ΠΛΗ30 ΜΑΘΗΜΑ 1.4

11
30 1: 1.4: !" # $% T(n)=aT(n/b)+f(n) &#’( )*( &#’( )*( ! "#$%&’"#$()%*+#$% !"#$%&’($%)* + , -(%.(’/%0 + 1 21 3 4 56#78$56#9:7;<6#7 = > ?2 + = 1 =@A B . +,( ’ #’( ’,$ ’ #’( -!: ! "#$%&’"#$()%*+#$% +!+- - $!( ’ -+! ’( #( $%( T(n)=aT(n/b)+f(n) +!+- . %( -+/( ’ -+! ’( #( $%( T(n)=aT(n/b)+f(n) +!+- ,( ’0,( MergeSort ,( 1#’( BinarySearch .. -"! 1. ! , ! "#$%&’"#$()%*+#$% -,’- %( ,( ,( -! %( ,( +!( +-!’ +, ! # !. +-!’ +, ! # ! . # %-’ ! + ’ ’ - ’( -’%-#( ’( -! ’ -’, ’( . -# 2# -,( * 2!-’ ’ $#. ’#’- ,’ - + -! ’ ’( ’( !( +%+- !-’ # ’ ! ’ ’: procedure recursive(n) ... ... --’#- * +-!2( *( ,(: BinarySearch ’ 1#’ ’$-! - ! ! MergeSort ’ ’0, -,( +! ... recursive(n-1) ... ... end procedure

Transcript of ΠΛΗ30 ΜΑΘΗΜΑ 1.4

Page 1: ΠΛΗ30 ΜΑΘΗΜΑ 1.4

30

1:

1.4:

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# $% T(n)=aT(n/b)+f(n)

&#'( )*(&#'( )*(

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+ , -(%.('/%0

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B

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+!+-

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

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%( -+/( ' -+! '( #( $%(

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

+!+- +!+-

,( '0,( MergeSort

,( 1#'( BinarySearch

.. -"!1. ! ,

!"#$%&'"#$()%*+#$%

-,'- %( ,( ,( -! %( ,( +!(

+-!' +, ! # !.+-!' +, ! # !.

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'( -! ' -', '(.

-# 2# -,( * 2!-' ' $#.

'#'- ,' - + -! ' '( '( !( +%+-

!-' # ' ! ' ':

procedure recursive(n)

...

...

--'#- * +-!2( *( ,(:

BinarySearch ' 1#' '$-! - ! !

MergeSort ' '0, -,( +!

...

recursive(n-1)

...

...

end procedure

Page 2: ΠΛΗ30 ΜΑΘΗΜΑ 1.4

.. -"!1. ! ,1. ,( 1#'( BinarySearch (1.&'*+" -' !)

, !"#$%&'"#$()%*+#$%

BinarySearch # &# 1#':

!(: 0,%( +!( A, '$-! x !(: 0,%( +!( A, '$-! x

30(:

' '$-! +$- ' +!, -+'%2-' % '

'$-! x ' +! .

' '$-! - +$- ' +!, -+'%2-' 0.

-' ! ' !: ,( -0-'1- ' -! '$-! '

+! !- +-+'-(:+! !- +-+'-(:

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' x -! ,'- +, ' -! '$-! ','- /$-

' ' ' +! +, ' $# %$ ' -! '$-!

' x -! - *'- +, ' -! '$-! ','- /$-

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.. -"!1. ! ,1. ,( 1#'( BinarySearch (2. )-()

- !"#$%&'"#$()%*+#$%

'" 2!-' ! +! '( 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 (x<A[middle]) then

pos=BinarySearch(A,x,start,middle-1)return posreturn pos

else if (x>middle) then

pos=BinarySearch(A,x,middle+1,finish)return pos

end if

end if

end procedure

.. -"!1. ! ,1. ,( 1#'( BinarySearch (3. - '%-()

. !"#$%&'"#$()%*+#$%

'-*- ' , /$'( ' '$-! 11 ' +!:1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

4#: BinarySearch(A,11,1,15): middle=(1+15) div 2=8. x<A[middle]

# 4#: BinarySearch(A,11,1,7) : middle=(1+7) div 2=4 x>A[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

# 4#: BinarySearch(A,11,5,7) : middle=(5+7) div 2=6 x<A[middle]

# 4#: BinarySearch(A,11,5,5) : middle=(5+5) div 2=5 x=A[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

start

start

start=finish

finish

finish

.. -"!1. ! ,1. ,( 1#'( BinarySearch (4. )

/ !"#$%&'"#$()%*+#$%

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Page 3: ΠΛΗ30 ΜΑΘΗΜΑ 1.4

.. -"!1. ! ,1. ,( 1#'( BinarySearch (4. )

0 !"#$%&'"#$()%*+#$%

- $-,'- +-!+'" *-' +, ' # $%:

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.. -"!1. ! ,2. ,( '0,( MergeSort (1.&'*+" -' !)

!"#$%&'"#$()%*+#$%

MergeSort # 0, - $-:

!(: +!( () A - n '$-! !(: +!( () A - n '$-!

30(: '0, '" '$-!" ' +! - *0 -

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!!

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!(.

.. -"!1. ! ,2. ,( '0,( MergeSort (2. )-()

!"#$%&'"#$()%*+#$%

'" 2!-' ! +! '( MergeSort - /-

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

if |A|<=2 then

else

middle=(start+finish) div 2A1=MergeSort(A,start,middle)A2=MergeSort(A,middle+1,finish)A=Merge(A1,A2)

end if

'# '-'* '( #( -! ,' +!( %$- ' +* 2 '$-!.

!' 2 %( #-( ' '0, ' '-* '-0* '* '!'$.

3+-' !-' $- '" * - ' ! Merge

end procedure

.. -"!1. ! ,2. ,( '0,( MergeSort (2. )-()

!"#$%&'"#$()%*+#$%

! Merge ' $- * # '0%" +"

+-! +-! "( -0#(:+-! +-! "( -0#(:

procedure Merge(A,)

i=1, j=1, k=1while (i<=n AND j<=m)

if (ai<bj) thenck=ai ; i=i+1

else

ck=bj ; j=j+1end if

+" -"*- ' ||=n, |B|=m

k=k+1end while

B Creturn C

end procedure

Page 4: ΠΛΗ30 ΜΑΘΗΜΑ 1.4

.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()

!"#$%&'"#$()%*+#$%

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.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()

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# 4# MergeSort(,1,16)

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1 2 3 4 5 6 7 8

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# 4# (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

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Page 5: ΠΛΗ30 ΜΑΘΗΜΑ 1.4

.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()

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# 4# (A,3,4): 0 ' ++!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

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.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()

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# 4# (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

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Page 6: ΠΛΗ30 ΜΑΘΗΜΑ 1.4

.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()

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# 4# (A,5,6): 0 ' ++!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

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# 4# (A,7,8): 0 ' ++!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

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Page 7: ΠΛΗ30 ΜΑΘΗΜΑ 1.4

.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()

, !"#$%&'"#$()%*+#$%

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.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()

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# 4# (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

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.. -"!1. ! ,2. ,( '0,( MergeSort (4. )

. !"#$%&'"#$()%*+#$%

++,'' '( '( Merge -!:

)()( mnnT +Θ=

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+! -! !'- +-!+ 8', -"*- ,'

>Θ+

+

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=2),(

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nnn≈

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Page 8: ΠΛΗ30 ΜΑΘΗΜΑ 1.4

. - ! #-"1. # T(n)=aT(n/b)+f(n)1. +! - ' - $!(

0 !"#$%&'"#$()%*+#$%

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+, !1- ' logba.

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-!'- ' ’ +-!+'"

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)( <

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nn

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)()( 3 ε−== nOnnf

)()( 3nnT Θ=

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

)()( 3nnT Θ=

. - ! #-"1. # T(n)=aT(n/b)+f(n)1. +! - ' - $!( (.’ +-!+'")

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2===== annfba b

)log()( 2 nnnT Θ=

Page 9: ΠΛΗ30 ΜΑΘΗΜΑ 1.4

. - ! #-"1. # T(n)=aT(n/b)+f(n)1. +! - ' - $!( (’ +-!+'")

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Page 10: ΠΛΗ30 ΜΑΘΗΜΑ 1.4

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Page 11: ΠΛΗ30 ΜΑΘΗΜΑ 1.4

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