ΠΛΗ30 ΜΑΘΗΜΑ 1.4
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Transcript of ΠΛΗ30 ΜΑΘΗΜΑ 1.4
30
1:
1.4:
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# $% T(n)=aT(n/b)+f(n)
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T(n)=aT(n/b)+f(n)
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T(n)=aT(n/b)+f(n)
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,( '0,( MergeSort
,( 1#'( BinarySearch
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-,'- %( ,( ,( -! %( ,( +!(
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procedure recursive(n)
...
...
--'#- * +-!2( *( ,(:
BinarySearch ' 1#' '$-! - ! !
MergeSort ' '0, -,( +!
...
recursive(n-1)
...
...
end procedure
.. -"!1. ! ,1. ,( 1#'( BinarySearch (1.&'*+" -' !)
, !"#$%&'"#$()%*+#$%
BinarySearch # &# 1#':
!(: 0,%( +!( A, '$-! x !(: 0,%( +!( A, '$-! x
30(:
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'$-! x ' +! .
' '$-! - +$- ' +!, -+'%2-' 0.
-' ! ' !: ,( -0-'1- ' -! '$-! '
+! !- +-+'-(:+! !- +-+'-(:
' -! '$-! -! ' x, -+'%2- ' % '.
' 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. )
/ !"#$%&'"#$()%*+#$%
4*'- +-!+'": -! ,' ' '$-! x 5!-' 5( '
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3'" T(n) ++,'' ,' +!( %$- ' n.
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.. -"!1. ! ,2. ,( '0,( MergeSort (1.&'*+" -' !)
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MergeSort # 0, - $-:
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.. -"!1. ! ,2. ,( '0,( MergeSort (2. )-()
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'" 2!-' ! +! '( MergeSort - /-
procedure MergeSort(A,start, finish)procedure MergeSort(A,start, finish)
if |A|<=2 then
else
middle=(start+finish) div 2A1=MergeSort(A,start,middle)A2=MergeSort(A,middle+1,finish)A=Merge(A1,A2)
end if
'# '-'* '( #( -! ,' +!( %$- ' +* 2 '$-!.
!' 2 %( #-( ' '0, ' '-* '-0* '* '!'$.
3+-' !-' $- '" * - ' ! Merge
end procedure
.. -"!1. ! ,2. ,( '0,( MergeSort (2. )-()
!"#$%&'"#$()%*+#$%
! Merge ' $- * # '0%" +"
+-! +-! "( -0#(:+-! +-! "( -0#(:
procedure Merge(A,)
i=1, j=1, k=1while (i<=n AND j<=m)
if (ai<bj) thenck=ai ; i=i+1
else
ck=bj ; j=j+1end if
+" -"*- ' ||=n, |B|=m
k=k+1end while
B Creturn C
end procedure
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
!"#$%&'"#$()%*+#$%
# 4# 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. ,( '0,( MergeSort (3. - '%-()
!"#$%&'"#$()%*+#$%
# 4# MergeSort(,1,16)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13
1 2 3 4 5 6 7 8
18 7 4 11 9 20 6 1
9 10 11 12 13 14 15 16
22 19 14 5 2 3 10 13
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
, !"#$%&'"#$()%*+#$%
# 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
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 1
9 10 11 12 13 14 15 16
22 19 14 5 2 3 10 13
1 2 3 4
18 7 4 11
5 6 7 8
9 20 6 118 7 4 11 9 20 6 1
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
- !"#$%&'"#$()%*+#$%
# 4# (A,1,4)1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13
1 2 3 4 5 6 7 8
18 7 4 11 9 20 6 1
9 10 11 12 13 14 15 16
22 19 14 5 2 3 10 13
1 2 3 4
18 7 4 11
5 6 7 8
9 20 6 118 7 4 11 9 20 6 1
1 2
18 7
3 4
4 11
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
. !"#$%&'"#$()%*+#$%
# 4# (A,1,2): 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
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 1
9 10 11 12 13 14 15 16
22 19 14 5 2 3 10 13
1 2 3 4
18 7 4 11
5 6 7 8
9 20 6 118 7 4 11 9 20 6 1
1 2
7 18
3 4
4 11
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
/ !"#$%&'"#$()%*+#$%
# 4# (A,3,4): 0 ' ++!1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13
1 2 3 4 5 6 7 8
18 7 4 11 9 20 6 1
9 10 11 12 13 14 15 16
22 19 14 5 2 3 10 13
1 2 3 4
18 7 4 11
5 6 7 8
9 20 6 118 7 4 11 9 20 6 1
1 2
7 18
3 4
4 11
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
0 !"#$%&'"#$()%*+#$%
# 4# (A,1,4): $- '" * ++"1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13
1 2 3 4 5 6 7 8
18 7 4 11 9 20 6 1
9 10 11 12 13 14 15 16
22 19 14 5 2 3 10 13
1 2 3 4
4 7 11 18
5 6 7 8
9 20 6 14 7 11 18 9 20 6 1
1 2
7 18
3 4
4 11
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
!"#$%&'"#$()%*+#$%
# 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
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 1
9 10 11 12 13 14 15 16
22 19 14 5 2 3 10 13
1 2 3 4
4 7 11 18
5 6 7 8
9 20 6 14 7 11 18 9 20 6 1
1 2
7 18
3 4
4 11
5 6
9 20
7 8
6 1
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
!"#$%&'"#$()%*+#$%
# 4# (A,5,6): 0 ' ++!1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13
1 2 3 4 5 6 7 8
18 7 4 11 9 20 6 1
9 10 11 12 13 14 15 16
22 19 14 5 2 3 10 13
1 2 3 4
4 7 11 18
5 6 7 8
9 20 6 14 7 11 18 9 20 6 1
1 2
7 18
3 4
4 11
5 6
9 20
7 8
6 1
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
!"#$%&'"#$()%*+#$%
# 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
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 1
9 10 11 12 13 14 15 16
22 19 14 5 2 3 10 13
1 2 3 4
4 7 11 18
5 6 7 8
9 20 6 14 7 11 18 9 20 6 1
1 2
7 18
3 4
4 11
5 6
9 20
7 8
1 6
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
!"#$%&'"#$()%*+#$%
# 4# (A,5,8): $- '" * ++"1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13
1 2 3 4 5 6 7 8
18 7 4 11 9 20 6 1
9 10 11 12 13 14 15 16
22 19 14 5 2 3 10 13
1 2 3 4
4 7 11 18
5 6 7 8
1 6 9 204 7 11 18 1 6 9 20
1 2
7 18
3 4
4 11
5 6
9 20
7 8
1 6
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
!"#$%&'"#$()%*+#$%
# 4# (A,1,8): $- '" * ++"1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
18 7 4 11 9 20 6 1 22 19 14 5 2 3 10 13
1 2 3 4 5 6 7 8
1 4 6 7 9 11 18 20
9 10 11 12 13 14 15 16
22 19 14 5 2 3 10 13
1 2 3 4
4 7 11 18
5 6 7 8
1 6 9 204 7 11 18 1 6 9 20
1 2
7 18
3 4
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5 6
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7 8
1 6
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
, !"#$%&'"#$()%*+#$%
'!'$ ! ,-( %( #-( ' (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 20
9 10 11 12 13 14 15 16
2 3 5 10 13 14 19 22
1 2 3 4
4 7 11 18
5 6 7 8
1 6 9 20
9 10 11 12
5 14 19 22
13 14 15 16
2 3 10 134 7 11 18 1 6 9 20
1 2
7 18
3 4
4 11
5 6
9 20
7 8
1 6
5 14 19 22 2 3 10 13
9 10
19 22
11 12
5 14
13 14
2 3
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10 13
.. -"!1. ! ,2. ,( '0,( MergeSort (3. - '%-()
- !"#$%&'"#$()%*+#$%
# 4# (A,1,16): $- '" * ++"1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 2 3 4 5 6 7 9 10 11 13 14 18 19 20 22
1 2 3 4 5 6 7 8
1 4 6 7 9 11 18 20
9 10 11 12 13 14 15 16
2 3 5 10 13 14 19 22
1 2 3 4
4 7 11 18
5 6 7 8
1 6 9 20
9 10 11 12
5 14 19 22
13 14 15 16
2 3 10 134 7 11 18 1 6 9 20
1 2
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3 4
4 11
5 6
9 20
7 8
1 6
5 14 19 22 2 3 10 13
9 10
19 22
11 12
5 14
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2 3
15 16
10 13
.. -"!1. ! ,2. ,( '0,( MergeSort (4. )
. !"#$%&'"#$()%*+#$%
++,'' '( '( Merge -!:
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