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Minimum Increments for Target Multiples in an Array

Difficulty: Hard

Problem Description

You are given two arrays, nums and target. In a single operation, you may increment any element of nums by 1. Return the minimum number of operations required so that each element in target has at least one multiple in nums.

Key Insights

  • Each element in target needs to have at least one multiple present in nums.
  • The operation consists of incrementing elements in nums, which allows us to transform them to meet the multiples condition.
  • We can calculate the required increments for each target element based on the smallest nums elements that can be transformed to satisfy the multiples condition.
  • The overall goal is to minimize the total number of increments across all target elements.

Space and Time Complexity

Time Complexity: O(n * m), where n is the length of nums and m is the length of target (which is at most 4). Space Complexity: O(1), as we are using a fixed amount of extra space regardless of input size.

Solution

The solution involves iterating through each element of the target array and determining the minimum increments needed for at least one of the elements in the nums array to be a multiple of that target element. We can do this by checking for each target element how many increments are needed to transform each num into a multiple of the target. The approach relies on basic arithmetic and a nested loop structure.

Code Solutions

def minIncrements(nums, target):
    increments = 0
    
    for t in target:
        min_increment = float('inf')
        
        for n in nums:
            if n % t == 0:
                min_increment = 0
                break
            else:
                # Calculate how much we need to increment n to make it a multiple of t
                increment_needed = t - (n % t)
                min_increment = min(min_increment, increment_needed)
        
        increments += min_increment
    
    return increments
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