The Maximum of Absolute Value Expression problem requires evaluating all pairs of indices to find the largest combined difference from two arrays. By converting absolute values into four sign-based expressions, you can efficiently compute the maximum without brute-force iteration. This method reduces complexity while directly leveraging the array plus math pattern of the problem.
Problem Statement
Given two integer arrays of equal length, compute the maximum possible value of |arr1[i]-arr1[j]| + |arr2[i]-arr2[j]| + |i-j| for any indices i and j. The arrays may contain negative or positive integers, and your solution must handle large input sizes efficiently using array and math insights.
Return a single integer representing this maximum. Constraints ensure array lengths between 2 and 40000 and element values between -10^6 and 10^6. Focus on optimizing absolute value calculations through combining sign patterns rather than direct nested loops.
Examples
Example 1
Input: arr1 = [1,2,3,4], arr2 = [-1,4,5,6]
Output: 13
Example details omitted.
Example 2
Input: arr1 = [1,-2,-5,0,10], arr2 = [0,-2,-1,-7,-4]
Output: 20
Example details omitted.
Constraints
- 2 <= arr1.length == arr2.length <= 40000
- -10^6 <= arr1[i], arr2[i] <= 10^6
Solution Approach
Transform Absolute Expressions
Rewrite |arr1[i]-arr1[j]| + |arr2[i]-arr2[j]| as one of four linear expressions by assigning positive or negative signs to arr1[i] and arr2[i]. This converts the problem into maximizing expressions like arr1[i]+arr2[i]+i - (arr1[j]+arr2[j]+j).
Compute Max-Min for Each Pattern
For each of the four sign combinations, track the maximum and minimum of the transformed values across the array. The maximum absolute expression value for that pattern is the difference between the max and min. Repeat for all patterns and take the overall maximum.
Aggregate and Return Result
After calculating the differences for all four patterns, return the largest value. This approach avoids nested loops and fully leverages the array plus math pattern for efficient computation, achieving linear time relative to array length.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity is O(n) since each array element is processed once per sign pattern. Space complexity is O(1) extra space beyond input arrays, as only running maxima and minima are stored.
What Interviewers Usually Probe
- Look for recognition that naive nested loops will exceed time limits on large arrays.
- Expect candidates to convert absolute values into a fixed set of sign combinations.
- Watch for the candidate correctly maintaining running maxima and minima per transformed pattern.
Common Pitfalls or Variants
Common pitfalls
- Attempting brute-force O(n^2) iteration, which fails for array lengths up to 40000.
- Forgetting to include the index difference |i-j| in the transformed expressions.
- Miscalculating sign combinations, leading to incorrect maximum values.
Follow-up variants
- Compute the minimum of |arr1[i]-arr1[j]| + |arr2[i]-arr2[j]| + |i-j| instead of the maximum.
- Extend to three or more arrays, adjusting the four-pattern approach to eight or more combinations.
- Handle dynamic updates to arrays, requiring efficient recalculation of maxima and minima.
How GhostInterview Helps
- GhostInterview identifies the correct sign patterns automatically, avoiding common manual mistakes with absolute values.
- It guides through efficient linear-time computation, bypassing brute-force failures for large arrays.
- The tool clarifies handling of the index component in the absolute expression, ensuring full correctness.
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 trick for Maximum of Absolute Value Expression?
Transform the absolute differences into four linear expressions using different sign combinations, then track max-min for each pattern.
Can this problem be solved with nested loops?
Technically yes, but nested loops are O(n^2) and will time out for large arrays up to length 40000.
How do index differences affect the calculation?
Include |i-j| in each transformed pattern by adding or subtracting the index value with the chosen signs to maintain correctness.
Is extra space needed for this solution?
No extra arrays are required; just maintain running maximum and minimum per pattern, keeping space O(1).
Does GhostInterview help with sign combination logic?
Yes, it highlights the correct four-pattern approach and ensures the array plus math strategy is applied efficiently.
Need direct help with Maximum of Absolute Value Expression instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture Maximum of Absolute Value Expression from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Stone Game II is a dynamic programming problem where Alice and Bob alternate taking stones from piles to maximize their total.
Open problem page#1093 Statistics from a Large SampleCalculate minimum, maximum, mean, median, and mode from a large sample represented by an array of counts.
Open problem page#1073 Adding Two Negabinary NumbersAdd two numbers represented in negabinary format and return the result in the same format.
Open problem page