algorithm first version

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found = false;//so far mid = approximate midpoint between 0 and finalIndex; if (key == a[mid]) { found = true; location = mid; } else if (key < a[mid]) search a[0] through a[mid - 1]; else if (key > a[mid]) search a[mid + 1] through a[finalIndex];

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Since the searchings of the shorter lists are smaller versions of the very task we are designing the algorithm to perform, this algorithm naturally lends itself to the use of recursion The smaller lists can be searched with recursive calls to the algorithm itself

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Thinking Recursively

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Our pseudocode is a bit too imprecise to be easily translated into C++ code The problem has to do with the recursive calls There are two recursive calls shown:

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search a[0] through a[mid - 1];

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search a[mid + 1] through a[finalIndex];

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To implement these recursive calls we need two more parameters A recursive call speci es that a subrange of the array is to be searched In one case it is the elements indexed by 0 through mid - 1 In the other case it is the elements indexed by mid + 1 through finalIndex The two extra parameters will specify the rst and last indexes of the search, so we will call them first and last Using these parameters for the lowest and highest indexes, instead of 0 and finalIndex, we can express the pseudocode more precisely, as follows:

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To search a[first] through a[last] do the following:

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found = false;//so far mid = approximate midpoint between first and last; if (key == a[mid]) { found = true; location = mid; } else if (key < a[mid]) search a[first] through a[mid - 1]; else if (key > a[mid]) search a[mid + 1] through a[last];

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algorithm first refinement

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To search the entire array, the algorithm would be executed with first set equal to 0 and last set equal to finalIndex The recursive calls will use other values for first and last For example, the rst recursive call would set first equal to 0 and last equal to the calculated value mid - 1 As with any recursive algorithm, we must ensure that our algorithm ends rather than producing in nite recursion If the sought-after number is found on the list, then there is no recursive call and the process terminates, but we need some way to detect when the number is not on the list On each recursive call the value of first is increased or the value of last is decreased If they ever pass each other and first actually becomes larger than last, we will know that there are no more indexes left to check and that the number key is not in the array If we add this test to our pseudocode, we obtain a complete solution, as shown in Display 135

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stopping case

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algorithm final version

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Recursion

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Display 135 Pseudocode for Binary Search

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int a[Some_Size_Value];

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ALGORITHM TO SEARCH a[first] THROUGH a[last]

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//Precondition: //a[first]<= a[first + 1] <= a[first + 2] <= <= a[last]

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TO LOCATE THE VALUE KEY:

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if (first > last) //A stopping case found = false; else { mid = approximate midpoint between first and last; if (key == a[mid]) //A stopping case { found = false; location = mid; } else if key < a[mid] //A case with recursion search a[first] through a[mid - 1]; else if key > a[mid] //A case with recursion search a[mid + 1] through a[last]; }

CODING

Now we can routinely translate the pseudocode into C++ code The result is shown in Display 136 The function search is an implementation of the recursive algorithm given in Display 135 A diagram of how the function performs on a sample array is given in Display 137 Notice that the function search solves a more general problem than the original task Our goal was to design a function to search an entire array, yet the search function will let us search any interval of the array by specifying the index bounds first and last This is common when designing recursive functions Frequently, it is necessary to solve a more general problem in order to be able to express the recursive algorithm In this case, we only wanted the answer in the case where first and last are set equal to 0 and finalIndex However, the recursive calls will set them to values other than 0 and finalIndex