Problem Description
You are given nums, an array of positive integers of size 2 * n. You must perform n operations on this array. In the ith operation (1-indexed), you will choose two elements, x and y, receive a score of i * gcd(x, y), and remove x and y from nums. Return the maximum score you can receive after performing n operations.
Key Insights
- The score is defined in terms of the greatest common divisor (GCD) of pairs of elements.
- The order of operations is crucial since the score increases with the operation index.
- A backtracking approach can be used to explore all possible pairs and maximize the score.
- The maximum number of operations is small (up to 7), making exhaustive search feasible.
Space and Time Complexity
Time Complexity: O(n * 2^n)
Space Complexity: O(n)
Solution
The solution employs a backtracking approach to explore all possible pairs of elements from the array. For each pair selected, we calculate the score based on the current operation index and the GCD of the selected elements. We then recursively call the function to evaluate the remaining elements. The process continues until all elements have been paired, allowing us to find the maximum score possible.
Data structures used:
- An array to hold the initial numbers.
- A set or array to keep track of selected elements to avoid pairing the same element again.