Remember that one “Guess the Number” game? Let’s look at the following scenario by applying the Binary Search algorithm in order for you to understand how Binary Search works.

The Guessing Game

Pick a number from 0 – 100 and depending on your guess, I’ll either say Correct, Higher, or Lower. What would you choose?

0

1

…

49

50

51

…

99

100

Well, 50 of course! Why? Best case you get it correct. Worst case you’ll either get a “Higher” or “Lower”. Now think about the following, if you got “Higher” you eliminated 0-49, or 51-100 if “Lower”, in other words, you eliminate HALF the possibilities.

50

51

…

74

75

76

…

99

100

Well, let’s say it was “Higher”. What would you guess after? 75. Since it’s in between 50 – 100. If you didn’t guess correctly, you’ll be facing the similar scenario as before. You’ll end up eliminating half the possibilities!

Binary Search Algorithm

The following is the recursive implementation.

int binarysearch(vector  vec, int low, int high, int key)
{
    //Not found
    if ( low > high )
    {
          return -1;
    }

    //Finding the middle index between high and low
    int mid = (low + high)/2;

    //Returning the index if key is found
    if ( vec[mid] == key )
    {
          return mid;
    }
    //If middle index value is larger than our key, we search the lower half of the vector
    //else we search the upper half of the vector
    else if ( vec[mid] > key )
    {
          return binarysearch(vec, low, mid-1, key);
    }
    else
    {
          return binarysearch(vec, mid+1, high, key);
    }
}

Example of Binary Search

Let’s say our container held the values 0-100 in its respective index values. Let’s say the value we are looking for is 57.

First iteration

50 < 57, so search through the upper half of our vector.

Second iteration

75 > 57, so search through the lower half of our vector.

Third iteration

63 > 57, so search through the lower half of our vector.

Fourth iteration

57 == 57!