Problem Description
You are given an array of positive integers nums. You need to select a subset of nums which satisfies the following condition: You can place the selected elements in a 0-indexed array such that it follows the pattern: [x, x², x⁴, ..., x^(k/2), x^k, x^(k/2), ..., x⁴, x², x]. Return the maximum number of elements in a subset that satisfies these conditions.
Key Insights
- The subset must be arranged in a specific symmetrical pattern that involves powers of a number
x. - For each unique number in the array, we need to find its power relationships with other numbers.
- The maximum subset size can be calculated by checking how many times each number can be represented as the square of another number.
- Hash maps can be used to keep track of the frequency of each number and their valid pairs.
Space and Time Complexity
Time Complexity: O(n) - where n is the number of elements in the input array, as we iterate through the array and perform constant-time operations. Space Complexity: O(n) - for storing the frequency of each number in a hash map.
Solution
To solve the problem, we can use a hash map to count the occurrences of each number. For each unique number x, we will check if x² exists in the hash map. If it does, we can form a subset with both x and x². We continue this process for every power of x until no further valid powers exist in the map. The maximum count of such pairs will give us the size of the largest subset that can be formed.