To solve the Maximum Score of a Good Subarray problem, binary search on the valid answer space is crucial for optimal subarray identification. The key lies in separating the prefix and suffix before and after index k, applying a two-pointer approach to maximize the score. Efficient implementation ensures the solution operates in linear time with constant space complexity.
Problem Statement
You are given an array of integers nums and an integer k. The score of a subarray (i, j) is defined as min(nums[i], ..., nums[j]) * (j - i + 1). A good subarray is one where i <= k <= j.
Return the maximum possible score of a good subarray. The challenge requires balancing two-pointer strategies with binary search over the valid answer space to achieve an optimal solution.
Examples
Example 1
Input: nums = [1,4,3,7,4,5], k = 3
Output: 15
The optimal subarray is (1, 5) with a score of min(4,3,7,4,5) * (5-1+1) = 3 * 5 = 15.
Example 2
Input: nums = [5,5,4,5,4,1,1,1], k = 0
Output: 20
The optimal subarray is (0, 4) with a score of min(5,5,4,5,4) * (4-0+1) = 4 * 5 = 20.
Constraints
- 1 <= nums.length <= 105
- 1 <= nums[i] <= 2 * 104
- 0 <= k < nums.length
Solution Approach
Binary Search on Valid Answer Space
Binary search helps narrow down the potential scores for subarrays efficiently. By testing different subarray sizes and checking the corresponding score, we can hone in on the maximum score without brute-forcing every possibility.
Two-Pointer Technique
By splitting the problem into two distinct segments—prefix before k and suffix after k—you can apply the two-pointer technique to evaluate possible subarrays while considering index k as part of the valid range.
Monotonic Stack for Score Calculation
A monotonic stack can be used to track the minimum value of the subarray dynamically as the pointers move. This helps calculate the score efficiently while maintaining the correct bounds for the subarray.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(n) |
| Space | O(1) |
The time complexity is O(n) because the solution processes the array in a linear scan using binary search and two-pointer techniques. Space complexity is O(1) as it uses constant extra space, making it highly efficient for large arrays.
What Interviewers Usually Probe
- The candidate demonstrates an understanding of binary search and two-pointer techniques.
- The candidate is able to explain how to handle index k as a fixed element in subarrays.
- The candidate can describe how to efficiently calculate subarray scores without unnecessary recalculations.
Common Pitfalls or Variants
Common pitfalls
- Failing to consider the entire valid subarray range that includes index k.
- Not efficiently handling the score calculation for varying subarray sizes.
- Confusing the logic behind subarray boundaries, leading to incorrect minimum score calculations.
Follow-up variants
- Handling larger arrays with more efficient search techniques.
- Exploring different ways to optimize the calculation of the minimum in subarrays.
- Incorporating additional constraints that may limit subarray size or range.
How GhostInterview Helps
- GhostInterview guides you through the binary search strategy, ensuring you understand the algorithm's key concepts.
- The solver suggests optimal approaches for applying two-pointer techniques, helping you break down the problem into manageable steps.
- GhostInterview provides explanations for common pitfalls, making sure you don't overlook key aspects of the problem like handling index k.
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 key pattern in Maximum Score of a Good Subarray?
The key pattern in this problem is binary search over the valid answer space, combined with two-pointer strategies to maximize subarray scores.
How do I approach finding the maximum score?
Use binary search to test different subarray sizes and apply the two-pointer technique to evaluate subarrays containing the element at index k.
What is the time complexity of the solution?
The time complexity is O(n), achieved through linear scanning with binary search and two-pointer techniques.
How can I avoid recalculating the subarray minimum?
Use a monotonic stack to dynamically track the minimum value of the subarray as the pointers move, avoiding redundant calculations.
What are the common mistakes when solving this problem?
Common mistakes include neglecting the valid subarray range containing index k or inefficient score calculations due to incorrect subarray boundaries.
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