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ΠΛΗ30 ΜΑΘΗΜΑ 1.5 (4in1)

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ΠΛΗ30 ΜΑΘΗΜΑ 1.5 (4in1)

  1. 1. 30 1: 1.5: T(n)=aT(n-b)+f(n) T(n)=T(n/a)+T(n/b)+f(n)T(n)=T(n/a)+T(n/b)+f(n) ! " # $ " !" #$ %& ' ( ( ) ( ' * (+( , ! # %- ' ( ( ) ( ' * (+( !" # !$ #% & ' " ($ #% . ! /" % /$ %- ' ( ( ) 0 * , ' ( 1) ( . ' ( ( 1 . 2 3 4 . %&" ! ' ! " !& ! ' ! " () : %)%( ! &! ! 3.6 * ! (%)+' ! " T(n)=T(n/a)+T(n/b)+f(n) " (% + , " * ! (%)+' ! " T(n)=T(n-1)+f(n) %)%( - !" # !$ #% & ' " ($ #% %)%( - " * ! (%)+' ! " T(n)=T(n/a)+T(n/b)+f(n) " (% + , " * ! (%)+' ! " T(n)=aT(n-b)+c %)%( " '% + * $ . * !/ * ! (%)+' ! " T(n)=T(n/a)+T(n/b)+f(n) B. ( + *) (/ T(n)=aT(n-b)+f(n) )!" # !$ #% & ' " ($ #% H (%)+' ! " " " T(n)=aT(n-b)+f(n) *) (! ( ! (% + , . (+(! ' ( $ ( " %( %!0 ( " '! " ! " ": f(n)=c, %&!( % $%!( T(n)=aT(n-b)+c % !() ! + ! " (% + , " % ' () ( ! % * $ ( .. a=1, % $%!( T(n)=T(n-b)+f(n) % ' +$ (! ( ($ + ( %( & ! &% % ' % !(+() % ++ * ! " ( & ' (% + , ". *( . ! " " * ( a, b, f(n) () ( !&" $+ ".
  2. 2. -. ( + *) (/ 1. T(n)=aT(n-b)+c !" # !$ #% & ' " ($ #% T(n)=aT(n-b)+c +$ (! ( ! (% + , " 1. 1 ' ( 3 (. * " ! " " " ( .! ' ( ! . ) 2. !) ! " ( " % ' % $%!( (! %& k (% + ,( " ( " ( ) ....3...)( +−⋅= bnTnT ( )+−⋅=*() & " ) 3. % + *)2 ' ( %&!( ! ! ( ( !/ &% ' n0 ' !( ! $ ! " " +$ ' ( /" % " k). . . n0=0 !&!( n/b 4. ! ! ! ! ' k ! & !$% ! ' 3 ! " 2. 5. % + * &" ! ' ) ! " % ' % ',(. ( ) .......)( +−⋅= kbnTnT 0nkbn =− B. ( + *) (/ 1. T(n)=aT(n-b)+c 1. %)+' ( ! (% + , " (- 1: 3 (. * " ! ' & ) *!" # !$ #% & ' " ($ #% ! 1 3 (. &2 ' ( ! & & 3 . " ' ( ! " % 4( " % ' % $%! ' . : +$ (!( ! : : ( ) = >+− = 0,1 0,225 )( n nnT nT αν αν : ( ) ( )[ ] ( ) ( )[ ] ( ) 22525652252655 2254522455 225)( 232 2 +⋅+⋅+−=+⋅++−= +⋅+−=++−= +−= nTnT nTnT nTnT B. ( + *) (/ 1. T(n)=aT(n-b)+c 1. %)+' ( ! (% + , " (- 2: !) ! 3 k) +!" # !$ #% & ' " ($ #% ! 2 3 ( ! ( ! ( % ' % $,( (! %& k (% + ,( " ( " *() & " (… ' ( …) ( ) .......)( +−⋅= kbnTnT ( ) ... 2252565)( 23 == =+⋅+⋅+−= nTnT ( ) 22525...2525 ... 21 +⋅+⋅++⋅+−= == −kk knT B. ( + *) (/ 1. T(n)=aT(n-b)+c 1. %)+' ( ! (% + , " (- 3: % + * &" ! ' k) ,!" # !$ #% & ' " ($ #% % + *)2 ' ( %&!( ! ! ( ( !/ &% ' n0 ' !( ! $ ! " " +$ ' ( /" % " k). (… ' ( …) 0nkbn =− (… ' ( …) ! ! &! 2/ 2 02 nk kn kn = = =−
  3. 3. B. ( + *) (/ 1. T(n)=aT(n-b)+c 1. %)+' ( ! (% + , " (- 4: ! ! ! ! ' k) -!" # !$ #% & ' " ($ #% ! ! $ ( ! k % ' 3 ( ! % ! % ' % ',( ! 3 2. % %( % +( . () &" & " ( ! ' !( ! $ ! " ". (… ' ( …) ! ! " k=n/2 ! (n) ' (: ( ) 22525...255 22525...2505)( 212/2/ 212/2/ +⋅+⋅++⋅+= +⋅+⋅++⋅+= − − nn nn TnT B. ( + *) (/ 1. T(n)=aT(n-b)+c 1. %)+' ( ! (% + , " (- 5: % + * &" ) ! ") !" # !$ #% & ' " ($ #% % ' % $%!( '% + *)2(! ( ! * / !& !$% ! ' '% + * $ ) ! " & / *(/ (! " % & ': (… ' ( …) = + − − = n i n i x x x 0 1 1 1 ! ! " k=n/2 ! (n) ' (: 5,055,1 )15(5,05 15 15 25 525 ]5...551[25 ]25...25252[5 22525...255)( 2/ 2/2/ 112/ 2/ 12/ 0 2/ 12/22/ 12/22/ 212/2/ −⋅= =−+= = − − += =+= =+++++= =⋅++⋅+⋅++= =+⋅+⋅++⋅+= +− − = − − − n nn n n n i in nn nn nn nT -. ( + *) (/ 2. T(n)=T(n-b)+f(n) !" # !$ #% & ' " ($ #% T(n)=T(n-1)+f(n) +$ (! ( ! (% + , " 1. . ' ( &+ '" ! '" $" & '" T(n), T(n-1),… ! %( )%!/ ! " " 2. ! ' ( ! " (4 0 ( " ! + 3. % + *)2 ' ( ! % ' % $%!( B. ( + *) (/ 2. T(n)=aT(n-b)+c 1. %)+' ( ! (% + , " (- 1: , !/ & / ) !" # !$ #% & ' " ($ #% ! 1 3 * . ' ( &+ '" ! '" $" & '" %& ! T(n) ! & T(n0) &% ' n0 () %( )%!/ ! " " : +$ (!( ! : : ( )1 3 , 0 ( ) 1, 0 T n n n T n n αν αν − + > = = : 1)0( 13)0()1( 23)1()2( ... )2(3)3()2( )1(3)2()1( 3)1()( = ⋅+= ⋅+= −+−=− −+−=− +−= T TT TT nnTnT nnTnT nnTnT
  4. 4. B. ( + *) (/ 2. T(n)=aT(n-b)+c 1. %)+' ( ! (% + , " (- 2: & ( ! + ) !" # !$ #% & ' " ($ #% ! 2 3 % ! ' ( ! + ! " ( " % ' ' ( * ,( ! % * $ ( 3 : 3)1()( +−= nnTnT 1)0( 13)0()1( 23)1()2( ... )2(3)3()2( )1(3)2()1( 3)1()( = ⋅+= ⋅+= −+−=− −+−=− +−= T TT TT nnTnT nnTnT nnTnT )(+ 11323...)2(3)1(33)( +⋅+⋅++−+−+= nnnnT B. ( + *) (/ 2. T(n)=aT(n-b)+c 1. %)+' ( ! (% + , " (- 3: % + * &" ! ' ) ! ") )!" # !$ #% & ' " ($ #% ! 3 3 '% + *)2 ' ( ! % ' % $%!( % ' ' /" () ! % & ". 5 . $ ! (4 " ) ! : = + = n i nn i 1 2 )1( = ++ = n i nnn i 1 2 6 )12)(1( = + Θ= n i kk ni 1 1 )( ( & ( '%& ( 4 ) ( ' ( …) 15,15,1 2 )1( 3131 ])1()2(...21[31 3)1(3)2(3...23131 11323...)2(3)1(33)( 2 1 ++= = + +=+= +−+−++++= +−+−++⋅+⋅+= +⋅+⋅++−+−+= = nn nn i nnn nnn nnnnT n i B. ( + *) (/ T(n)=T(n/a)+T(n/b)+f(n) !" # !$ #% & ' " ($ #% H (%)+' ! " " " T(n)=T(n/a)+T(n/b)+f(n) *) (! : ( (. * ! " ! &! ! " 3.6 ( ) ( ! " ( ) 1 11 ≤+ ba 1 11 >+ ba ( ! " ( ) % ( " '% + * $ . * !/ ! % ) ( (. &2 ' ( % ! , % & " ! 2 ! ( ! ! 1>+ ba -. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) *!" # !$ #% & ' " ($ #% T(n)=T(n/a)+ (n/b)+f(n) +$ (! ( ! ! ' ' " 1. %!'4 ! ' ' 0 1+ (/ ( ! 2 (%)%( 2. ( ( & 3 ( 0 ' ( %& (" % 4( " *) ! ( %& ! f(n))2. ( ( & 3 ( 0 ' ( %& (" % 4( " *) ! ( %& ! f(n)) 3. % + * &" % 4(/ (%)%( ( ' /" *(/ (! % & ") 4. % + * &" ! ' $, '" ! ' ' ( ) logcn ( c ! (+ ! %& ! a,b) 5. T(n)=! !/ % 4(/ &+/ !/ (% % /
  5. 5. B. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) 1. %)+' ( ! " (- 1: %!'4 ' 2 (%)%( ) +!" # !$ #% & ' " ($ #% ! 1 3 %!$ ' ( ! 0 + (/ ( . )2 ! " & ! '" $" & '" (&%/" * & ! ' + ( " ! !) ! & 0 ). : +$ (!( ! : = >++ = 1,1 1, 32)( n nn n T n T nT αν αν : =1,1 nαν )(nT 2 n T 3 n T 4 n T 3963 2642 32 )( nn T n T n T nn T n T n T n n T n TnT ++= ++= ++= 5 6 n T 6 n T 9 n T B. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) 1. %)+' ( ! " (- 2: 4( " ( ( & 3 ) ,!" # !$ #% & ' " ($ #% ! 2 3 ( 0 ' ( ( ( & 3 %& (" % 4( " *) ! ( '! ! + ( )2(! %& ! & % ' ' ( ( . ) ( ! !0 ! " ! f(n) ) (… ' ( …) )(nT 2 n T 3 n T 4 n T 6 n T 6 n T 9 n T n 2 n 3 n 4 n 6 n 6 n 9 n 3963 2642 32 )( nn T n T n T nn T n T n T n n T n TnT ++= ++= ++= 5 B. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) 1. %)+' ( ! " (- 3: 4( " (%)%( ) -!" # !$ #% & ' " ($ #% ! 3 3 % ! ' ( (%)%( ! " % 4( " * " 3*( + . &! % ! " 3* ) ( &! () *(/ (! % & ". ! ( %& (" % 4( " *) ! ! (%)%( i. (… ' ( …) )(nT 2 n T 3 n T 4 n T 6 n T 6 n T 9 n T n 2 n 3 n 4 n 6 n 6 n 9 n % %( 0: n % %( 1: 6 5 6 2 6 3 32 nnnnn =+=+ % %( 2: 2 2 6 5 ... 9664 nnnnn ==+++ 6 ! (%)%( i *) ! % 4( "i i n 6 5 B. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) 1. %)+' ( ! " (- 4: % + * &" $, '" ') !" # !$ #% & ' " ($ #% ! 4 3 '% + *)2 ' ( ! $, " ! ' '. $, " ! ' ' )2(! %& % " & " %& ! '" n/a kai n/b .! ( % * *) ( ) " ( ! n0, + +$ ! " ! (4) / n/min{a,b}=n0 %( ! $, " ! ' ' )2(! %& ! &!( %& ! '" $ % ! " '* ( () c () &!( " %& ! '" $ % ! " ( +. c=min{a,b} ) %(! &! ! $, " ! ' ' () logcn.% ! " ( +. c=min{a,b} ) %(! &! ! $, " ! ' ' () logcn. (… ' ( …) $, " ! ' ' () log2n
  6. 6. B. ( + *) (/ 2. T(n)=T(n/a)+T(n/b)+f(n) 1. %)+' ( ! " (- 5: % + * &" ) ! ") !" # !$ #% & ' " ($ #% ! 5 3 '% + *)2 ' ( ! % +'%+ &! ! /" ! !/ % 4(/ &+/ !/ (% % / . () % ! *(/ (! % & ". 6 % ' ( ! !$% : (… ' ( …) = + − − = n i n i x x x 0 1 1 1 ' (%0" % 4( " () : ( ) log log 0 0 log 1 log 0 log 1 5 ( ) 5 6 6 5 1 5 6 56 1 6 6 0,83 6 i n ni i ii i n i n i n n T n n n n n n = = + = + = = = − = = = − = ⋅ − -. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) 2. %)+' ( ! . * ! !" # !$ #% & ' " ($ #% T(n)=T(n/a)+ (n/b)+f(n) +$ (! ( ! !/ . * !/ 1. % + * &" !/ . * ! " ( ! (* +$!( %& ! a b ! (0 ' ) ". 2. % + * &" / . * ! " ( ! &!( %& ! a b ! (0 ' ) ". 3. ! !/ . * () ) ( ! / . * ' ( ' %!/! ( !) ! " ' ! " % +'%+ &! ! ". ++ 0" " -. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) 2. %)+' ( ! . * ! !" # !$ #% & ' " ($ #% * . ' ( ! " $ " ( " / !/ % )/ '% + *) ' ( ! / ! !/ . * . / . * % $,( ( ! &!( %& ! a,b ! !/ . * % $,( ( ! (* +$!( %& ! a,b : '% + *) (!( ' %!/! ( !) ! " ": 2 43 )( n n T n TnT ++= : / . * % $,( %& ! (%)+' ! " " ": 1 ! !/ . * % $,( %& ! (%)+' ! " " ": 2 ( ) 2 3 n n nΑ = Α + 2 ( ) 2 4 n n nΚ = Κ + -. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) 2. %)+' ( ! . * ! (1. % + * &" ! ' / . * ! ") )!" # !$ #% & ' " ($ #% '% + * &" ! ' / . * ! " *) ( ( ! (0 ' ) ". !" #$ %& 2 3 2)( n n AnA += 63,02loglog,)(,3,2 2 ===== annfba7 /: $( : * % ! ( (>0 + * / '% ( c<1 ! ! 0 !(: 6 $( * 2/9 8 c < 1. 6 %& ! ’ %( )%!/ ! ' (/ ! " 1' ) " %(! &! : 6 63,02loglog,)(,3,2 3 2 ===== annfba b )()( 63,02 ε+ Ω== nnnf )()( 2 nnA Θ= ccn n cn n ncf n fncf b n af ≤⇔≤⇔≤⇔≤⇔≤ 9 2 3 2 3 2)( 2 4)( 2 2 2 2 2 )()( 2 nnT Ο=
  7. 7. -. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) 2. %)+' ( ! . * ! (2. % + * &" ! ' !/ . * ! ") !" # !$ #% & ' " ($ #% '% + * &" ! ' / . * ! " *) ( ( ! (0 ' ) ". ' &" #$ %& 2 4 2)( n n KnK += 5,02loglog,)(,4,2 2 ===== annfba7 /: $( : * % ! ( (>0 + * / '% ( c<1 ! ! 0 !(: 6 $( * 1/8 8 c < 1. 6 %& ! ’ %( )%!/ ! ' (/ ! " 1' ) " %(! &! : 6 5,02loglog,)(,4,2 4 2 ===== annfba b )()( 5,02 ε+ Ω== nnnf )()( 2 nnK Θ= cccn n cn n ncf n fncf b n af ≤⇔≤⇔≤⇔≤⇔≤⇔≤ 8 1 16 2 4 2 4 2)( 2 4)( 3 2 2 3 2 )()( 2 nnT Ω= -. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) 2. %)+' ( ! . * ! (3. ' % * ! ' %!/! % +/! ) *!" # !$ #% & ' " ($ #% ! / . * ! !/ . * () ) , !&!( ' ( ' %!/! ( !) ! " % +'%+ &! ! " ! '. (... (!)*+ %…) ' (%0" %& ! % * $ ( : )()( 2 nnT Ο= ! . * ! () . (! , " !/ . * !/ ( % !$ ( ! $( : ' (%0" )()( 2 nnT Ο= )()( 2 nnT Ω= )()( 2 nnT Θ= -. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) 3. %)+' ( ! ! &! ! 3.6 +!" # !$ #% & ' " ($ #% T(n)=T(n/a)+ (n/b)+f(n) +$ (! ( ! ! &! ! 3.6 ! ' 3 3+) ' 3.6 % + *)2 ' ( ! % &! ! ba 11 + 1. !&!( 2. !&!( 3. !&!( ! &! ! 3.6 ( % !$ ( % ( '% (/! ( " ba + 1 11 <+ ba ))(()( nfnT Θ= 1 11 =+ ba )log)(()( nnfnT ⋅Θ= 1 11 >+ ba B. ( + *) (/ 3. T(n)=T(n/a)+T(n/b)+f(n) 3. %)+' ( ! ! &! ! 3.6 ( ()* ! ) ,!" # !$ #% & ' " ($ #% (. * ! " ! &! ! " 3.6 () % +$ ($ + &! " ) ( ! ! +$ ( % (" ". : '% + *) (!( ' %!/! ( !) ! " ": : 2 43 )( n n T n TnT ++= : $( : %& ! !.3.6 $( : : '% + *) (!( ' %!/! ( !) ! " ": : $( : %& ! !.3.6 $( : 1 12 7 12 3 12 4 4 1 3 1 <=+=+ )()( 2 nnT Θ= n n T n TnT ++= 3 2 3 )( 1 3 3 3 2 3 1 ==+ )log()( nnnT ⋅Θ=
  8. 8. . ( " . * 1 % + *) !( ! 3 +$ !/ 0 ( ! (% + , ": -!" # !$ #% & ' " ($ #% ( )4 3 5, 0 ) ( ) 0, 0 T n n A T n n αν αν − + > = = 2 0, 0 ( 1) 2 , 0 ) ( ) 1, 0 n T n n n B T n n αν αν αν = − + > = = . ( " . * 2 % + *) !( ' %!/! ( !) !/ 0 % 0 ! " ! ": !" # !$ #% & ' " ($ #% = >++ = 1,1 1, 43)() 2 n nn n T n T nTA αν αν = >++ = = 1,1 1, 3 2 3)() 1,1 n nn n T n T nTB n αν αν αν . ( " . * 3 5 % () !( ! '% + * $ . * !/ * ! (%)+' ! " " !" # !$ #% & ' " ($ #% 2 3 2 2 )( n n T n TnT ++= . ( " . * 4 % + *) !( ' %!/! ( !) !/ 0 !" # !$ #% & ' " ($ #% log 44 3 )() n n T n TnTA ++= 2 3 23 2 )() 65 2 )() n n T n TnTC n n T n TnTB ++= ++=
  9. 9. . ( " . * 5 ! (%)+' ( &" % 3+ ! " ' ( ! ( " ! ()" +*& $". • (A1) % 0! " +*& " * (% +$ ( % &3+ (* '" n, (% +$( (%! '% % 3+ ! (* '" n/3 ! ' ' 2( ! " +$ ( " ! '" ( & n3. !" # !$ #% & ' " ($ #% • (A2) ($!( " +*& " * (% +$ ( % &3+ (* '" n, (% +$( '% % 3+ ! (* '" n/2 ! ' ' 2( ! " +$ ( " ! '" ( & n. • (A3) ! )! " +*& " (% +$( '% % &3+ (* '" n-1 3 ) ( ! +$ ! ' $ % 3+ ! " ( & n3. 3 ( $ ' %!/! ) & (%)+' " ! ' % 3+ ! " * ! ( +*& , (% + 4(!( ! ! $!( +*& * ! (%)+' ! ' % 3+ ! ". 9 : % ()!( % (!( &! = Θ= n i ni 1 43 )(

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