 4(/ (%)%( ( ' /" *(/ (! % & ")
4. % + * &" ! ' $, '" ! ' ' ( ) logcn ( c ! (+ ! %& !
a,b)
5. T(n)=! !/ % 4(/ &+/ !/ (% % /

B. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
1. %)+' ( ! " (- 1: %!'4 ' 2 (%)%( )
+!" # !$ #% & ' " ($ #%
! 1 3 %!$ ' ( ! 0 + (/ ( . )2 ! " &
! '" $" & '" (&%/" * & ! ' + ( " ! !) !
& 0 ).
:
+$ (!( ! :
=
>++
=
1,1
1,
32)(
n
nn
n
T
n
T
nT
αν
αν
:
=1,1 nαν
)(nT
2
n
T
3
n
T
4
n
T
3963
2642
32
)(
nn
T
n
T
n
T
nn
T
n
T
n
T
n
n
T
n
TnT
++=
++=
++=
5
6
n
T
6
n
T
9
n
T
B. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
1. %)+' ( ! " (- 2: 4( " ( ( & 3 )
,!" # !$ #% & ' " ($ #%
! 2 3 ( 0 ' ( ( ( & 3 %& (" % 4( " *) ! ( '! !
+ ( )2(! %& ! & % ' ' ( ( . ) ( ! !0 ! "
! f(n) )
(… ' ( …)
)(nT
2
n
T
3
n
T
4
n
T
6
n
T
6
n
T
9
n
T
n
2
n
3
n
4
n
6
n
6
n
9
n
3963
2642
32
)(
nn
T
n
T
n
T
nn
T
n
T
n
T
n
n
T
n
TnT
++=
++=
++=
5
B. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
1. %)+' ( ! " (- 3: 4( " (%)%( )
-!" # !$ #% & ' " ($ #%
! 3 3 % ! ' ( (%)%( ! " % 4( " * " 3*( + .
&! % ! " 3* ) ( &! () *(/ (! % & ". ! ( %& ("
% 4( " *) ! ! (%)%( i.
(… ' ( …)
)(nT
2
n
T
3
n
T
4
n
T
6
n
T
6
n
T
9
n
T
n
2
n
3
n
4
n
6
n
6
n
9
n
% %( 0: n
% %( 1:
6
5
6
2
6
3
32
nnnnn
=+=+
% %( 2: 2
2
6
5
...
9664
nnnnn
==+++
6 ! (%)%( i *) ! % 4( "i
i
n
6
5
B. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
1. %)+' ( ! " (- 4: % + * &" $, '" ')
!" # !$ #% & ' " ($ #%
! 4 3 '% + *)2 ' ( ! $, " ! ' '.
$, " ! ' ' )2(! %& % " & " %& ! '" n/a kai n/b .! ( %
* *) ( ) " ( ! n0, + +$ ! " ! (4) / n/min{a,b}=n0
%( ! $, " ! ' ' )2(! %& ! &!( %& ! '" $
% ! " '* ( () c () &!( " %& ! '" $
% ! " ( +. c=min{a,b} ) %(! &! ! $, " ! ' ' () logcn.% ! " ( +. c=min{a,b} ) %(! &! ! $, " ! ' ' () logcn.
(… ' ( …)
$, " ! ' ' () log2n

B. ( + *) (/
2. T(n)=T(n/a)+T(n/b)+f(n)
1. %)+' ( ! " (- 5: % + * &" ) ! ")
!" # !$ #% & ' " ($ #%
! 5 3 '% + *)2 ' ( ! % +'%+ &! ! /" ! !/ % 4(/ &+/
!/ (% % / . () % ! *(/ (! % & ". 6 % ' (
! !$% :
(… ' ( …)
=
+
−
−
=
n
i
n
i
x
x
x
0
1
1
1
' (%0" % 4( " () :
( )
log log
0 0
log 1
log
0
log 1
5
( ) 5
6 6
5
1
5 6
56 1
6
6 0,83 6
i
n ni
i ii i
n
i
n
i
n
n
T n n
n n
n n
= =
+
=
+
= = =
−
= = =
−
= ⋅ −
-. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
2. %)+' ( ! . * !
!" # !$ #% & ' " ($ #%
T(n)=T(n/a)+ (n/b)+f(n) +$ (! ( ! !/
. * !/
1. % + * &" !/ . * ! " ( ! (* +$!( %& ! a b !
(0 ' ) ".
2. % + * &" / . * ! " ( ! &!( %& ! a b ! (0
' ) ".
3. ! !/ . * () ) ( ! / . * ' ( ' %!/!
( !) ! " ' ! " % +'%+ &! ! ". ++ 0" "
-. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
2. %)+' ( ! . * !
!" # !$ #% & ' " ($ #%
* . ' ( ! " $ " ( " / !/ % )/ '% + *) ' (
! / ! !/ . * . / . * % $,( ( ! &!( %& !
a,b ! !/ . * % $,( ( ! (* +$!( %& ! a,b
:
'% + *) (!( ' %!/! ( !) ! " ": 2
43
)( n
n
T
n
TnT ++=
:
/ . * % $,( %& ! (%)+' ! " " ":
1 ! !/ . * % $,( %& ! (%)+' ! " " ":
2
( ) 2
3
n
n nΑ = Α +
2
( ) 2
4
n
n nΚ = Κ +
-. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
2. %)+' ( ! . * ! (1. % + * &" ! ' / . * ! ")
)!" # !$ #% & ' " ($ #%
'% + * &" ! ' / . * ! " *) ( ( ! (0 ' ) ".
!" #$ %& 2
3
2)( n
n
AnA +=
63,02loglog,)(,3,2 2
===== annfba7 /:
$( : * % ! ( (>0
+ * / '% ( c<1 ! ! 0 !(:
6 $( * 2/9 8 c < 1.
6 %& ! ’ %( )%!/ ! ' (/ ! " 1' ) " %(! &! :
6
63,02loglog,)(,3,2 3
2
===== annfba b
)()( 63,02 ε+
Ω== nnnf
)()( 2
nnA Θ=
ccn
n
cn
n
ncf
n
fncf
b
n
af ≤⇔≤⇔≤⇔≤⇔≤
9
2
3
2
3
2)(
2
4)( 2
2
2
2
2
)()( 2
nnT Ο=

-. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
2. %)+' ( ! . * ! (2. % + * &" ! ' !/ . * ! ")
!" # !$ #% & ' " ($ #%
'% + * &" ! ' / . * ! " *) ( ( ! (0 ' ) ".
' &" #$ %& 2
4
2)( n
n
KnK +=
5,02loglog,)(,4,2 2
===== annfba7 /:
$( : * % ! ( (>0
+ * / '% ( c<1 ! ! 0 !(:
6 $( * 1/8 8 c < 1.
6 %& ! ’ %( )%!/ ! ' (/ ! " 1' ) " %(! &! :
6
5,02loglog,)(,4,2 4
2
===== annfba b
)()( 5,02 ε+
Ω== nnnf
)()( 2
nnK Θ=
cccn
n
cn
n
ncf
n
fncf
b
n
af ≤⇔≤⇔≤⇔≤⇔≤⇔≤
8
1
16
2
4
2
4
2)(
2
4)( 3
2
2
3
2
)()( 2
nnT Ω=
-. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
2. %)+' ( ! . * ! (3. ' % * ! ' %!/! % +/! )
*!" # !$ #% & ' " ($ #%
! / . * ! !/ . * () ) , !&!( ' ( ' %!/! ( !)
! " % +'%+ &! ! " ! '.
(... (!)*+ %…)
' (%0" %& ! % * $ ( :
)()( 2
nnT Ο=
! . * ! () . (! , " !/ . * !/ ( % !$ ( !
$( :
' (%0"
)()( 2
nnT Ο=
)()( 2
nnT Ω=
)()( 2
nnT Θ=
-. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
3. %)+' ( ! ! &! ! 3.6
+!" # !$ #% & ' " ($ #%
T(n)=T(n/a)+ (n/b)+f(n) +$ (! ( !
! &! ! 3.6 ! ' 3 3+) '
3.6
% + *)2 ' ( ! % &! !
ba
11
+
1. !&!(
2. !&!(
3. !&!( ! &! ! 3.6 ( % !$ ( % ( '% (/!
( "
ba
+
1
11
<+
ba
))(()( nfnT Θ=
1
11
=+
ba
)log)(()( nnfnT ⋅Θ=
1
11
>+
ba
B. ( + *) (/
3. T(n)=T(n/a)+T(n/b)+f(n)
3. %)+' ( ! ! &! ! 3.6 ( ()* ! )
,!" # !$ #% & ' " ($ #%
(. * ! " ! &! ! " 3.6 () % +$ ($ + &! " ) ( ! !
+$ ( % (" ".
:
'% + *) (!( ' %!/! ( !) ! " ":
:
2
43
)( n
n
T
n
TnT ++=
:
$( : %& ! !.3.6 $( :
:
'% + *) (!( ' %!/! ( !) ! " ":
:
$( : %& ! !.3.6 $( :
1
12
7
12
3
12
4
4
1
3
1
<=+=+ )()( 2
nnT Θ=
n
n
T
n
TnT ++=
3
2
3
)(
1
3
3
3
2
3
1
==+ )log()( nnnT ⋅Θ=

. ( "
. * 1
% + *) !( ! 3 +$ !/ 0 ( ! (% + , ":
-!" # !$ #% & ' " ($ #%
( )4 3 5, 0
) ( )
0, 0
T n n
A T n
n
αν
αν
− + >
=
=
2
0, 0
( 1) 2 , 0
) ( )
1, 0
n
T n n n
B T n
n
αν
αν
αν
=
− + >
=
=
. ( "
. * 2
% + *) !( ' %!/! ( !) !/ 0 % 0 ! " !
":
!" # !$ #% & ' " ($ #%
=
>++
=
1,1
1,
43)()
2
n
nn
n
T
n
T
nTA
αν
αν
=
>++
=
=
1,1
1,
3
2
3)()
1,1
n
nn
n
T
n
T
nTB
n
αν
αν
αν
. ( "
. * 3
5 % () !( ! '% + * $ . * !/ * ! (%)+' ! "
"
!" # !$ #% & ' " ($ #%
2
3
2
2
)( n
n
T
n
TnT ++=
. ( "
. * 4
% + *) !( ' %!/! ( !) !/ 0
!" # !$ #% & ' " ($ #%
log
44
3
)() n
n
T
n
TnTA ++=
2
3
23
2
)()
65
2
)()
n
n
T
n
TnTC
n
n
T
n
TnTB
++=
++=

. ( "
. * 5
! (%)+' ( &" % 3+ ! " ' ( ! ( " ! ()" +*& $".
• (A1) % 0! " +*& " * (% +$ ( % &3+ (* '" n, (% +$(
(%! '% % 3+ ! (* '" n/3 ! ' ' 2( ! " +$ ( "
! '" ( & n3.
!" # !$ #% & ' " ($ #%
• (A2) ($!( " +*& " * (% +$ ( % &3+ (* '" n, (% +$(
'% % 3+ ! (* '" n/2 ! ' ' 2( ! " +$ ( "
! '" ( & n.
• (A3) ! )! " +*& " (% +$( '% % &3+ (* '" n-1 3 ) ( !
+$ ! ' $ % 3+ ! " ( & n3.
3 ( $ ' %!/! ) & (%)+' " ! ' % 3+ ! " * ! (
+*& , (% + 4(!( ! ! $!( +*& * ! (%)+' ! ' % 3+ ! ".
9 : % ()!( % (!( &!
=
Θ=
n
i
ni
1
43
)(