7224066-Algorithmic-Problem-Solving-by-Roland-Backhouse.pdf

February 14, 2017 | Author: מוחמד טאהא | Category: N/A
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problem solving using an algorthims...

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Contents 1 Introduction                                                                          2 Invariants                 "            "  #      % & '              "  "

          ( "               )  %'          *            

       

       

       

3 Crossing a River                        +                    , 

  -.        /  %'                         0                 - 1 "    2                     3   " (   3        )  4              (  . 5        ) 

&                 

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

       

1   5 ! ! $   ) $  21

            

        

            

            

            

            

            

             

            

            

            

            

            

            

            

            

            

            

            

            

) *  ! ! $  ( ) $



  

 4 Games ( #   ,           ( -           (            (  6        (  +  5   (   / ,         ((   . ,             (( &

&           (( 7 8 &

 9    (( #          ((( "  #%: +      (() ;   #%: +   () 

&               (*            

             

             

             

             

             

             

             

             

             

             

5 Knights and Knaves ) 6  &          5         5  +4 "       ?  "                                  

9 Knight’s Circuit $  /#   $ 5      $     $( 3         $)  . >       . .&         &     +    4   %            .            +   '           &             6      

  4  4  Abstraction "        &   4   -    p    

 .    4    c    

 .  "   .           .      "  A     

 - K  L .        .   G K LH    &        .      8         > .             &    4  

9 "   &    &     "   

 5& 4         .    .   "         K    L           . 

      K/4L    /4     &  .      I         

     &  & & >             5  .    

         &   

  ' 4  .     

 "  ./  &

&  "  .      &     4 .        4          

"  &   &    S0    5  S1  S2  S3    { 3C || } S1 { 3H || 3W } , { 3H || 3W } S2 { 3W || 3H } ,

{ 3W || 3H } S3 { || 3C } .

"  5  S1     .         4     4                  .  "  5  S2      . S1     4     .   4      +&   5  S3      . S2     4  &          S1 .4 & S2 .4 & S3  4  4  & S1 ; S2 ; S3  4      D  .     .     G&     . H    A  G&       H "            4 4  ' &

&

 &   

 .  4   & &

&   &

   .  . S3 & &

     .  . S1  8. 4     . S3      .    4 .         .   . 4   S1           S3  & .  - 4         .   S1  S2         4              .   & G8          &  .  .  &H =  4             



  



      {

3C ||

}

1C,2H |2W| ;

{

1C,2H || 2W

}

1C,2H |1W| 1W ;

{

2C,1H || 1W

}

3H |2W| 1W {

3H || 3W

} .

"   { 3C || } 1C,2H |2W| ; 1C,2H |1W| 1W ; 3H |2W| 1W

{ 3H || 3W } .

       5  S3     . S1 J {

3W || 3H

}

1W |2W| 3H ;

{

1W || 2C,1H }

1W |1W| 1C,2H ;

{

2W || 1C,2H }

|2W| 1C,2H {

|| 3C

}

.

-  4 .  4     .   S2  -       &

       ?   . S2      .            A      .  "    5  . S2  .4 S1   .4 & S3  "   . S2      4  .         5  8.        &

    &    .4 . J {

3H || 3W

}

T1 ;

1C |1C| 1C

;

T2 {

3W || 3H

} .

        



  

 $   



?  &      C 1C |1C| 1C C  "  &   .//      //. I 4    

&  "    4      &

 5  .   T1  T2  8.       .    .         &    1C || 2C       2C || 1C  @ / .       . .            &    2C || 1C       1C || 2C  "        /.    &    4      +& T1     8  . D 4  J {

3H || 3W

}

3H |1W| 2W ;

{

1C,2H || 2W

}

1C |2H| 2W {

1C || 2C }

.

&

 & . T2 4 J {

2C || 1C }

2W |2H| 1C ;

{

2W || 1C,2H }

2W |1W| 3H {

3W || 3H

} .

+&  &    4          D/    J {

3C ||

}

1C,2H |2W| ; 1C,2H |1W| 1W ; 3H |2W| 1W ;

{

3H || 3W

}

3H |1W| 2W ; 1C |2H| 2W ;

{

1C || 2C }

1C |1C| 1C ;

{

2C || 1C }

2W |2H| 1C ; 2W |1W| 3H

        



  

(

      ;

{

3W || 3H

}

1W |2W| 3H ; 1W |1W| 1C,2H ; |2W| 1C,2H {

|| 3C

}

.

G8           4 "          &         &  "    

       H

3.3.5

A Review

  4   4          D/        &  .&   4   F      /.    "         '   &

& 4   .         

 .  4  &      /      5  4    A    K   L     ?    4 5   . &  .&   .     { p } S { q }    .& 4           "  A     " 1  & . &  

   &          4     &   



     .    &   #               '        +   . .       .4    J Exercise 3.1 (Five-couple Problem) "   A D       &  '  .    3  4            2 Exercise 3.2 (Four-couple Problem) ;.&   &

& 4   .          & /      &

  "        . D       &  '/   .          &

  3          "  .4  &  . +     A  & 4 & 4       .  A       G'  H  & 

.&    . A       . . B       A 4     4&     .     &  .   

        



  

   % &'     

)

G8       &

      .   &

  "       .  .       .       &

 .   .       .H 2 Exercise 3.3  4   .       '  . 4         .          4   .       '  .           '         =J          "     J

   . .          "    &    2

3.4

Rule of Sequential Composition

"  { p } S { q }  4  .    D/         .  .&        8    !

  G "& =        4    5 . . & .&     .    I  4  .   A     H       A &   4          "  4     A &  /     p         A &      q         .      8. S      p  q   .     

pSq

   .      .& & p . '  .   S  '  . S       .4      4 .& & q  + '           r   d .  

 M & 

 N 4    N = 0 ,

        



  

*

     

G   & 0   4H    M = N×d + r ∧ 0 ≤ r < N .

8.     S     A  .       { N = 0 } S { M = N×d + r ∧ 0 ≤ r < N } .

   .   & 5 I       '   .          &    5  "  . S1  S2  S3    S1 ; S2 ; S3        '  & A '  S1    '  S2     '  S3  "      "       . S1  S2  S3   5         4                 8    .      4       p     q      r &   "    .     S .&    A 

pSq     &  S  S1 ; S2    S1  S2  .&    A 

pS r 1



rS q 2

.

"     r        . S1    . S2  8.                4      "   4  4      D/     "    

     3C ||    || 3C  "     3H || 3W  3W || 3H 4           A    "   F 4& .     . 5   "    .      &       &     &       S1 I        .&         A   S2  &     &    A   S2 I          . .&      r       S1                    J         



  

   % &'     



+  4     8       &     4        & &    4    "    4  & 4       &    "  .   F   .       I 4  4     &        .   4 "  A   1        2      5     . 10  "     .    .           4&  4        4    .      4  17  8  &     A       .       "  A       4  4      4 4  4     4          & 4       .    4       &            4 4       D& "  5  4    &       & "          '              4    4 4    G8 1 4&         &      4 4     '  (H ;      A .       4  .   .   p  q    .   .

p |5,10| q

     ' &

&        .   5  .  S1  S2   p  q    { 1,2,5,10 || } S1 { p,5,10 || q } { p,5,10 || q } p |5,10| q { p || q,5,10 }

 { p || q,5,10 } S2 { || 1,2,5,10 } .

-  &       Exercise 3.4 (The Torch Problem)                  .   .  .         A   t1        t2      t3     . t4      t1 ≤ t2 ≤ t3 ≤ t4  +    .   .    

         B &          & .4   4   &      & &     .4 4 J

        



  

!

     

(a) "      1  1  3   3  (b) "      1  4  4   5 

=J 8    A      G4  t1  t2  t3  t4  1  2  5  10   &H          &    4 4    =4   1 4&    &   &      .    &  .         B 4         .     4       .     8         & 4    .4  G?    GH  G H 4 &    H (a) =4 &                   .   2 G8          H =4 & .     2 (b)

    

 .     D&    4   

 .  4   & &     

 .  4       2 -  .   .  4 &       2 G,        &  H

(c) ?4 .   4      4           

 -     F 2 %      .   =     .  .           .         (d) ,          ;        4  &  

?     '          A      "    D             &   &    4      &     "     4 .& 4      4  5 2 Exercise 3.5   /.            G'  (  H "  4       F 4& .  .       =4 & 4&   2 2

        



  

 & (

3.5

$

Summary

8    4   /.   4      /.     &       #    &

             .          /      /  '          .      4    4.    '&  & 4  & 4 &         .      I   4 

  .       &     4      L "    4      ' 

 . K>L  F .   ' 

 . KL "     G 3  2 H    4 &              KL 4       ' 

  K>L "               .         4 ' 

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  5  &       .  ?     .     .   .      4       15  .   .     11  .              ' 

 .   8   . 4            



  

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1

4

5

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 2 

  26 F      D    G     .   4H . 15  11            165  "          F #    4    
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