ΠΛΗ30 ΜΑΘΗΜΑ 1.5

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ΠΛΗ30 ΜΑΘΗΜΑ 1.5

Transcript of ΠΛΗ30 ΜΑΘΗΜΑ 1.5

  • 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)

  • .

    B.

    1. T(n)=aT(n-b)+c

    1.

    2. T(n)=T(n-1)+f(n)

    1.

    2 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    3. T(n)=T(n/a)+T(n/b)+f(n)

    1.

    2.

    3. .3.6

    .

  • .

    :

    3.6 T(n)=T(n/a)+T(n/b)+f(n) T(n)=T(n-1)+f(n)

    3 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+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)

    4 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    H T(n)=aT(n-b)+f(n) .

    :

    f(n)=c, T(n)=aT(n-b)+c

    ..

    a=1, T(n)=T(n-b)+f(n)

    .

    a, b, f(n) .

  • . 1. T(n)=aT(n-b)+c

    5 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    T(n)=aT(n-b)+c

    1. 3 ( )

    2. k ( ( ) ....3...)( += bnTnT

    ( )+= )3. ( n0

    k). .. n0=0 n/b

    4. k 2.

    5. .

    ( ) .......)( += kbnTnT

    0nkbn =

  • B. 1. T(n)=aT(n-b)+c1. ( 1: 3 )

    6 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    1 3 .

    : :

    :

    ( )

    =

    >+=

    0,10,225)(

    n

    nnTnT

    :

    ( )( )[ ] ( )( )[ ] ( ) 22525652252655

    2254522455225)(

    232

    2

    +++=+++=

    ++=++=

    +=

    nTnTnTnT

    nTnT

  • B. 1. T(n)=aT(n-b)+c1. ( 2: k)

    7 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    2 k (

    ()

    ( ) .......)( += kbnTnT

    ( )...

    2252565)( 23==

    =+++= nTnT

    ( ) 22525...2525...

    21 +++++=

    ==

    kk knT

  • B. 1. T(n)=aT(n-b)+c1. ( 3: k)

    8 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    ( n0 k).

    ()

    0nkbn =

    ()

    2/2

    02

    nkknkn

    =

    =

    =

  • B. 1. T(n)=aT(n-b)+c1. ( 4: k)

    9 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    k 2. .

    ()

    k=n/2 (n) :( )

    22525...25522525...2505)(

    212/2/

    212/2/

    +++++=

    +++++=

    nn

    nn TnT

  • B. 1. T(n)=aT(n-b)+c1. ( 5: )

    10 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    :

    ()

    =

    +

    =

    n

    i

    ni

    x

    xx

    0

    1

    11

    k=n/2 (n) :

    5,055,1)15(5,05

    151525

    525

    ]5...551[25]25...25252[5

    22525...255)(

    2/

    2/2/

    112/2/

    12/

    02/

    12/22/

    12/22/

    212/2/

    =

    =+=

    =

    +=

    =+=

    =+++++=

    =+++++=

    =+++++=

    +

    =

    n

    nn

    nn

    n

    iin

    nn

    nn

    nnnT

  • . 2. T(n)=T(n-b)+f(n)

    11 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    T(n)=T(n-1)+f(n)

    1. T(n), T(n-1),

    2.

    3.

  • B. 2. T(n)=aT(n-b)+c1. ( 1: )

    12 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    1 T(n) T(n0) n0 : :

    :

    ( )1 3 , 0( )1, 0T n n n

    T nn

    + >=

    =:

    1)0(13)0()1(23)1()2(

    ...

    )2(3)3()2()1(3)2()1(

    3)1()(

    =

    +=

    +=

    +=

    +=

    +=

    TTTTT

    nnTnTnnTnT

    nnTnT

  • B. 2. T(n)=aT(n-b)+c1. ( 2: )

    13 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    2 :

    3)1()( += nnTnT

    1)0(13)0()1(23)1()2(

    ...

    )2(3)3()2()1(3)2()1(

    3)1()(

    =

    +=

    +=

    +=

    +=

    +=

    TTTTT

    nnTnTnnTnT

    nnTnT

    )(+

    11323...)2(3)1(33)( ++++++= nnnnT

  • B. 2. T(n)=aT(n-b)+c1. ( 3: )

    14 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    3 . :

    =

    +=

    n

    i

    nni1 2

    )1(

    =

    ++=

    n

    i

    nnni1

    2

    6)12)(1(

    =

    +=n

    i

    kk ni1

    1)(( )

    ()

    15,15,12

    )1(3131

    ])1()2(...21[313)1(3)2(3...23131

    11323...)2(3)1(33)(

    2

    1

    ++=

    =

    ++=+=

    ++++++=

    ++++++=

    ++++++=

    =

    nn

    nni

    nnn

    nnn

    nnnnT

    n

    i

  • B. T(n)=T(n/a)+T(n/b)+f(n)

    15 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    H T(n)=T(n/a)+T(n/b)+f(n) :

    3.6 ( )

    ( )

    111 +ba

    111 >+ba

    ( )

    , !

    1>+ba

  • . 3. T(n)=T(n/a)+T(n/b)+f(n)

    16 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

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

    1. 2 2. ( f(n))2. ( f(n))3. ( )4. ( logcn c

    a,b)5. T(n)=

  • B. 3. T(n)=T(n/a)+T(n/b)+f(n)1. ( 1: 2 )

    17 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    1 ( ).

    : :

    =

    >+

    +

    =

    1,1

    1,32)(

    n

    nnnTnT

    nT

    :

    =1,1 n

    )(nT

    2nT

    3nT

    4nT 3963

    2642

    32)(

    nnTnTnT

    nnTnTnT

    nnTnTnT

    +

    +

    =

    +

    +

    =

    +

    +

    =

    6nT

    6nT

    9nT

    N N N NO O O O

  • B. 3. T(n)=T(n/a)+T(n/b)+f(n)1. ( 2: )

    18 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    2 ( f(n) )

    ()

    )(nT

    2nT

    3nT

    4nT

    6nT

    6nT

    9nT

    n

    2n

    3n

    4n

    6n

    6n

    9n 3963

    2642

    32)(

    nnTnTnT

    nnTnTnT

    nnTnTnT

    +

    +

    =

    +

    +

    =

    +

    +

    =

    N N N NO O O O

  • B. 3. T(n)=T(n/a)+T(n/b)+f(n)1. ( 3: )

    19 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    3 . .

    i.()

    )(nT

    2nT

    3nT

    4nT

    6nT

    6nT

    9nT

    n

    2n

    3n

    4n

    6n

    6n

    9n

    0: n

    1:6

    56

    26

    332

    nnnnn=+=+

    2: 22

    65

    ...

    9664nnnnn

    ==+++

    i iin

    65

  • B. 3. T(n)=T(n/a)+T(n/b)+f(n)1. ( 4: )

    20 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    4 . n/a kai n/b n0, n/min{a,b}=n0

    c (. c=min{a,b} ) logcn. (. c=min{a,b} ) logcn.

    ()

    log2n

  • B. 2. T(n)=T(n/a)+T(n/b)+f(n)1. ( 5: )

    21 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    5 . .

    :

    ()

    =

    +

    =

    n

    i

    ni

    x

    xx

    0

    1

    11

    :

    ( )

    log log

    0 0

    log 1

    log

    0

    log 1

    5( ) 56 6

    5 15 656 16

    6 0,83 6

    in ni

    i ii i

    n

    in

    i

    n

    nT n n

    n n

    n n

    = =

    +

    =

    +

    = = =

    = = =

    =

  • . 3. T(n)=T(n/a)+T(n/b)+f(n)2.

    22 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

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

    1. a b .

    2. a b .

    3. .

  • . 3. T(n)=T(n/a)+T(n/b)+f(n)2.

    23 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    .

    a,b a,b: : 2

    43)( nnTnTnT +

    +

    =

    : :

    :

    2( ) 23n

    n n = +

    2( ) 24n

    n n = +

  • . 3. T(n)=T(n/a)+T(n/b)+f(n)2. (1. )

    24 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    .

    2

    32)( nnAnA +

    =

    63,02loglog,)(,3,2 2 ===== annfba:: >0 c

  • . 3. T(n)=T(n/a)+T(n/b)+f(n)2. (2. )

    25 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    .

    2

    42)( nnKnK +

    =

    5,02loglog,)(,4,2 2 ===== annfba:: >0 c

  • . 3. T(n)=T(n/a)+T(n/b)+f(n)2. (3. /)

    26 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    ,

    .

    (...) :

    )()( 2nnT =

    , !

    :

    )()( 2nnT =

    )()( 2nnT =

    )()( 2nnT =

  • . 3. T(n)=T(n/a)+T(n/b)+f(n)3. 3.6

    27 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    T(n)=T(n/a)+(n/b)+f(n) 3.6 3.6

    ba11

    +

    1.

    2.

    3. 3.6

    ba+

    111 +ba

  • B. 3. T(n)=T(n/a)+T(n/b)+f(n)3. 3.6 ()

    28 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    3.6 .

    : :

    :

    2

    43)( nnTnTnT +

    +

    =

    :: .3.6 :

    : :

    :: .3.6 :

    1127

    123

    124

    41

    31

  • . 1 :

    29 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    ( )4 3 5, 0) ( )0, 0T n n

    A T nn

    + >=

    =

    2

    0, 0

    ( 1) 2 , 0) ( )1, 0

    n

    T n n nB T n

    n

    =

    + >=

    =

  • . 2

    :

    30 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    =

    >+

    +

    =

    1,1

    1,43)()

    2

    n

    nnnTnT

    nTA

    =

    >+

    +

    =

    =

    1,1

    1,3

    23)()

    1,1

    n

    nnnTnT

    nTB

    n

  • . 3

    31 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    2

    32

    2)( nnTnTnT +

    +

    =

  • . 4

    32 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    log44

    3)() nnTnTnTA +

    +

    =

    2

    3

    232)()

    652)()

    nnTnTnTC

    nnTnTnTB

    +

    +

    =

    +

    +

    =

  • . 5 .

    (A1) n, n/3 n3.

    33 , 30, 1.5: T(n)=T(n/a)+T(n/b)+f(n) T(n)=aT(n-b)+f(n)

    (A2) n, n/2 n.

    (A3) n-1 n3.

    , .

    : =

    =n

    ini

    1

    43 )(