PHP Program to Find the smallest missing number

Given a sorted array of n distinct integers where each integer is in the range from 0 to m-1 and m > n. Find the smallest number that is missing from the array. 

Examples: 

Input: {0, 1, 2, 6, 9}, n = 5, m = 10 
Output: 3

Input: {4, 5, 10, 11}, n = 4, m = 12 
Output: 0

Input: {0, 1, 2, 3}, n = 4, m = 5 
Output: 4

Input: {0, 1, 2, 3, 4, 5, 6, 7, 10}, n = 9, m = 11 
Output: 8

Thanks to Ravichandra for suggesting the following two methods.

Method 1 (Use Binary Search) 

For i = 0 to m-1, do binary search for i in the array. If i is not present in the array then return I.

Time Complexity: O(m log n) 

Method 2 (Linear Search) 

If arr[0] is not 0, return 0. Otherwise traverse the input array starting from index 0, and for each pair of elements a[i] and a[i+1], find the difference between them. if the difference is greater than 1 then a[i]+1 is the missing number. 

Time Complexity: O(n)

Method 3 (Use Modified Binary Search) 

In the standard Binary Search process, the element to be searched is compared with the middle element and on the basis of comparison result, we decide whether to search is over or to go to left half or right half. 

In this method, we modify the standard Binary Search algorithm to compare the middle element with its index and make decision on the basis of this comparison.

• If the first element is not same as its index then return first index
• Else get the middle index say mid
  • If arr[mid] greater than mid then the required element lies in left half.
  • Else the required element lies in right half.

<?php
// PHP program to find the
// smallest elements missing
// in a sorted array.

// function that returns 
// smallest elements missing
// in a sorted array.
function findFirstMissing($array, $start, $end)
{
    if ($start > $end)
        return $end + 1;

    if ($start != $array[$start])
        return $start;

    $mid = ($start + $end) / 2;

    // Left half has all 
    // elements from 0 to mid
    if ($array[$mid] == $mid)
        return findFirstMissing($array, 
                                $mid + 1, 
                                $end);

    return findFirstMissing($array, 
                            $start, 
                            $mid);
}

    // Driver Code
    $arr = array (0, 1, 2, 3, 4, 5, 6, 7, 10);
    $n = count($arr);
    echo "Smallest missing element is " ,
          findFirstMissing($arr, 2, $n - 1);
        
?>

Smallest missing element is 8

Complexity Analysis:

• Time Complexity: O(Logn)
• Auxiliary Space: O(logn) where logn is the size of the recursive stack tree

Please refer complete article on Find the smallest missing number for more details!