30

2:

2.2:

! !

" #$

!""

# $%&%

' ()*+

,-

,-

./(

. % "

! & " '(:

%(%')

(-)

%(%') *

'&" !&')(! +, !

'&" !&')(! +, !

+ .. +( Fibonacci

%(%')

+ .. +!), %++%+!! %,

+ .. '$'! -! ." /%+(

B. ',('&- &')(! +(,

2 ' " !&+$! ' ' '&- % -&

% &'(, , '- ')+(' !'" ' +(:

'&" (' *!(+'' ( 2.1)

( 2.2)

!'" , 0%+! , +, ( 2.3)

/%& )')' '&- !'" +(, % '( '

$+.

B. ',(1. !

!

1 -&' - %2+ % -&' '3" ) ':

) , *-+ ! , %- 4: ( +$!' %2+

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

!", ' )".

/%%2+ ,: %+" , %%2+ , % /%%2+ ,: %+" , %%2+ , %

%-%' +$!' '( ()+)" %+, , % -'

%2+" )

%+% ' %- 2+" : 1 +$' %++- 5- ()

%%2+" ' % -+'! &' &

% ! '( +$!!

( +$' % '+$ ' + ' %2+"

6'' % ' %+(7' + '+$ ' %2+" !!

%'$ ')'!' +$!' 3%"!' '!

B. ',(1. !

"

!'!' - + $ !$ ' '3"

2" :

1. '5' - ) + % +$' %2+

2. (' )" !&-! % %+(7' 2-+ ! +$! ('%(+! %

%, % ,)%, % ,)

3. %! 4' !&$ ' !"' !'" +(

)$ % !$.

4. ' 2! )" !&-!, !'7' ))!( '%(+! %

%2+" !' + '+$ ' ('%(+! % , % %,)

5. (' '%+% + % ' '%(+!" %2+"

6. /%+(7' %+%+ '%+% $ +(

*. ',(1. !1. +( Fibonacci

#

*: (' - 5! n. %+! '( n-

Fibonacci.

%4 15 ( Fibonacci:

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

1 1 2 3 5 8 13 21 34 55 89 144 233 377 610

. % +( (7' -!, )" !&-!:

1 1 2 3 5 8 13 21 34 55 89 144 233 377 610

>+

===

2,

21,1

21 nff

nnf

nn

n

*. ',(1. !1. +( Fibonacci (1. $%,! )$ +()

$

8 ) + % +%'( %%, ))!( '(

'3":

procedure FibRec(n)

if n=1 or n=2 then

return 1

else

O + -&' %+%+ % %'5' %

)" !&-!:

a=FibRec(n-1)

b=FibRec(n-2)

c=a+b

return c

end if

end procedure

(1), 1 2( )

( 1) ( 2) (1), 2

n nT n

T n T n n

= ==

+ + >

*. ',(1. !1. +( Fibonacci (1. $%,! )$ +()

%

)" !&-! T(n)=T(n-1)+T(n-2)+1 %'( %!'! '( -!,

5 ,

!"

!&$' '( ) '( 1) '( 2) 1 '( 1) '( 1) 1T n T n T n T n T n= + + + +

1)1(2)( += nAnA'%4 +$' )"

1)1(2)( += nAnA

)2(12

12)1(2

122...2)0(2

...

122...2)(2

...

122)3(2

12)2(2

1)1(2)(

11

21

21

23

2

nn

n

nn

kk

A

knA

nA

nA

nAnA

=

+=

=+++++=

==

=+++++=

==

=+++=

=++=

=+=

+

)2()( nnT =

*. ',(1. !1. +( Fibonacci (1. $%,! )$ +()

&

)" !&-! T(n)=T(n-1)+T(n-2)+1 %'( %!'! '( -!,

5 ,

#" !"

!&$' '( ) '( 1) '( 2) 1 '( 2) '( 2) 1T n T n T n T n T n= + + + +

1)2(2)( += nKnK'%4 +$' )"

1)2(2)( += nKnK

)2(12

12)1(2

122...2)0(2

...

122...2)2(2

...

122)6(2

12)4(2

1)2(2)(

2/11)2/(

21)2/(2/

21

23

2

nn

n

nn

kk

K

knK

nK

nK

nKnK

=

+=

=+++++=

==

=+++++=

==

=+++=

=++=

=+=

+

)2()( 2/nnT =

*. ',(1. !1. +( Fibonacci (1. $%,! )$ +()

&&

%+%+ +( '(

( %+$ +!!

, % ' -+'! )" ))!(. .&. n=5:

)2()( 2/nnT = )2()( nOnT =

)5(fib

)4(fib

)3(fib )2(fib

)2(fib )1(fib )1(fib )0(fib

)1(fib )0(fib

)3(fib

)2(fib )1(fib

)1(fib )0(fib

*. ',(1. !1. +( Fibonacci (3."' ..)

&

%+'$' !&$ !"' !'" ' +

$ !$:

/ / .. +( Fibonacci:

) , *-+ ! , %- 4: %+(!'

+$!, %+(7' +$! ! )$ -!, %$' %%2+"-

. .

/%%2+ ,: %+" , %%2+ ,

%-%' +$!' '( (!'- %-%' +$!' n

%%2+" )

%+% ' %- 2+" : 7! '

+$!' %++- 5- () %%2+" , +, % ' -+'!

)" ))!(.

*. ',(1. !1. +( Fibonacci (4. )!( %(+!)

&

) % ! % '(':

+$!' %2+ % , % %,, )+)"

+$!' ))& %%2+" :

Fib(1)

Fib(2) Fib(2)

Fib(n)1 2 3 4 5 6 n

1 1 fib(n)

*. ',(1. !1. +( Fibonacci (5.%+% + 6.+%+ )

&

8 + )$ % !$ % +%'(

%%, ))!( '( '3":

procedure FibSeq(n)

A[1]=1

A[2]=1

for i=3 to n

%+%+ +( '( (n).

A[i]=A[i-1]+A[i-2]

end for

return A[n]

end procedure

*. ',(1. !2. +!), ++%+!! ,

&!

*: () %(' 1,2,,n % %( Ai '(

)! ! di-1 x di. ' % !' (' %++%+!!

- 1 x 2 x x n

$%:

A1 x A2 x A3 % 1: 5x8, A2: 8x4, A3: 4x2

A1 x A2 x A3 x A4 % 1: 10x6, A2: 6x3, A3: 3x8, A4: 8x2

+:

5" +: .' %3' % ! ' % )'3:

4 A1 x A2 = C1 -%' C1 x A3 ... %3' % ( '(:

d0d1d2 + d0d2d3 + + d0dn-1dn + $ !$: %-+'3' !' '

%( ( %++%+!!( , %,. % &'

! " 2'+ (,! ! %+%+ .

*. ',(1. !2. +!), ++%+!! ,

&"

)$' ( -&' 3( - + !$ !' ,%++%+!!4. )':

A1 x A2 x A3 % 1: 5x8, A2: 8x4, A3: 4x2 %3' %$ (:

(A1 x A2) x A3 %+! A1 x A2: 5x8x4=160 %3' %$% ' %+! A1 x A2: 5x8x4=160 %3' %$% '

%( 5x4, -! , C

%+! C x A3: 5x4x2=40 %3' %$% ' %(-' 5 x 2

$+ 200 %3'

A1 x (A2 x A3) %+! A2 x A3: 8x4x2=64 %3' %$% '

%( 8x2, -! , C

%+! 1 x C: 5x8x2=80 %3' %$% ' %(-' 5 x 2

$+ 144 %3'

0 ' ++ !' -, + 4' %++- %3'!

*. ',(1. !2. +!), ++%+!! ,

%'!! ' %(' %& %'!! ' % '

%++%+!!, '+$ ' 5'+ ! %3':

A1 x A2 x A3 x A4 %'( %'' %'( , '3":

((A1 A2) A3) A4)

((A1A2)(A3A4)) ((A1A2)(A3A4))

(A1(A2A3))A4 A1((A2A3)A4)

A1(A2(A3A4))

, '%+-3' +$ ' %'' %(! &,( &'! '(

'+' "!' ' ( 3'&,! ;

*. ',(1. !2. +!), ++%+!! , (1. %(+! ' )")

&$

'%(+! ' )" %2+" !'5 ! ' , '3":

',$' -&' ))!( (i,j) % -&' ") ! '( )+'$'

% %(' , ! +( AiAj %'

2-+ ! %'' %(!.

8! , 4 ' ! %: A1A2An-1An. '%+-3' +$ '

% ':1 2 n-1 n

% ':

A1(A2An)

(A1A2)(A3An)

(A1An-2)(An-1An)

(A1An-1)An %+! 2-+ ! %'' %(! ' %%2+

(' ' +"! ))!( T(i,j)

" " -&' -&' - %( (0 %3')

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

&%

+ % +%'( %%, ))!( ' )" '(

+:

procedure MatMult(Ai,Ai+1,,Aj)

if i=j then

return 0

else

' 2! %%, 2-+ ! %+" , %++%+!!4

+( %, Ai,,Aj )()' % )" !&-!:

for k=i to j-1

ck=MatMult(Ai,,Ak)+MatMult(Ak+1,,Aj)+di-1dkdjend for

end if

return ck k=i,,j-1

end procedure

{ }

*. ',(1. !2. +!), ++%+!! , (4.' %(+! /%%2/ ,)

&

%5!(!' ' % !' %-%' ' %3'% $' % % %%2+" '3 '%(+! '%%2+" .

.&. ',"!' %++%+!7' 5 %(', ' %3' %-% ( ' '3" !':

& !( ' '3" !':

-%' %4 %+(!' ' 1 %(, ' 2 %,, ' 3 %, ...

'+( (i,j) %'$' 2-+ ! %+" %3', % % $ ! %+/! +( %, Ai,Ai+1,Aj

& !

&

!

*. ',(1. !2. +!), ++%+!! , (4.' %(+! /%%2/ ,)

)' -+'! %++%+!! %, 1234,

1: 5x3, A2: 3x4, A3: 4x8, A4:8x2, A5: 2x3

&

!

' '

'

'' ($()"

'

''( ($)%"

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

$$

& '&

'&'!(( )"

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

&"

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

&&$

*. ',(1. !2. +!), ++%+!! , (5. %+% +)

+ % +%'( %%, ))!( !' %3', '(

+:

procedure DP_MatMult(A1,A2,,An)

for i=1 to n

m[i,i]=0

end for

for p=2 to nfor p=2 to n

for i=2 to n-p+1

j=i+p-1

m[i,j]=+

for k=1 to j-1

q=M[i,k]+M[k+1,j]+d[i-1]*d[k]*d[j]

if (q>M[i,j]) then M[i,j]=q

end for

end for

end for

return M[1,n]

end procedure

*. ',(1. !2. +!), ++%+!! , (6. +! +%+ )

+'( +! %+%+ :

%4 for -&' %+%+ (n)

)$ '%' for +%$ ))!( %+!$ , '+4

)4 %, '()' ! ))!( '%(+!. '+ '(

n2/2, %3' 4 , for, ' '+'! $ (n2) 5-.

%3' % ( !' ' '%+: '( (n)

'%4 '+" %+%+ '( O(n3)

*. ',(1. !2. +!), ++%+!! , (%'! $!)

!

+ % -&' !'!':

/%+(7' 2-+ ! +$!, %'$ " 2-+ !

%'' %"! ' +( %, % ! &'( !' -

!- '+( %(.

'%! -5' 2-+ ! +$!, $ ' 5' %, (' 2-+ !

%'' %"!

+ %%$' '+5 4) , 4! ' '

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

*. ',(1. !2. +!), ++%+!! , (%'! $!)

"

%%$' + , '3":

procedure DP_MatMult(A1,A2,,An)

for i=1 to n

m[i,i]=0

end for

for p=2 to n

for i=2 to n-p+1

H ' +" % %! -' !2+(7' !'( )&,!$ %++%+!!$ %, AiAi+1Aj (' ! !'( s[i,j]=k, )+)" 2-+ ! %'' %(! '( (AiAi+1..k)(Ak+1Aj)

for i=2 to n-p+1

j=i+p-1

m[i,j]=+

for k=1 to j-1

q=M[i,k]+M[k+1,j]+d[i-1]*d[k]*d[j]

if (q>M[i,j]) then M[i,j]=q , s[i,j]=k

end for

end for

end for

return M[1,n]

e