Minimize Length of Array Using Operations

Problem Description

You are given a 0-indexed integer array nums containing positive integers. Your task is to minimize the length of nums by performing the following operations any number of times (including zero):

1. Select two distinct indices i and j from nums, such that nums[i] > 0 and nums[j] > 0.
2. Insert the result of nums[i] % nums[j] at the end of nums.
3. Delete the elements at indices i and j from nums.

Return an integer denoting the minimum length of nums after performing the operation any number of times.

Key Insights

• The operation reduces the number of elements while potentially introducing new elements (the result of the modulo operation).
• The modulo operation yields results that are less than the divisor, which can help reduce the array size.
• The process continues until all elements become zero or cannot produce any new positive integers.
• The minimum length achievable is determined by the number of distinct positive integers in the array.

Space and Time Complexity

Time Complexity: O(n * log(n)), where n is the number of elements in the array. This accounts for sorting and potential operations. Space Complexity: O(n) for storing unique values.

Solution

To solve the problem, we can utilize a set data structure to track distinct positive integers in the array. The idea is to:

1. Iterate through the input array and add all positive integers to a set.
2. The size of the set at the end of the iteration represents the minimum length of the array after performing all valid operations, as each distinct integer can potentially reduce to zero through the modulo operation with others.

This approach ensures we only retain unique positive integers, as any repeated integers will not contribute to further reductions.

Code Solutions