ΠΛΗ30 ΜΑΘΗΜΑ 1.5
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ΠΛΗ30 ΜΑΘΗΜΑ 1.5
Transcript of ΠΛΗ30 ΜΑΘΗΜΑ 1.5
-
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)
-
.
B.
1. T(n)=aT(n-b)+c
1.
2. T(n)=T(n-1)+f(n)
1.
2 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
3. T(n)=T(n/a)+T(n/b)+f(n)
1.
2.
3. .3.6
.
-
.
:
3.6 T(n)=T(n/a)+T(n/b)+f(n) T(n)=T(n-1)+f(n)
3 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+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)
4 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
H T(n)=aT(n-b)+f(n) .
:
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
5 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
T(n)=aT(n-b)+c
1. 3 ( )
2. k ( ( ) ....3...)( += bnTnT
( )+= )3. ( n0
k). .. n0=0 n/b
4. k 2.
5. .
( ) .......)( += kbnTnT
0nkbn =
-
B. 1. T(n)=aT(n-b)+c1. ( 1: 3 )
6 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
1 3 .
: :
:
( )
=
>+=
0,10,225)(
n
nnTnT
:
( )( )[ ] ( )( )[ ] ( ) 22525652252655
2254522455225)(
232
2
+++=+++=
++=++=
+=
nTnTnTnT
nTnT
-
B. 1. T(n)=aT(n-b)+c1. ( 2: k)
7 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
2 k (
()
( ) .......)( += kbnTnT
( )...
2252565)( 23==
=+++= nTnT
( ) 22525...2525...
21 +++++=
==
kk knT
-
B. 1. T(n)=aT(n-b)+c1. ( 3: k)
8 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
( n0 k).
()
0nkbn =
()
2/2
02
nkknkn
=
=
=
-
B. 1. T(n)=aT(n-b)+c1. ( 4: k)
9 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
k 2. .
()
k=n/2 (n) :( )
22525...25522525...2505)(
212/2/
212/2/
+++++=
+++++=
nn
nn TnT
-
B. 1. T(n)=aT(n-b)+c1. ( 5: )
10 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
:
()
=
+
=
n
i
ni
x
xx
0
1
11
k=n/2 (n) :
5,055,1)15(5,05
151525
525
]5...551[25]25...25252[5
22525...255)(
2/
2/2/
112/2/
12/
02/
12/22/
12/22/
212/2/
=
=+=
=
+=
=+=
=+++++=
=+++++=
=+++++=
+
=
n
nn
nn
n
iin
nn
nn
nnnT
-
. 2. T(n)=T(n-b)+f(n)
11 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
T(n)=T(n-1)+f(n)
1. T(n), T(n-1),
2.
3.
-
B. 2. T(n)=aT(n-b)+c1. ( 1: )
12 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
1 T(n) T(n0) n0 : :
:
( )1 3 , 0( )1, 0T n n n
T nn
+ >=
=:
1)0(13)0()1(23)1()2(
...
)2(3)3()2()1(3)2()1(
3)1()(
=
+=
+=
+=
+=
+=
TTTTT
nnTnTnnTnT
nnTnT
-
B. 2. T(n)=aT(n-b)+c1. ( 2: )
13 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
2 :
3)1()( += nnTnT
1)0(13)0()1(23)1()2(
...
)2(3)3()2()1(3)2()1(
3)1()(
=
+=
+=
+=
+=
+=
TTTTT
nnTnTnnTnT
nnTnT
)(+
11323...)2(3)1(33)( ++++++= nnnnT
-
B. 2. T(n)=aT(n-b)+c1. ( 3: )
14 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
3 . :
=
+=
n
i
nni1 2
)1(
=
++=
n
i
nnni1
2
6)12)(1(
=
+=n
i
kk ni1
1)(( )
()
15,15,12
)1(3131
])1()2(...21[313)1(3)2(3...23131
11323...)2(3)1(33)(
2
1
++=
=
++=+=
++++++=
++++++=
++++++=
=
nn
nni
nnn
nnn
nnnnT
n
i
-
B. T(n)=T(n/a)+T(n/b)+f(n)
15 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
H T(n)=T(n/a)+T(n/b)+f(n) :
3.6 ( )
( )
111 +ba
111 >+ba
( )
, !
1>+ba
-
. 3. T(n)=T(n/a)+T(n/b)+f(n)
16 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
T(n)=T(n/a)+(n/b)+f(n)
1. 2 2. ( f(n))2. ( f(n))3. ( )4. ( logcn c
a,b)5. T(n)=
-
B. 3. T(n)=T(n/a)+T(n/b)+f(n)1. ( 1: 2 )
17 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
1 ( ).
: :
=
>+
+
=
1,1
1,32)(
n
nnnTnT
nT
:
=1,1 n
)(nT
2nT
3nT
4nT 3963
2642
32)(
nnTnTnT
nnTnTnT
nnTnTnT
+
+
=
+
+
=
+
+
=
6nT
6nT
9nT
N N N NO O O O
-
B. 3. T(n)=T(n/a)+T(n/b)+f(n)1. ( 2: )
18 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
2 ( f(n) )
()
)(nT
2nT
3nT
4nT
6nT
6nT
9nT
n
2n
3n
4n
6n
6n
9n 3963
2642
32)(
nnTnTnT
nnTnTnT
nnTnTnT
+
+
=
+
+
=
+
+
=
N N N NO O O O
-
B. 3. T(n)=T(n/a)+T(n/b)+f(n)1. ( 3: )
19 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
3 . .
i.()
)(nT
2nT
3nT
4nT
6nT
6nT
9nT
n
2n
3n
4n
6n
6n
9n
0: n
1:6
56
26
332
nnnnn=+=+
2: 22
65
...
9664nnnnn
==+++
i iin
65
-
B. 3. T(n)=T(n/a)+T(n/b)+f(n)1. ( 4: )
20 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
4 . n/a kai n/b n0, n/min{a,b}=n0
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: )
21 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
5 . .
:
()
=
+
=
n
i
ni
x
xx
0
1
11
:
( )
log log
0 0
log 1
log
0
log 1
5( ) 56 6
5 15 656 16
6 0,83 6
in ni
i ii i
n
in
i
n
nT n n
n n
n n
= =
+
=
+
= = =
= = =
=
-
. 3. T(n)=T(n/a)+T(n/b)+f(n)2.
22 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
T(n)=T(n/a)+(n/b)+f(n)
1. a b .
2. a b .
3. .
-
. 3. T(n)=T(n/a)+T(n/b)+f(n)2.
23 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
.
a,b a,b: : 2
43)( nnTnTnT +
+
=
: :
:
2( ) 23n
n n = +
2( ) 24n
n n = +
-
. 3. T(n)=T(n/a)+T(n/b)+f(n)2. (1. )
24 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
.
2
32)( nnAnA +
=
63,02loglog,)(,3,2 2 ===== annfba:: >0 c
-
. 3. T(n)=T(n/a)+T(n/b)+f(n)2. (2. )
25 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
.
2
42)( nnKnK +
=
5,02loglog,)(,4,2 2 ===== annfba:: >0 c
-
. 3. T(n)=T(n/a)+T(n/b)+f(n)2. (3. /)
26 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
,
.
(...) :
)()( 2nnT =
, !
:
)()( 2nnT =
)()( 2nnT =
)()( 2nnT =
-
. 3. T(n)=T(n/a)+T(n/b)+f(n)3. 3.6
27 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
T(n)=T(n/a)+(n/b)+f(n) 3.6 3.6
ba11
+
1.
2.
3. 3.6
ba+
111 +ba
-
B. 3. T(n)=T(n/a)+T(n/b)+f(n)3. 3.6 ()
28 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
3.6 .
: :
:
2
43)( nnTnTnT +
+
=
:: .3.6 :
: :
:: .3.6 :
1127
123
124
41
31
-
. 1 :
29 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
( )4 3 5, 0) ( )0, 0T n n
A T nn
+ >=
=
2
0, 0
( 1) 2 , 0) ( )1, 0
n
T n n nB T n
n
=
+ >=
=
-
. 2
:
30 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
=
>+
+
=
1,1
1,43)()
2
n
nnnTnT
nTA
=
>+
+
=
=
1,1
1,3
23)()
1,1
n
nnnTnT
nTB
n
-
. 3
31 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
2
32
2)( nnTnTnT +
+
=
-
. 4
32 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
log44
3)() nnTnTnTA +
+
=
2
3
232)()
652)()
nnTnTnTC
nnTnTnTB
+
+
=
+
+
=
-
. 5 .
(A1) n, n/3 n3.
33 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)
(A2) n, n/2 n.
(A3) n-1 n3.
, .
: =
=n
ini
1
43 )(
