ΠΛΗ30 ΜΑΘΗΜΑ 2.1
29
ΠΛΗ30 ΕΝΟΤΗΤΑ 2: ΣΧΕ∆ΙΑΣΗ ΑΛΓΟΡΙΘΜΩΝ Μάθηµα 2.1: ∆ιαίρει και Βασίλευε ∆ηµήτρης Ψούνης
-
Upload
dimitris-psounis -
Category
Documents
-
view
32 -
download
1
description
1) Διαίρει και Βασίλευε1.1) Ο αλγόριθμος MergeSort (Ταξινόμηση με Συγχώνευση)1.2) Ο αλγόριθμος QuickSort (Γρήγορη Ταξινόμηση)1.3) Ο αλγόριθμος QuickSelect (Γρήγορη Επιλογή)1.4) Ο αλγόριθμος Strassen για τον πολλαπλασιασμό πινάκωνΑσκήσεις
Transcript of ΠΛΗ30 ΜΑΘΗΜΑ 2.1
-
30 2:
2.1:
-
.
1.
1. MergeSort( )
2. QuickSort ( )
3. QuickSelect ( )
4. Strassen /
.
2 , 30, 2.1:
.
1.
-
.
:
(-)
3 , 30, 2.1:
(QuickSort) k- Strassen
-
B.
4 , 30, 2.1:
2 , :
( 2.1)
( 2.2)
( 2.3)
.
-
B. 1.
5 , 30, 2.1:
:
1. : .
2. : ( )
3. : , .
-
. 1. 1. MergeSort
6 , 30, 2.1:
MergeSort ( ) :
1. : :
],...,[ 1 naaA =],...,[ 2/11 naaA = ],...,[ 12/2 nn aaA +=
:
2. : MergeSort.
3. : Merge ( 1.4)
2/11 n 12/2 nn +
)log(...)(2
2)( nnnn
TnT ==+
=
-
. 1. 2. QuickSort
7 , 30, 2.1:
QuickSort ( ) :
1. : : 1 1
],...,[ 1 naaA =1 1
2 1
2. : ( 1 2) QuickSort.
3. : A = [A1]1[A2]
-
. 1. 2. QuickSort (1.)
8 , 30, 2.1:
QuickSort:procedure QuickSort(A,start,finish)
if startodigo)B[j]=A[k]; j=j-1
else B[i]=A[k]; i=i+1
end forB[i]=odigo; =return pos;
end procedure
-
. 1. 2. QuickSort (2. )
9 , 30, 2.1:
Partition: B.1
B.2
1 2 3 4 5 6 7 8
9 7 4 11 18 20 6 1
1 2 3 4 5 6 7 8
7k i j1 2 3 4 5 6 7 8
9 7 4 11 18 20 6 1
1 2 3 4 5 6 7 8
7 4k i j
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
B.3
B.4
B.5
B.6
B.7
1 2 3 4 5 6 7 8
9 7 4 11 18 20 6 1
1 2 3 4 5 6 7 8
7 4 11k i j
1 2 3 4 5 6 7 8
9 7 4 11 18 20 6 1
1 2 3 4 5 6 7 8
7 4 18 11k i j
1 2 3 4 5 6 7 8
9 7 4 11 18 20 6 1
1 2 3 4 5 6 7 8
7 4 20 18 11k i j
1 2 3 4 5 6 7 8
9 7 4 11 18 20 6 1
1 2 3 4 5 6 7 8
7 4 6 20 18 11k i j
1 2 3 4 5 6 7 8
9 7 4 11 18 20 6 1
1 2 3 4 5 6 7 8
7 4 6 1 20 18 11k i j1 2 3 4 5 6 7 8
7 4 6 1 9 20 18 11
-
QuickSort (2. )
10 , 30, 2.1:
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
7 4 11 9 6 1 14 5 2 3 10 13 18 19 22 20
1 2 3 4 5 6 7 8 9 10 11 12
7 4 11 9 6 1 14 5 2 3 10 13
4 6 1 5 2 3 7 13 10 14 9 11
14 15 16
19 22 20
19 20 22
1 2 3 4 5 6
4 6 1 5 2 3
1 2 3 4 5 6
8 9 10 11 12
13 10 14 9 11
10 9 11 13 14
15 16
20 22
20 221 2 3 4 5 6 10 9 11 13 14 20 22
16
22
22
1 2 3
1 2 3
1 3 2
5 6
5 6
5 6
8 9 10
10 9 11
9 10 11
12
14
14
6
6
6
2 3
3 2
2 3
2
2
2
8
9
9
10
11
11
1 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. QuickSort (3. )
11 , 30, 2.1:
Partition (n) :
. :
)log(...)(2)(.
nnnn
TnT
==+=
: . :
?
)log(...)(2
2)( nnnn
TnT ==+
=
( ) )(...)(1)( 2.
nnnTnT ==+=
-
. 1. 2. QuickSort (3. )
12 , 30, 2.1:
: :
[ ] [ ] [ ]nTnTnnTnnTTnT
)()0()1(...)()2()1()()1()0()(
+++++++++=
:
T(n)=(nlogn)
[ ] [ ] [ ]n
nT )( =
[ ]n
nnnTTTnT
)()(...)1()0(2)(
++++=
[ ])(
)(...)1()0(2)( n
n
nTTTnT ++++=
-
. 1. 3. QuickSelect
13 , 30, 2.1:
: n . k-
: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
5- 5 10- 11
: : ( MergeSort) k- . : (nlogn)
QuickSelect: : /: O(n)
-
. 1. 3. QuickSelect ()
14 , 30, 2.1:
o partition. k.
Partition [A1] apos [A2] k=pos apos. kpos 2 (k-n1-1)- ( n1 1)
-
. 1. 3. QuickSelect (1.)
15 , 30, 2.1:
:
procedure QuickSelect(A,start,finish,k)if start>finish then
return 0elseelse
pos=Partition(A,start,finish)if k=pos then
return A[pos]else if kpos then
return QuickSelect(A,pos+1,finish,k-pos)end if
end ifend procedure
-
3. QuickSelect(2. k=12)
16 , 30, 2.1:
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
7 4 11 9 6 1 14 5 2 3 10 13 18 19 22 20
1 2 3 4 5 6 7 8 9 10 11 12
7 4 11 9 6 1 14 5 2 3 10 13
4 6 1 5 2 3 7 13 10 14 9 11
8 9 10 11 12
13 10 14 9 11
10 9 11 13 1410 9 11 13 14
12
14
14
-
. 1. 3. QuickSelect (3. )
17 , 30, 2.1:
Partition (n) :
. :
)(...)()(.
nnn
TnT
==+=
: . :
.
)(...)(2
)( nnn
TnT ==+
=
( ) )(...)(1)( 2.
nnnTnT ==+=
-
. 1. 3. QuickSelect (4. )
18 , 30, 2.1:
: 5- 5- . . ( ) ( ) .
..:
5-
9 ( )
16
13
1 2 3 4 5
18 7 4 11 9
6 7 8 9 10
20 6 1 22 19
11 12 13 14 15
14 5 2 3 10
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
9 19 5 13
-
. 1. 3. QuickSelect (4. )
19 , 30, 2.1:
5 : n T(n)
n/5 ( (1)).
n/5 .
)()1(5
nn =
n/5 . 5- :
, 3n/10 ( ). QuickSelect 7n/10 .
)(...)(5
)(.
nnn
TnT
==+
=
-
. 1. 3. QuickSelect (4. )
20 , 30, 2.1:
:
procedure QuickSelect(A,start,finish,k)if start=finish then
return A[start]elseelse
m 5-.swap(A[m],A[start])pos=Partition(A,start,finish)if k=pos then
return A[pos]else if kpos then
return QuickSelect(A,pos+1,finish,k-pos)end if
end ifend procedure
-
. 1. 3. QuickSelect (4. )
21 , 30, 2.1:
:
n: (n) (n) Partition (n)
7n/10, (7n/10) :
, T(n)=O(n)
)(...)(10
7)(
.
nnn
TnT
==+
=
-
. 1. 4. Strassen
22 , 30, 2.1:
: nxn .
++++
=
=
==
2222122121221121
2212121121121111
2221
1211
2221
1211
babababa
babababa
bb
bb
aa
aaC
BAC
: 12. (n2).
Strassen: : : (n2,81)
. : (n2,38)
++ 222212212122112122212221 bababababbaa
-
. 1. 4. Strassen
O :
23 , 30, 2.1:
procedure MultMatrix(A,B)
for (i=1 to n)for (j=1 to n)
C[i][j]=0;
:
C[i][j]=0;for (k=1 to n)
C[i][j]=C[i][j]+A[i][k]*B[k][j];end for
end for end for
return Cend procedure
( ) )(...)1()1()( 3nnnnnT ==+=
-
. 1. 4. Strassen
Strassen 70: ( n )
:
24 , 30, 2.1:
= 1211
AA
AAA
= 1211
BB
BBB
n/2 x n/2
O Strassen C=AxB :
2221 AA
=2221 BB
B
=
2221
1211
CC
CCC
-
. 1. 4. Strassen
1 : :
25 , 30, 2.1:
2222122122
2122112121
2212121112
2112111111
BABAC
BABAC
BABAC
BABAC
+=+=+=+=
nxn / . : 8 / n/2 x n/2 : . : 4 nx2 x nx2
n, 8 n/2 (n2)
:
2222122122 BABAC +=
)(...)(2
8)( 32 nnn
TnTremMasterTheo
==+
=
-
. 1. 4. Strassen
1 : :
26 , 30, 2.1:
2222122122
2122112121
2212121112
2112111111
BABAC
BABAC
BABAC
BABAC
+=+=+=+=
nxn / . : 8 / n/2 x n/2 : . : 4 nx2 x nx2
n, 8 n/2 (n2)
:
2222122122 BABAC +=
)(...)(2
8)( 32 nnn
TnTremMasterTheo
==+
=
-
. 1. 4. Strassen 2 (Strassen): :
27 , 30, 2.1:
))((
))((
21123
11112
1112221122211
BBAAM
BAM
BAM
BBBAAAM
===
++=
742121
652112
3211
MMMMC
MMMMC
MMMMC
MMC
+++=++=+++=
+=
)(
)(
))((
22221121126
111222215
BBBBAM
BAAAAM
BBAAM
+=+=+=
nxn / . : 7 / n/2 x n/2 : . : 24 nx2 x nx2
n, 7 n/2 (n2)
:
(n2,37). / (n2)
))(( 122221114 BBAAM =
)(...)(2
7)( 81,22 nnn
TnTremMasterTheo
==+
=
542122 MMMMC +++=)( 21122211227 BBBBAM +=
-
. 1
:
I. A n , n.
28 , 30, 2.1:
II. B n n-1 , , O(n).
III. n n/3 (n2) .
?
-
. 2
:
I. A n , 1/4 , ( n/4) ( n)
29 , 30, 2.1:
( n/4) ( n) n.
II. B n n/2 , , (n3).
III. n n/2 O(n2) .
?
