Watch 10 video solutions for Maximum Alternating Sum of Squares, a medium level problem involving Array, Greedy, Sorting. This walkthrough by Developer Coder has 362 views views. Want to try solving it yourself? Practice on FleetCode or read the detailed text solution.
You are given an integer array nums. You may rearrange the elements in any order.
The alternating score of an array arr is defined as:
score = arr[0]2 - arr[1]2 + arr[2]2 - arr[3]2 + ...Return an integer denoting the maximum possible alternating score of nums after rearranging its elements.
Example 1:
Input: nums = [1,2,3]
Output: 12
Explanation:
A possible rearrangement for nums is [2,1,3], which gives the maximum alternating score among all possible rearrangements.
The alternating score is calculated as:
score = 22 - 12 + 32 = 4 - 1 + 9 = 12
Example 2:
Input: nums = [1,-1,2,-2,3,-3]
Output: 16
Explanation:
A possible rearrangement for nums is [-3,-1,-2,1,3,2], which gives the maximum alternating score among all possible rearrangements.
The alternating score is calculated as:
score = (-3)2 - (-1)2 + (-2)2 - (1)2 + (3)2 - (2)2 = 9 - 1 + 4 - 1 + 9 - 4 = 16
Constraints:
1 <= nums.length <= 105-4 * 104 <= nums[i] <= 4 * 104Problem Overview: You are given an array and must arrange the numbers to maximize an alternating expression of squared values: a1^2 - a2^2 + a3^2 - a4^2 .... The order of elements can be rearranged, so the goal is to place numbers in positions that maximize the final alternating sum.
Approach 1: Brute Force Permutations (O(n!))
The most direct approach tries every possible permutation of the array. For each permutation, compute the alternating sum of squares by iterating through the sequence and adding or subtracting nums[i]^2 depending on the index parity. Track the maximum result across all permutations. This guarantees correctness but becomes infeasible quickly because the number of permutations grows factorially. Time complexity is O(n!) and space complexity is O(n) for recursion or permutation storage.
Approach 2: Greedy Sorting (O(n log n))
The key observation: positions with a + sign should contain the largest squared values, while - positions should contain the smallest. Since squaring preserves order for non-negative magnitude comparisons, maximizing the expression reduces to selecting the largest values for the positive slots and the smallest for the negative slots. First compute or compare based on squared values, then sort the array. Assign the largest elements to indices contributing positively (0, 2, 4, ...) and the smallest elements to negative positions (1, 3, 5, ...). This greedy placement ensures the positive contributions dominate the subtraction terms.
This approach uses standard sorting followed by a single pass to compute the alternating sum. Sorting takes O(n log n) time and the final scan takes O(n). Extra space is O(1) if the array is sorted in place. The method relies on greedy reasoning: maximize positive contributions and minimize negative ones.
Conceptually, this problem combines array manipulation with greedy ordering strategies. Understanding how value magnitude affects the alternating expression is the main trick. Related techniques appear in problems involving rearranging values to optimize expressions using greedy algorithms, ordering elements with sorting, and iterating efficiently over an array.
Recommended for interviews: The greedy sorting approach is what interviewers expect. Mentioning the brute force permutation strategy shows you understand the search space, but recognizing that the largest squares should occupy positive positions demonstrates algorithmic insight. Implementing the sorted greedy solution with O(n log n) time and O(1) extra space is typically considered the optimal answer.
| Approach | Time | Space | When to Use |
|---|---|---|---|
| Brute Force Permutations | O(n!) | O(n) | Useful only for understanding the problem or very small arrays |
| Greedy with Sorting | O(n log n) | O(1) | General optimal approach; sort values then assign largest squares to positive positions |