الحد الأقصى من المرايا التي يمكنها نقل الضوء من الأسفل إلى اليمين

يتم إعطاء مصفوفة مربعة تمثل فيها كل خلية إما فراغًا أو عائقًا. يمكننا وضع المرايا في وضع فارغ. سيتم وضع جميع المرايا بزاوية 45 درجة، أي يمكنها نقل الضوء من الأسفل إلى اليمين في حالة عدم وجود أي عائق في طريقها. 

في هذا السؤال، علينا حساب عدد المرايا التي يمكن وضعها في مصفوفة مربعة يمكنها نقل الضوء من الأسفل إلى اليمين. 

أمثلة: 

Output for above example is 2. In above diagram mirror at (3 1) and (5 5) are able to send light from bottom to right so total possible mirror count is 2. 

يمكننا حل هذه المشكلة عن طريق التحقق من موضع هذه المرايا في المصفوفة، فالمرآة التي يمكنها نقل الضوء من الأسفل إلى اليمين لن يكون لها أي عائق في طريقها، أي. 
إذا كانت هناك مرآة في الفهرس (i j) إذن 
لن يكون هناك أي عائق عند الفهرس (k j) لجميع k i < k <= N 
لن يكون هناك عائق عند الفهرس (i k) لجميع k j < k <= N 
مع الأخذ في الاعتبار المعادلتين أعلاه، يمكننا العثور على العائق الموجود في أقصى اليمين في كل صف في تكرار واحد لمصفوفة معينة ويمكننا العثور على العائق السفلي في كل عمود في تكرار آخر لمصفوفة معينة. بعد تخزين هذه المؤشرات في مصفوفة منفصلة يمكننا التحقق من كل مؤشر فيما إذا كان لا يفي بأي شرط من العوائق أم لا ثم نزيد العدد وفقًا لذلك. 

يوجد أدناه حل مطبق على المفهوم أعلاه والذي يتطلب وقت O(N^2) ومساحة إضافية O(N).

C++
   // C++ program to find how many mirror can transfer   // light from bottom to right   #include          using     namespace     std  ;   // method returns number of mirror which can transfer   // light from bottom to right   int     maximumMirrorInMatrix  (  string     mat  []     int     N  )   {      // To store first obstacles horizontally (from right)      // and vertically (from bottom)      int     horizontal  [  N  ]     vertical  [  N  ];      // initialize both array as -1 signifying no obstacle      memset  (  horizontal       -1       sizeof  (  horizontal  ));      memset  (  vertical       -1       sizeof  (  vertical  ));      // looping matrix to mark column for obstacles      for     (  int     i  =  0  ;     i   <  N  ;     i  ++  )      {      for     (  int     j  =  N  -1  ;     j  >=  0  ;     j  --  )      {      if     (  mat  [  i  ][  j  ]     ==     'B'  )      continue  ;      // mark rightmost column with obstacle      horizontal  [  i  ]     =     j  ;      break  ;      }      }      // looping matrix to mark rows for obstacles      for     (  int     j  =  0  ;     j   <  N  ;     j  ++  )      {      for     (  int     i  =  N  -1  ;     i  >=  0  ;     i  --  )      {      if     (  mat  [  i  ][  j  ]     ==     'B'  )      continue  ;      // mark leftmost row with obstacle      vertical  [  j  ]     =     i  ;      break  ;      }      }      int     res     =     0  ;     // Initialize result      // if there is not obstacle on right or below      // then mirror can be placed to transfer light      for     (  int     i     =     0  ;     i      <     N  ;     i  ++  )      {      for     (  int     j     =     0  ;     j      <     N  ;     j  ++  )      {      /* if i > vertical[j] then light can from bottom    if j > horizontal[i] then light can go to right */      if     (  i     >     vertical  [  j  ]     &&     j     >     horizontal  [  i  ])      {      /* uncomment this code to print actual mirror    position also    cout  < < i  < < ' '  < < j  < < endl; */      res  ++  ;      }      }      }      return     res  ;   }   // Driver code to test above method   int     main  ()   {      int     N     =     5  ;      // B - Blank O - Obstacle      string     mat  [  N  ]     =     {  'BBOBB'        'BBBBO'        'BBBBB'        'BOOBO'        'BBBOB'      };      cout      < <     maximumMirrorInMatrix  (  mat       N  )      < <     endl  ;      return     0  ;   }   
Java
   // Java program to find how many mirror can transfer   // light from bottom to right   import     java.util.*  ;   class   GFG      {      // method returns number of mirror which can transfer      // light from bottom to right      static     int     maximumMirrorInMatrix  (  String     mat  []       int     N  )         {      // To store first obstacles horizontally (from right)      // and vertically (from bottom)      int  []     horizontal     =     new     int  [  N  ]  ;      int  []     vertical     =     new     int  [  N  ]  ;      // initialize both array as -1 signifying no obstacle      Arrays  .  fill  (  horizontal       -  1  );      Arrays  .  fill  (  vertical       -  1  );          // looping matrix to mark column for obstacles      for     (  int     i     =     0  ;     i      <     N  ;     i  ++  )         {      for     (  int     j     =     N     -     1  ;     j     >=     0  ;     j  --  )         {      if     (  mat  [  i  ]  .  charAt  (  j  )     ==     'B'  )      {      continue  ;      }      // mark rightmost column with obstacle      horizontal  [  i  ]     =     j  ;      break  ;      }      }      // looping matrix to mark rows for obstacles      for     (  int     j     =     0  ;     j      <     N  ;     j  ++  )         {      for     (  int     i     =     N     -     1  ;     i     >=     0  ;     i  --  )         {      if     (  mat  [  i  ]  .  charAt  (  j  )     ==     'B'  )         {      continue  ;      }      // mark leftmost row with obstacle      vertical  [  j  ]     =     i  ;      break  ;      }      }      int     res     =     0  ;     // Initialize result      // if there is not obstacle on right or below      // then mirror can be placed to transfer light      for     (  int     i     =     0  ;     i      <     N  ;     i  ++  )      {      for     (  int     j     =     0  ;     j      <     N  ;     j  ++  )         {      /* if i > vertical[j] then light can from bottom    if j > horizontal[i] then light can go to right */      if     (  i     >     vertical  [  j  ]     &&     j     >     horizontal  [  i  ]  )      {      /* uncomment this code to print actual mirror    position also    cout  < < i  < < ' '  < < j  < < endl; */      res  ++  ;      }      }      }      return     res  ;      }   // Driver code   public     static     void     main  (  String  []     args  )      {      int     N     =     5  ;      // B - Blank O - Obstacle      String     mat  []     =     {  'BBOBB'        'BBBBO'        'BBBBB'        'BOOBO'        'BBBOB'      };      System  .  out  .  println  (  maximumMirrorInMatrix  (  mat       N  ));   }   }   /* This code is contributed by PrinciRaj1992 */   
Python3
   # Python3 program to find how many mirror can transfer   # light from bottom to right   # method returns number of mirror which can transfer   # light from bottom to right   def   maximumMirrorInMatrix  (  mat     N  ):   # To store first obstacles horizontally (from right)   # and vertically (from bottom)   horizontal   =   [  -  1   for   i   in   range  (  N  )]   vertical   =   [  -  1   for   i   in   range  (  N  )];   # looping matrix to mark column for obstacles   for   i   in   range  (  N  ):   for   j   in   range  (  N   -   1     -  1     -  1  ):   if   (  mat  [  i  ][  j  ]   ==   'B'  ):   continue  ;   # mark rightmost column with obstacle   horizontal  [  i  ]   =   j  ;   break  ;   # looping matrix to mark rows for obstacles   for   j   in   range  (  N  ):   for   i   in   range  (  N   -   1     -  1     -  1  ):   if   (  mat  [  i  ][  j  ]   ==   'B'  ):   continue  ;   # mark leftmost row with obstacle   vertical  [  j  ]   =   i  ;   break  ;   res   =   0  ;   # Initialize result   # if there is not obstacle on right or below   # then mirror can be placed to transfer light   for   i   in   range  (  N  ):   for   j   in   range  (  N  ):          ''' if i > vertical[j] then light can from bottom    if j > horizontal[i] then light can go to right '''   if   (  i   >   vertical  [  j  ]   and   j   >   horizontal  [  i  ]):          ''' uncomment this code to print actual mirror    position also'''   res  +=  1  ;   return   res  ;   # Driver code to test above method   N   =   5  ;   # B - Blank O - Obstacle   mat   =   [  'BBOBB'     'BBBBO'     'BBBBB'     'BOOBO'     'BBBOB'   ];   print  (  maximumMirrorInMatrix  (  mat     N  ));   # This code is contributed by rutvik_56.   
C#
   // C# program to find how many mirror can transfer   // light from bottom to right   using     System  ;       class     GFG      {      // method returns number of mirror which can transfer      // light from bottom to right      static     int     maximumMirrorInMatrix  (  String     []  mat       int     N  )         {      // To store first obstacles horizontally (from right)      // and vertically (from bottom)      int  []     horizontal     =     new     int  [  N  ];      int  []     vertical     =     new     int  [  N  ];      // initialize both array as -1 signifying no obstacle      for     (  int     i     =     0  ;     i      <     N  ;     i  ++  )         {      horizontal  [  i  ]  =-  1  ;      vertical  [  i  ]  =-  1  ;      }          // looping matrix to mark column for obstacles      for     (  int     i     =     0  ;     i      <     N  ;     i  ++  )         {      for     (  int     j     =     N     -     1  ;     j     >=     0  ;     j  --  )         {      if     (  mat  [  i  ][  j  ]     ==     'B'  )      {      continue  ;      }      // mark rightmost column with obstacle      horizontal  [  i  ]     =     j  ;      break  ;      }      }      // looping matrix to mark rows for obstacles      for     (  int     j     =     0  ;     j      <     N  ;     j  ++  )         {      for     (  int     i     =     N     -     1  ;     i     >=     0  ;     i  --  )         {      if     (  mat  [  i  ][  j  ]     ==     'B'  )         {      continue  ;      }      // mark leftmost row with obstacle      vertical  [  j  ]     =     i  ;      break  ;      }      }      int     res     =     0  ;     // Initialize result      // if there is not obstacle on right or below      // then mirror can be placed to transfer light      for     (  int     i     =     0  ;     i      <     N  ;     i  ++  )      {      for     (  int     j     =     0  ;     j      <     N  ;     j  ++  )         {      /* if i > vertical[j] then light can from bottom    if j > horizontal[i] then light can go to right */      if     (  i     >     vertical  [  j  ]     &&     j     >     horizontal  [  i  ])      {      /* uncomment this code to print actual mirror    position also    cout  < < i  < < ' '  < < j  < < endl; */      res  ++  ;      }      }      }      return     res  ;      }   // Driver code   public     static     void     Main  (  String  []     args  )      {      int     N     =     5  ;      // B - Blank O - Obstacle      String     []  mat     =     {  'BBOBB'        'BBBBO'        'BBBBB'        'BOOBO'        'BBBOB'      };      Console  .  WriteLine  (  maximumMirrorInMatrix  (  mat       N  ));   }   }   // This code is contributed by Princi Singh   
JavaScript
    <  script  >   // JavaScript program to find how many mirror can transfer   // light from bottom to right   // method returns number of mirror which can transfer   // light from bottom to right   function     maximumMirrorInMatrix  (  mat       N  )      {      // To store first obstacles horizontally (from right)      // and vertically (from bottom)      var     horizontal     =     Array  (  N  ).  fill  (  -  1  );      var     vertical     =     Array  (  N  ).  fill  (  -  1  );          // looping matrix to mark column for obstacles      for     (  var     i     =     0  ;     i      <     N  ;     i  ++  )         {      for     (  var     j     =     N     -     1  ;     j     >=     0  ;     j  --  )         {      if     (  mat  [  i  ][  j  ]     ==     'B'  )      {      continue  ;      }      // mark rightmost column with obstacle      horizontal  [  i  ]     =     j  ;      break  ;      }      }      // looping matrix to mark rows for obstacles      for     (  var     j     =     0  ;     j      <     N  ;     j  ++  )         {      for     (  var     i     =     N     -     1  ;     i     >=     0  ;     i  --  )         {      if     (  mat  [  i  ][  j  ]     ==     'B'  )         {      continue  ;      }      // mark leftmost row with obstacle      vertical  [  j  ]     =     i  ;      break  ;      }      }      var     res     =     0  ;     // Initialize result      // if there is not obstacle on right or below      // then mirror can be placed to transfer light      for     (  var     i     =     0  ;     i      <     N  ;     i  ++  )      {      for     (  var     j     =     0  ;     j      <     N  ;     j  ++  )         {      /* if i > vertical[j] then light can from bottom    if j > horizontal[i] then light can go to right */      if     (  i     >     vertical  [  j  ]     &&     j     >     horizontal  [  i  ])      {      /* uncomment this code to print actual mirror    position also    cout  < < i  < < ' '  < < j  < < endl; */      res  ++  ;      }      }      }      return     res  ;   }   // Driver code   var     N     =     5  ;   // B - Blank O - Obstacle   var     mat     =     [  'BBOBB'        'BBBBO'        'BBBBB'        'BOOBO'        'BBBOB'   ];   document  .  write  (  maximumMirrorInMatrix  (  mat       N  ));    <  /script>    

الإخراج
2  

تعقيد الوقت: O(n 2 ).
المساحة المساعدة: O(n)

 

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