Missing Number

To solve this coding challenge, we need to determine which number is missing from a given array of
n
distinct numbers in the range
[0, n]
. This problem can be efficiently solved using a mathematical approach that leverages the property of the sum of the first
n
natural numbers.

Explanation

The sum of the first
n
natural numbers is given by the formula:
\[ S = \frac{n \times (n + 1)}{2} \] where
n
is the length of the given array of numbers.
Given that the numbers in the array range from
0
to
n
and are distinct, the sum of the numbers in the array should be equal to this calculated sum if no number were missing. However, since one number is missing, the actual sum of the numbers in the array will be less than this calculated sum by the value of the missing number.
Therefore, the missing number can be found by: \[ \text{Missing Number} = S - \text{Actual Sum} \] where
Actual Sum
is the sum of the elements in the given array.

Pseudocode with Comments

                                            
# Begin by calculating the expected sum using the formula.
expected_sum = n * (n + 1) // 2
# The sum function provides the actual sum of numbers in the array.
actual_sum = sum(nums)
# The missing number is simply the difference between the expected_sum and actual_sum.
missing_number = expected_sum - actual_sum
# Return the missing number to the caller.
return missing_number

                                        

Detailed Steps in Pseudocode

  1. Calculate the length of the input array:
    • This will tell us the value of
      n
      , which is the upper bound of our range
      [0, n]
      .
      •                                             
        n = length_of(nums)  # Get the length of nums array
        
                                                
  2. Compute the expected sum of the first
    n
    natural numbers:
    • Use the formula for the sum of the first
      n
      numbers.
      •                                             
        expected_sum = n * (n + 1) // 2  # Calculate the expected sum
        
                                                
  3. Calculate the actual sum of the elements in the array:
    • Sum all the numbers present in the given array using a summation function.
      •                                             
        actual_sum = sum(nums)  # Compute the sum of all elements in the array
        
                                                
  4. Determine the missing number:
    • The missing number is the difference between the expected sum and the actual sum computed.
      •                                             
        missing_number = expected_sum - actual_sum  # Derive the missing number
        
                                                
Thus, by following this highly efficient method, you can determine the missing number with a time complexity of \( O(n) \) and space complexity of \( O(1) \). This approach leverages basic arithmetic operations and avoids unnecessary use of additional data structures.