# 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 following two methods.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 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)
Thanks to yasein and Jams for suggesting this method.
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.

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

## C++

 `// C++ program to find the smallest elements ` `// missing in a sorted array. ` `#include ` `using` `namespace` `std; ` ` `  `int` `findFirstMissing(``int` `array[],  ` `                    ``int` `start, ``int` `end) ` `{ ` `    ``if` `(start > end) ` `        ``return` `end + 1; ` ` `  `    ``if` `(start != array[start]) ` `        ``return` `start; ` ` `  `    ``int` `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 ` `int` `main() ` `{ ` `    ``int` `arr[] = {0, 1, 2, 3, 4, 5, 6, 7, 10}; ` `    ``int` `n = ``sizeof``(arr)/``sizeof``(arr); ` `    ``cout << ``"Smallest missing element is "` `<< ` `        ``findFirstMissing(arr, 0, n-1) << endl; ` `} ` ` `  `// This code is contributed by ` `// Shivi_Aggarwal  `

## C

 `// C program to find the smallest elements missing ` `// in a sorted array. ` `#include ` ` `  `int` `findFirstMissing(``int` `array[], ``int` `start, ``int` `end) ` `{ ` `    ``if` `(start  > end) ` `        ``return` `end + 1; ` ` `  `    ``if` `(start != array[start]) ` `        ``return` `start; ` ` `  `    ``int` `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 program to test above function ` `int` `main() ` `{ ` `    ``int` `arr[] = {0, 1, 2, 3, 4, 5, 6, 7, 10}; ` `    ``int` `n = ``sizeof``(arr)/``sizeof``(arr); ` `    ``printf``(``"Smallest missing element is %d"``, ` `           ``findFirstMissing(arr, 0, n-1)); ` `    ``return` `0; ` `} `

## Java

 `class` `SmallestMissing  ` `{ ` `    ``int` `findFirstMissing(``int` `array[], ``int` `start, ``int` `end)  ` `    ``{ ` `        ``if` `(start > end) ` `            ``return` `end + ``1``; ` ` `  `        ``if` `(start != array[start]) ` `            ``return` `start; ` ` `  `        ``int` `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 program to test the above function ` `    ``public` `static` `void` `main(String[] args)  ` `    ``{ ` `        ``SmallestMissing small = ``new` `SmallestMissing(); ` `        ``int` `arr[] = {``0``, ``1``, ``2``, ``3``, ``4``, ``5``, ``6``, ``7``, ``10``}; ` `        ``int` `n = arr.length; ` `        ``System.out.println(``"First Missing element is : "` `                ``+ small.findFirstMissing(arr, ``0``, n - ``1``)); ` `    ``} ` `} `

## Python3

 `# Python3 program to find the smallest ` `# elements missing in a sorted array. ` ` `  `def` `findFirstMissing(array, start, end): ` ` `  `    ``if` `(start > end): ` `        ``return` `end ``+` `1` ` `  `    ``if` `(start !``=` `array[start]): ` `        ``return` `start; ` ` `  `    ``mid ``=` `int``((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 program to test above function ` `arr ``=` `[``0``, ``1``, ``2``, ``3``, ``4``, ``5``, ``6``, ``7``, ``10``] ` `n ``=` `len``(arr) ` `print``(``"Smallest missing element is"``, ` `      ``findFirstMissing(arr, ``0``, n``-``1``)) ` ` `  `# This code is contributed by Smitha Dinesh Semwal  `

## C#

 `// C# program to find the smallest ` `// elements missing in a sorted array. ` `using` `System; ` ` `  `class` `GFG ` `{ ` `    ``static` `int` `findFirstMissing(``int` `[]array, ` `                            ``int` `start, ``int` `end)  ` `    ``{ ` `        ``if` `(start > end) ` `            ``return` `end + 1; ` ` `  `        ``if` `(start != array[start]) ` `            ``return` `start; ` ` `  `        ``int` `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 program to test the above function ` `    ``public` `static` `void` `Main()  ` `    ``{ ` `        ``int` `[]arr = {0, 1, 2, 3, 4, 5, 6, 7, 10}; ` `        ``int` `n = arr.Length; ` `         `  `        ``Console.Write(``"smallest Missing element is : "` `                    ``+ findFirstMissing(arr, 0, n - 1)); ` `    ``} ` `} ` ` `  `// This code is contributed by Sam007  `

## PHP

 ` ``\$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); ` `         `  `// This code Contributed by Ajit. ` `?> `

Output:

`Smallest missing element is 8`

Note: This method doesn’t work if there are duplicate elements in the array.

Time Complexity: O(Logn)

Please write comments if you find the above codes/algorithms incorrect, or find other ways to solve the same problem.

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Improved By : jit_t, Shivi_Aggarwal

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