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Smallest Missing Non-negative Integer After Operations

Difficulty: Medium

Problem Description

You are given a 0-indexed integer array nums and an integer value. In one operation, you can add or subtract value from any element of nums. The MEX (minimum excluded) of an array is the smallest missing non-negative integer in it. Return the maximum MEX of nums after applying the mentioned operation any number of times.

Key Insights

  • You can manipulate elements of the array nums by adding or subtracting the value.
  • The goal is to maximize the MEX, which is the smallest non-negative integer not present in the modified array.
  • The operations allow for a wide range of values to be created, specifically multiples of value.
  • You can generate all non-negative integers up to a certain limit by modifying the elements appropriately.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(n)

Solution

To find the maximum MEX, we can use a set to keep track of all the reachable non-negative integers that can be formed by the operations on nums. For each number in nums, we can calculate its possible adjusted values by adding or subtracting multiples of value. We will collect all unique non-negative integers in a set. Finally, we will iterate from 0 upwards to find the smallest integer that is not in the set, which will be the maximum MEX.

Code Solutions

def maximum_mex(nums, value):
    reachable = set()
    
    # Calculate all reachable non-negative integers
    for num in nums:
        if num < 0:
            continue
        for i in range(0, 10**9 + 1, value):
            adjusted_value = num + i
            if adjusted_value >= 0:
                reachable.add(adjusted_value)
            adjusted_value = num - i
            if adjusted_value >= 0:
                reachable.add(adjusted_value)

    # Find the maximum MEX
    mex = 0
    while mex in reachable:
        mex += 1
    return mex
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