11. Recursion_outside

This sometimes requires we define additional

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This sometimes requires we define additional  paramaters that are passed to the method. For example, we defined the array reversal  method as ReverseArray( A, i,  j ), not  ReverseArray( A ). ©  2010 Goodrich, Tamassia
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Using Recursion 7 Computing Powers The power function, p(x,n)=x n , can be defined  recursively: This leads to an power function that runs in  O(n) time (for we make n recursive calls). We can do better than this, however. - = = else ) 1 , ( 0 if 1 ) , ( n x p x n n x p ©  2010 Goodrich, Tamassia
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Using Recursion 8 Recursive Squaring We can derive a more efficient linearly  recursive algorithm by using repeated squaring: For example, 2 4 = 2 ( 4 / 2 ) 2 = ( 2 4 / 2 ) 2 = ( 2 2 ) 2 = 4 2 = 16 2 5 = 2 1 +( 4 / 2 ) 2 = 2 ( 2 4 / 2 ) 2 = 2 ( 2 2 ) 2 = 2 ( 4 2 ) = 32 2 6 = 2 ( 6 / 2)2 = ( 2 6 / 2 ) 2 = ( 2 3 ) 2 = 8 2 = 64 2 7 = 2 1 +( 6 / 2 ) 2 = 2 ( 2 6 / 2 ) 2 = 2 ( 2 3 ) 2 = 2 ( 8 2 ) = 128 . = - = even is 0 if odd is 0 if 0 if ) 2 / , ( ) 2 / ) 1 ( , ( 1 ) , ( 2 2 x x x n x p n x p x n x p ©  2010 Goodrich, Tamassia
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Using Recursion 9 Recursive Squaring Method Algorithm  Power ( x, n ):       Input:  A number  and integer  n =  0       Output:  The value  x n      if  = 0 then return  1      if  is odd  then y  =  Power ( x,  ( n -  1) 2) return  ∙  y      else y =  Power ( x, n/  2) return   y ©  2010 Goodrich, Tamassia
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Using Recursion 10 Analysis Algorithm  Power ( x, n ):       Input:  A number  and  integer  n =  0       Output:  The value  x n      if  = 0 then return  1      if  is odd  then y  =  Power ( x,  ( n -  1) 2) return  x ∙ y ∙ y      else y =  Power ( x, n/  2) return  y ∙ y It is important that we use a  variable twice here rather than  calling the method twice. Each time we make a recursive  call we halve the value of n;  hence, we make log n recursive  calls. That is, this method runs  in O(log n) time. ©  2010 Goodrich, Tamassia
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Using Recursion 11 Tail Recursion Tail recursion occurs when a linearly recursive  method makes its recursive call as its last step. The array reversal method is an example. Such methods can be easily converted to non- recursive methods (which saves on some resources). Example: Algorithm  IterativeReverseArray( A, i, j  ):       Input:  An array  and nonnegative integer indices  and  j       Output:  The reversal of the elements in  starting at index  and ending at  j      while  i <  j  do Swap  A [ ] and  A ] i  = i  + 1 j  = j -  1      return ©  2010 Goodrich, Tamassia
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Using Recursion 12 Binary Recursion
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