The goal is to maximize the score by selecting pairs from an array and using their GCD. This can be approached using state transition dynamic programming, where we recursively calculate the best score while considering all possible pairs. The problem is a good test of dynamic programming and backtracking skills.
Problem Statement
You are given an array nums of size 2 * n. Your task is to perform n operations on this array, where in the ith operation, you pick two numbers, compute their GCD, and multiply it by i. The total score is the sum of these values across all operations.
The challenge lies in maximizing the score by selecting the optimal pairs to compute the GCD and considering the multiplication factor in each operation. Your task is to return the maximum score you can achieve after n operations.
Examples
Example 1
Input: nums = [1,2]
Output: 1
The optimal choice of operations is: (1 * gcd(1, 2)) = 1
Example 2
Input: nums = [3,4,6,8]
Output: 11
The optimal choice of operations is: (1 * gcd(3, 6)) + (2 * gcd(4, 8)) = 3 + 8 = 11
Example 3
Input: nums = [1,2,3,4,5,6]
Output: 14
The optimal choice of operations is: (1 * gcd(1, 5)) + (2 * gcd(2, 4)) + (3 * gcd(3, 6)) = 1 + 4 + 9 = 14
Constraints
- 1 <= n <= 7
- nums.length == 2 * n
- 1 <= nums[i] <= 106
Solution Approach
State Transition Dynamic Programming
The problem can be solved efficiently using state transition dynamic programming. We can represent the state of the array and recursively calculate the maximum score by picking pairs of elements. By keeping track of the pairs already chosen, we avoid redundant calculations.
Backtracking with Memoization
Backtracking can be used to explore all possible pairings of elements in the array. With memoization, we store intermediate results to avoid redundant calculations, significantly reducing time complexity compared to a pure brute force solution.
Bitmasking
A bitmasking approach allows us to represent which elements of the array have been paired, making it easier to explore all valid pairings. This technique can be used to generate all possible pair combinations and calculate the maximum score.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity depends on the approach. A dynamic programming solution with bitmasking can achieve a time complexity of O(n * 2^n), where n is the size of the array. The space complexity is O(n * 2^n) due to the storage of intermediate states.
What Interviewers Usually Probe
- Check for the candidate's ability to apply dynamic programming in combinatorial problems.
- Assess how efficiently the candidate handles memoization and backtracking to reduce computational overhead.
- Look for understanding of bitmasking techniques and how it can be applied to problems involving subsets.
Common Pitfalls or Variants
Common pitfalls
- Failing to efficiently memoize subproblems, leading to redundant recalculations.
- Not correctly handling the state transitions in dynamic programming, which can cause incorrect results or excessive time complexity.
- Overcomplicating the problem with unnecessary optimizations or not fully exploring all pairings.
Follow-up variants
- Consider the problem with different constraints, such as larger arrays or additional scoring rules.
- Allow multiple solutions for pairing, where the goal could be to find the second-highest score instead of the maximum.
- Modify the problem to return all possible score configurations and have the candidate find the one with the maximum score.
How GhostInterview Helps
- GhostInterview offers step-by-step explanations of dynamic programming approaches and how to manage state transitions effectively.
- The tool provides solutions and hints for memoization and backtracking, helping candidates understand how to optimize recursive solutions.
- GhostInterview provides detailed insights into bitmasking, explaining how to implement this technique and optimize it for problems like this.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the optimal approach for maximizing score in the "Maximize Score After N Operations" problem?
The optimal approach is state transition dynamic programming, where you explore all possible pairings of elements in the array while maximizing the score based on their GCD values.
Can backtracking be used to solve this problem?
Yes, backtracking can be used with memoization to efficiently explore all possible pairings while avoiding redundant calculations.
How does bitmasking help in solving the "Maximize Score After N Operations" problem?
Bitmasking allows you to represent which elements have been paired, making it easier to explore all valid pair combinations and calculate the maximum score.
What is the time complexity of the optimal solution?
The time complexity of the optimal solution using dynamic programming with bitmasking is O(n * 2^n), where n is the size of the array.
How can I avoid redundant calculations in this problem?
Memoization and backtracking are key to avoiding redundant calculations by storing intermediate results and only recalculating when necessary.
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