#1131

Medium

LeetCode Problem Workspace

Maximum of Absolute Value Expression

Calculate the largest sum of absolute differences across two arrays and their indices using an efficient pattern-based approach.

array math

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.

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.

terminal

Solution

Solution 1: Mathematics + Enumeration

Let's denote $x_i = arr1[i]$, $y_i = arr2[i]$. Since the size relationship between $i$ and $j$ does not affect the value of the expression, we can assume $i \ge j$. Then the expression can be transformed into:

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class Solution:
    def maxAbsValExpr(self, arr1: List[int], arr2: List[int]) -> int:
        dirs = (1, -1, -1, 1, 1)
        ans = -inf
        for a, b in pairwise(dirs):
            mx, mi = -inf, inf
            for i, (x, y) in enumerate(zip(arr1, arr2)):
                mx = max(mx, a * x + b * y + i)
                mi = min(mi, a * x + b * y + i)
                ans = max(ans, mx - mi)
        return ans

Continue Practicing

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