Find the Prefix-MEX Array for given Array

Given an arrayA[] of N elements, the task is to create a Prefix-MEX array for this given array. Prefix-MEX array B[] of an array A[] is created such that MEX of A[0] till A[i] is B[i]. 

MEX of an array refers to the smallest missing non-negative integer of the array.

Examples:

Input: A[] = {1, 0, 2, 4, 3}
Output: 0 2 3 3 5
Explanation: In the array A, elements 
Till 1st index, elements are [1] and mex till 1st index is 0.
Till 2nd index, elements are [1, 0] and mex till 2nd index is 2.
Till 3rd index, elements are [ 1, 0, 2] and mex till 3rd index is 3.
Till 4th index, elements are [ 1, 0, 2, 4] and mex till 4th index is 3.
Till 5th index, elements are [ 1, 0, 2, 4, 3] and mex till 5th index is 5.
So our final array B would be [0, 2, 3, 3, 5].

Input: A[] = [ 1, 2, 0 ]
Output: [ 0, 0, 3 ]
Explanation: In the array A, elements 
Till 1st index, elements are [1] and mex till 1st index is 0.
Till 2nd index, elements are [1, 2] and mex till 2nd index is 0.
Till 3rd index, elements are [ 1, 2, 0] and mex till 3rd index is 3.
So our final array B would be [0, 0, 3].

Naive Approach: The simplest way to solve the problem is:

For each element at i^th (0 ≤ i < N)index of the array A[], find MEX from 0 to i and store it at B[i].

Follow the steps mentioned below to implement the idea:

• Iterate over the array from i = 0 to N-1:
  • For every i^th index in array A[]: 
    • Run a for loop to find the MEX of index 0 till i.
    • Then store this MEX at index i in the resultant array B[i].
• Return the resultant array B[] at the end.

Time Complexity: O(N²)
Auxiliary Space: O(N)

Efficient Approach: This approach is based on the usage of Set data structure.

A set stores data in sorted order. We can take advantage of that and store all the non-negative integers till the maximum value of the array. Then traverse through each array element and remove the visited data from set. The smallest remaining element will be the MEX for that index.

Follow the steps below to implement the idea:

• Find the maximum element of the array A[].
• Create a set and store the numbers from 0 to the maximum element in the set.
• Traverse through the array from i = 0 to N-1: 
  • For each element, erase that element from the set.
  • Now find the smallest element remaining in the set.
  • This is the prefix MEX for the i^th element. Store this value in the resultant array.
• Return the resultant array as the required answer.

Below is the implementation of the above approach. 

0 2 3 3 5 

Time Complexity: O(N * log N )

• O(N) for iterating the vector, and 
• O(log N) for inserting and deleting the element from the set.

Auxiliary Space: O(N)

Efficient Approach 2: This approach is based on using an array and a pointer to keep track of the MEX.

Follow these steps mentioned below to implement this idea:

• Find the maximum element of the array A[].
• Create a boolean array of size equal to the maximum element + 1, B[] with all values initialised as 0.
• Create a variable to track the current MEX.
• Traverse through the A[] from i = 0 to N-1:
  • For each element, set the value in B[] at index equal to the value of the element at i in A[] to true.
  • Update the current MEX by increasing the variable until value in B[] at MEX is true.
  • Store this value in the resultant array.
• Return the resultant array as the required answer.

Below is the implementation of the above approach.

0 0 3 4 4 6 

Time Complexity: O(N)

• O(N) for iterating the vector.
• O(N) for updating the MEX. Important thing to note is that the inner while loop can run only N times independent of the outer for loop.

Space Complexity: O(N)