To solve the Maximum Width Ramp problem, use a two-pointer approach combined with a monotonic stack. Start by iterating through the array with a second pointer, and track potential ramp widths using a stack. Efficiently determine the maximum width with O(n) time complexity.
Problem Statement
Given an integer array nums, a ramp is defined as a pair of indices (i, j) such that i < j and nums[i] <= nums[j]. The width of the ramp is calculated as j - i. Your task is to return the maximum width of such a ramp from the array.
If no ramp exists in the array, return 0. For example, in the array nums = [6,0,8,2,1,5], the maximum width ramp is achieved between indices 1 and 5, giving a width of 4.
Examples
Example 1
Input: nums = [6,0,8,2,1,5]
Output: 4
The maximum width ramp is achieved at (i, j) = (1, 5): nums[1] = 0 and nums[5] = 5.
Example 2
Input: nums = [9,8,1,0,1,9,4,0,4,1]
Output: 7
The maximum width ramp is achieved at (i, j) = (2, 9): nums[2] = 1 and nums[9] = 1.
Constraints
- 2 <= nums.length <= 5 * 104
- 0 <= nums[i] <= 5 * 104
Solution Approach
Two-pointer Scanning
A two-pointer technique is used to find the maximum width ramp efficiently. First, scan the array with the second pointer starting at the end, and use the stack to store indices of elements that could form valid ramps.
Monotonic Stack for Tracking Indices
Use a monotonic stack to maintain a list of indices where elements are in non-decreasing order. This allows for efficient checking of possible ramp widths between the stack and the second pointer as you scan the array.
Maximizing the Ramp Width
For each valid pair (i, j) where nums[i] <= nums[j], calculate the width j - i. Continuously update the maximum width found while iterating through the array.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | O(n) |
| Space | O(n) |
The time complexity is O(n) due to a single scan of the array using the two-pointer technique, and the space complexity is O(n) for storing indices in the stack. This ensures the algorithm scales efficiently with large inputs.
What Interviewers Usually Probe
- Candidate understands the two-pointer technique for optimizing problems like this one.
- Look for clarity in the candidate's explanation of the monotonic stack's role in solving the problem.
- Assess how well the candidate relates the problem to patterns like invariant tracking and ramp-width maximization.
Common Pitfalls or Variants
Common pitfalls
- Failing to use the monotonic stack efficiently, leading to incorrect ramp width calculations.
- Not recognizing that the two-pointer technique must be combined with a stack for optimal performance.
- Overcomplicating the problem by using a brute-force approach that results in O(n^2) time complexity.
Follow-up variants
- Consider variations where the ramp definition might change, such as requiring nums[i] > nums[j] or other constraints on the array.
- Explore edge cases with minimum and maximum array sizes to ensure the algorithm handles these scenarios effectively.
- Consider extending the problem to allow ramps in multidimensional arrays or with additional constraints.
How GhostInterview Helps
- GhostInterview helps by guiding you through the two-pointer approach with clear steps for implementation.
- The solver assists in visualizing the stack usage, highlighting key points for identifying valid ramps.
- You’ll get assistance with common pitfalls, including how to avoid brute-force solutions and focus on optimal algorithms.
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 time complexity of the Maximum Width Ramp problem?
The time complexity is O(n) due to a single pass through the array using the two-pointer technique, with a stack operation for each element.
How does the two-pointer technique work in the Maximum Width Ramp problem?
The two-pointer technique allows you to efficiently track potential ramps by iterating through the array, using a stack to track indices and calculate the maximum width.
What is a monotonic stack, and how does it help solve the Maximum Width Ramp problem?
A monotonic stack maintains a sequence of indices in non-decreasing order, helping efficiently track potential ramp widths and ensuring the correct pair of indices is selected for maximizing the width.
How does GhostInterview assist with solving the Maximum Width Ramp problem?
GhostInterview guides you step-by-step through the problem, offering clear explanations of the two-pointer and stack techniques, helping you avoid common mistakes.
Can the solution for the Maximum Width Ramp problem be optimized further?
The current solution is already optimal with O(n) time complexity, but variations of the problem with different constraints may require alternative strategies.
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