ΠΛΗ30 - ΜΑΘΗΜΑ 1.4

30 1:

1.4:

T(n)=aT(n/b)+f(n)

.

.

1.

2 , 30, 1.4: T(n)=aT(n/b)+f(n)

1.

1. BinarySearch

2. MergeSort

2.

.

1. T(n)=aT(n/b)+f(n)

1.

2. 2.

.

.

:

3 , 30, 1.4: T(n)=aT(n/b)+f(n)

T(n)=aT(n/b)+f(n)

T(n)=aT(n/b)+f(n)

MergeSort BinarySearch

. 1.

4 , 30, 1.4: T(n)=aT(n/b)+f(n)

. .

.

.

:procedure recursive(n)

...

...

:

BinarySearch MergeSort

...

recursive(n-1)...

...

end procedure

. 1. 1. BinarySearch (1. )

5 , 30, 1.4: T(n)=aT(n/b)+f(n)

BinarySearch : : A, x : A, x:

,

x .

, 0.

: : :

x, .

x

x


6 , 30, 1.4: T(n)=aT(n/b)+f(n)

BinarySearch procedure BinarySearch(A,x,start,finish)procedure BinarySearch(A,x,start,finish)

if start>finish then return 0

else middle=(start+finish) div 2if (x==A[middle]) then

return middleelse if (xmiddle) thenpos=BinarySearch(A,x,middle+1,finish)return pos

end ifend if

end procedure


7 , 30, 1.4: T(n)=aT(n/b)+f(n)

11 :1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

: BinarySearch(A,11,1,15): middle=(1+15) div 2=8. xA[middle]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 3 5 7 11 13 17 21 23 27 31 33 37 41 43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 3 5 7 11 13 17 21 23 27 31 33 37 41 43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 3 5 7 11 13 17 21 23 27 31 33 37 41 43

start

start

finish

finish

: BinarySearch(A,11,5,7) : middle=(5+7) div 2=6 x


8 , 30, 1.4: T(n)=aT(n/b)+f(n)

: x . .

T(n)=5 T(n)=(1). : x :

T(n) n. 8

( 2 )

1

n

n, T(n),

1 . :

n=0 1 ( )

12

n

12

n

12nT

912

)( +

=

nTnT


9 , 30, 1.4: T(n)=aT(n/b)+f(n)

:

= 0,1 n

,

:

:

>+

=

= 0,912

0,1)(

nnT

n

nT

12

n

2n

:

:

>+

=

= 0,92

0,1)(

nnT

n

nT

)(log)( nnT =

. 1. 2. MergeSort (1. )

10 , 30, 1.4: T(n)=aT(n/b)+f(n)

MergeSort : : () A n : () A n :

: :

.

. 1. 2. MergeSort (2. )

11 , 30, 1.4: T(n)=aT(n/b)+f(n)

MergeSort procedure MergeSort(A,start, finish)procedure MergeSort(A,start, finish)

if |A|

. 1. 2. MergeSort (2. )

12 , 30, 1.4: T(n)=aT(n/b)+f(n)

Merge : :procedure Merge(A,)

i=1, j=1, k=1while (i

. 1. 2. MergeSort (3. )

13 , 30, 1.4: T(n)=aT(n/b)+f(n)

MergeSort(,1,16)1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 9 10 13

. 1. 2. MergeSort (3. )

14 , 30, 1.4: T(n)=aT(n/b)+f(n)

MergeSort(,1,16)1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

18 7 4 11 9 20 6 19 10 11 12 13 14 15 16

22 19 14 5 2 3 10 13

. 1. 2. MergeSort (3. )

15 , 30, 1.4: T(n)=aT(n/b)+f(n)

(A,1,8)1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

18 7 4 11 9 20 6 19 10 11 12 13 14 15 16

22 19 14 5 2 3 10 13

1 2 3 4

18 7 4 115 6 7 8

9 20 6 118 7 4 11 9 20 6 1

. 1. 2. MergeSort (3. )

16 , 30, 1.4: T(n)=aT(n/b)+f(n)

(A,1,4)1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

18 7 4 11 9 20 6 19 10 11 12 13 14 15 16

22 19 14 5 2 3 10 13

1 2 3 4

18 7 4 115 6 7 8

9 20 6 118 7 4 11 9 20 6 1

1 2

18 73 4

4 11

. 1. 2. MergeSort (3. )

17 , 30, 1.4: T(n)=aT(n/b)+f(n)

(A,1,2): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

18 7 4 11 9 20 6 19 10 11 12 13 14 15 16

22 19 14 5 2 3 10 13

1 2 3 4

18 7 4 115 6 7 8

9 20 6 118 7 4 11 9 20 6 1

1 2

7 183 4

4 11

. 1. 2. MergeSort (3. )

18 , 30, 1.4: T(n)=aT(n/b)+f(n)

(A,3,4): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

18 7 4 11 9 20 6 19 10 11 12 13 14 15 16

22 19 14 5 2 3 10 13

1 2 3 4

18 7 4 115 6 7 8

9 20 6 118 7 4 11 9 20 6 1

1 2

7 183 4

4 11

. 1. 2. MergeSort (3. )

19 , 30, 1.4: T(n)=aT(n/b)+f(n)

(A,1,4): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

18 7 4 11 9 20 6 19 10 11 12 13 14 15 16

22 19 14 5 2 3 10 13

1 2 3 4

4 7 11 185 6 7 8

9 20 6 14 7 11 18 9 20 6 1

1 2

7 183 4

4 11

. 1. 2. MergeSort (3. )

20 , 30, 1.4: T(n)=aT(n/b)+f(n)

(A,5,8)1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

18 7 4 11 9 20 6 19 10 11 12 13 14 15 16

22 19 14 5 2 3 10 13

1 2 3 4

4 7 11 185 6 7 8

9 20 6 14 7 11 18 9 20 6 1

1 2

7 183 4

4 11

5 6

9 207 8

6 1

. 1. 2. MergeSort (3. )

21 , 30, 1.4: T(n)=aT(n/b)+f(n)

(A,5,6): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

18 7 4 11 9 20 6 19 10 11 12 13 14 15 16

22 19 14 5 2 3 10 13

1 2 3 4

4 7 11 185 6 7 8

9 20 6 14 7 11 18 9 20 6 1

1 2

7 183 4

4 11

5 6

9 207 8

6 1

. 1. 2. MergeSort (3. )

22 , 30, 1.4: T(n)=aT(n/b)+f(n)

(A,7,8): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

18 7 4 11 9 20 6 19 10 11 12 13 14 15 16

22 19 14 5 2 3 10 13

1 2 3 4

4 7 11 185 6 7 8

9 20 6 14 7 11 18 9 20 6 1

1 2

7 183 4

4 11

5 6

9 207 8

1 6

. 1. 2. MergeSort (3. )

23 , 30, 1.4: T(n)=aT(n/b)+f(n)

(A,5,8): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

18 7 4 11 9 20 6 19 10 11 12 13 14 15 16

22 19 14 5 2 3 10 13

1 2 3 4

4 7 11 185 6 7 8

1 6 9 204 7 11 18 1 6 9 20

1 2

7 183 4

4 11

5 6

9 207 8

1 6

. 1. 2. MergeSort (3. )

24 , 30, 1.4: T(n)=aT(n/b)+f(n)

(A,1,8): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

1 4 6 7 9 11 18 209 10 11 12 13 14 15 16

22 19 14 5 2 3 10 13

1 2 3 4

4 7 11 185 6 7 8

1 6 9 204 7 11 18 1 6 9 20

1 2

7 183 4

4 11

5 6

9 207 8

1 6

. 1. 2. MergeSort (3. )

25 , 30, 1.4: T(n)=aT(n/b)+f(n)

(9,16)1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13

1 2 3 4 5 6 7 8

1 4 6 7 9 11 18 209 10 11 12 13 14 15 16

2 3 5 10 13 14 19 22

1 2 3 4

4 7 11 185 6 7 8

1 6 9 209 10 11 12

5 14 19 2213 14 15 16

2 3 10 134 7 11 18 1 6 9 20

1 2

7 183 4

4 11

5 6

9 207 8

1 6

5 14 19 22 2 3 10 13

9 10

19 2211 12

5 14

13 14

2 315 16

10 13

. 1. 2. MergeSort (3. )

26 , 30, 1.4: T(n)=aT(n/b)+f(n)

(A,1,16): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 9 10 11 13 14 18 19 20 22

1 2 3 4 5 6 7 8

1 4 6 7 9 11 18 209 10 11 12 13 14 15 16

2 3 5 10 13 14 19 22

1 2 3 4

4 7 11 185 6 7 8

1 6 9 209 10 11 12

5 14 19 2213 14 15 16

2 3 10 134 7 11 18 1 6 9 20

1 2

7 183 4

4 11

5 6

9 207 8

1 6

5 14 19 22 2 3 10 13

9 10

19 2211 12

5 14

13 14

2 315 16

10 13

. 1. 2. MergeSort (4. )

27 , 30, 1.4: T(n)=aT(n/b)+f(n)

Merge :)()( mnnT +=

MergeSort :

>+

+

=== 2),(

22

21),1()(

nnnTnT

nn

nT

nnn

=

)()( mnnT +=

:

. : (n)=(nlogn)

222nnn

=

>+

=== 2),(

22

21),1()(

nnnT

nn

nT

. 1. T(n)=aT(n/b)+f(n)

28 , 30, 1.4: T(n)=aT(n/b)+f(n)

T(n)=aT(n/b)+f(n) :: ,

, ( ).

: , ..

( 2 3 ..) .

. 1. T(n)=aT(n/b)+f(n)1.

29 , 30, 1.4: T(n)=aT(n/b)+f(n)

(Master Theorem) : : :

a1, b>1 , f(n) . :

) >0, :)()( log = abnOnf

)()( nfbn

aTnT +

=

( )abnnT log)( =) :

) >0 c

. 1. T(n)=aT(n/b)+f(n)1.

30 , 30, 1.4: T(n)=aT(n/b)+f(n)

:

a,b f(n) a,b f(n) logba. f(n) :

B

abnnf log)(

. 1. T(n)=aT(n/b)+f(n)1. ( )

31 , 30, 1.4: T(n)=aT(n/b)+f(n)

: :: :

::

: >0 :

nnTnT +

=

28)(

)()( 3 == nOnnf

)()( 3nnT =

38loglog,)(,2,8 2 ===== annfba b

)()( 3nnT =

. 1. T(n)=aT(n/b)+f(n)1. ( )

32 , 30, 1.4: T(n)=aT(n/b)+f(n)

: :: :

::

:

:

2

39)( nnTnT +

=

)()( 22 nnnf ==

)log()( 2 nnnT =

29loglog,)(,3,9 32 ===== annfba b

)log()( 2 nnnT =

. 1. T(n)=aT(n/b)+f(n)1. ( )

33 , 30, 1.4: T(n)=aT(n/b)+f(n)

2 , c>0 . )(nfcnfa c>0 . c a, b f(n) f(n/b).: :

::

: >0 c

. 1. T(n)=aT(n/b)+f(n)2.

34 , 30, 1.4: T(n)=aT(n/b)+f(n)

T(n)=aT(n/b)+f(n), T(n)=aT(n/b)+f(n), :

1. 3 ( )

2. k ( )

.......)( 3 +

=

bnTnT

.......)( +

= kb

nTnT

nn

=3. ( n0 k). .. n0=1 k=logbn

4. k 2.

5. .

0nbn

k =

. 1. T(n)=aT(n/b)+f(n)2. ( 1: 3 )

35 , 30, 1.4: T(n)=aT(n/b)+f(n)

1 3 . .

,

: :

:

=

>+

=

1,1

1,3

5)(n

nnnT

nT

+

= nnTnT 5)( 5)( nnTnT +=

=+++

=++

+

=

++

=+

+

=

+

=

nnnnTnnnnT

nnnTnnnT

nTnT

35

35

35

35

3355

35

35

3355

35)(

22

33

232

22

2

232

2

335

3

335

3

35)(

nnTnT

nnTnT

nTnT

+

=

+

=

+

=

. 1. T(n)=aT(n/b)+f(n)2. ( 2: k)

36 , 30, 1.4: T(n)=aT(n/b)+f(n)

2 k ( .......)( += nTnT(

()

nnnnnT

nnnnT

kk

kk +++++

=

==

=+++

=

35

35...

35

35

...

35

35

35

22

11

22

33

.......)( +

= kbTnT

3333

. 1. T(n)=aT(n/b)+f(n)2. ( 3: k)

37 , 30, 1.4: T(n)=aT(n/b)+f(n)

( n0 k).

0nbn

k =

k).

()

nk 13

=

=

nkkn

n

n

k

k

3

33

33

log3loglog

3loglog3

=

=

=

=

. 1. T(n)=aT(n/b)+f(n)2. ( 4: k)

38 , 30, 1.4: T(n)=aT(n/b)+f(n)

k 2. 2. .

()

k=log3n (n) :

nnn

nnnnnTnT

n

n

n

n +++++

=

35

35...

35

35)( 221log1logloglog 3333

( )n

nnn

nnnnT

n

nn

n

nn

+++++=

+++++=

35

35...

355

35

35...

3515

22

1log1loglog

22

1log1loglog

3

33

3

33

. 1. T(n)=aT(n/b)+f(n)2. ( 5: )

39 , 30, 1.4: T(n)=aT(n/b)+f(n)

.

:

()

nn

nnnn

nnnn

nT

ni

nn in

n

nn

n

nn

5555

35...

35

355

35

35...

355)(

33

3

33

3

33

1loglog1loglog

1log1log

22log

22

1log1loglog

=+=+=

=

+++++=

=+++++=

=

+

=

n

i

ni

x

xx

0

1

11

nnn

nn

nn

nnn

n

n

iinn

i

in

n

i inn

i iin

5,166,15,15166,1

166,15

66,15355

355

355

333

3

3333

3333

loglog11log

log

1log

0log1log

0log

1log

0log1log

0log

+=

+=

=+=

+=

=+=+=

+

=

=

=

=

nnnT nn 5,166,15,15)( 33 loglog +=

. 1

:

40 , 30, 1.4: T(n)=aT(n/b)+f(n)

:

4

3

2

28)()

28)()

28)()

nnTnTC

nnTnTB

nnTnTA

+

=

+

=

+

=

. 2

:

41 , 30, 1.4: T(n)=aT(n/b)+f(n)

3

2

25)()

25)()

nnTnTB

nnTnTA

+

=

+

=

. 3 :

42 , 30, 1.4: T(n)=aT(n/b)+f(n)

>+

=

=

>+

=

1,3

4)()

1,1

1,2

6)()

2 nnnT

nTB

n

nnnT

nTA

=

1,1

3n

Documents

ΠΛΗ30 - ΜΑΘΗΜΑ 1.4