To solve Max Value of Equation, iterate through the points while maintaining a running state using a sliding window approach. The goal is to maximize the value of the equation yi + yj + |xi - xj| for valid pairs of points. A priority queue can help efficiently track the best values of yi - xi while ensuring |xi - xj| <= k.
Problem Statement
You are given an array of points, where each point is represented by its x and y coordinates, sorted by x-values. The task is to find the maximum value of the equation yi + yj + |xi - xj| for any valid pair of points, where |xi - xj| <= k and 1 <= i < j <= points.length.
The equation is subject to the constraint that |xi - xj| must be less than or equal to k. Your solution should return the maximum possible value of the equation for any such pair of points in the array.
Examples
Example 1
Input: points = [[1,3],[2,0],[5,10],[6,-10]], k = 1
Output: 4
The first two points satisfy the condition |xi - xj| <= 1 and if we calculate the equation we get 3 + 0 + |1 - 2| = 4. Third and fourth points also satisfy the condition and give a value of 10 + -10 + |5 - 6| = 1. No other pairs satisfy the condition, so we return the max of 4 and 1.
Example 2
Input: points = [[0,0],[3,0],[9,2]], k = 3
Output: 3
Only the first two points have an absolute difference of 3 or less in the x-values, and give the value of 0 + 0 + |0 - 3| = 3.
Constraints
- 2 <= points.length <= 105
- points[i].length == 2
- -108 <= xi, yi <= 108
- 0 <= k <= 2 * 108
- xi < xj for all 1 <= i < j <= points.length
- xi form a strictly increasing sequence.
Solution Approach
Sliding Window with Priority Queue
Use a sliding window to maintain valid pairs of points such that |xi - xj| <= k. A priority queue can be used to track the best yi - xi values in the window, ensuring we can calculate the equation efficiently.
Efficient Lookup
While iterating through the points, for each point, calculate the potential maximum value using the priority queue. Keep the window size within the constraint of k to avoid unnecessary computations.
Update with Running State
For every new point processed, update the priority queue with the new point's value and ensure the previous points that no longer satisfy the |xi - xj| <= k constraint are removed from the window.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity is O(n log n), where n is the number of points, due to the use of the priority queue. Space complexity is O(n) for storing the points and maintaining the priority queue.
What Interviewers Usually Probe
- Look for an understanding of how to maintain a sliding window.
- Check if the candidate correctly uses a priority queue for efficient lookups.
- Evaluate if the candidate handles the constraint |xi - xj| <= k appropriately while iterating.
Common Pitfalls or Variants
Common pitfalls
- Failing to properly manage the sliding window size could lead to incorrect results.
- Not using the priority queue effectively might result in inefficient or incorrect computation of the maximum value.
- Ignoring the constraint |xi - xj| <= k could cause invalid pairs to be considered.
Follow-up variants
- Increase the problem's complexity by adding additional constraints on the y-values of points.
- Consider modifying the equation to include additional terms or weights for each point.
- Allow for dynamic updates to the points array and re-calculate the maximum value accordingly.
How GhostInterview Helps
- GhostInterview provides step-by-step guidance to implement the sliding window and priority queue efficiently.
- With detailed hints on managing the sliding window and updating the state, GhostInterview ensures you're ready for edge cases.
- GhostInterview helps you focus on the core problem-solving techniques, like maintaining a running state, without being overwhelmed by implementation details.
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 main pattern used in the Max Value of Equation problem?
The problem uses the sliding window technique combined with a priority queue to efficiently find the maximum value of the given equation.
How do I handle the constraint |xi - xj| <= k in Max Value of Equation?
By maintaining a sliding window that ensures only valid pairs of points with |xi - xj| <= k are considered.
What data structure is best for solving Max Value of Equation?
A priority queue is ideal for maintaining the best values of yi - xi within the sliding window.
What is the time complexity of the Max Value of Equation solution?
The time complexity is O(n log n) due to the use of the priority queue for efficient lookups and updates.
How can GhostInterview help with Max Value of Equation?
GhostInterview provides tailored guidance, helping you implement the sliding window and priority queue efficiently, ensuring a clean solution.
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