#2567

Medium

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Minimum Score by Changing Two Elements

The problem asks for the minimum score after changing two elements of an array using a greedy approach.

Problem Statement

You are given an integer array nums, and you need to return the minimum score possible by changing two elements of the array. The score is defined as the difference between the maximum and minimum values of the array after two elements have been altered. The goal is to find a way to minimize this difference efficiently.

The problem relies on a greedy approach where you select the two elements to change in such a way that the maximum and minimum values of the array are as close as possible. The key observation is that modifying the largest or smallest values of the array can significantly impact the score.

Examples

Example 1

Input: nums = [1,4,7,8,5]

Output: 3

Example 2

Input: nums = [1,4,3]

Output: 0

Constraints

3 <= nums.length <= 105
1 <= nums[i] <= 109

Solution Approach

Greedy Choice

A greedy strategy works here by aiming to minimize the difference between the maximum and minimum values in the array. The best approach is to change the largest and smallest elements in order to reduce the spread between them. Sorting the array helps quickly identify which elements to alter.

Invariant Validation

After making the changes, it's important to validate that the new array satisfies the required conditions. The validation step ensures that the new score is the minimum possible, and it often involves checking if the modified values hold the invariant of minimizing the range.

Edge Case Considerations

Consider edge cases where the array has a very small length or contains elements with values at the extremes of the allowed range. Handling these cases ensures the algorithm works efficiently under all constraints and prevents unnecessary computations.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity is dependent on the sorting step, which is O(n log n), and space complexity is O(n) for the array processing. The greedy approach and invariant validation add minimal overhead.

What Interviewers Usually Probe

The candidate focuses on adjusting the extremes of the array for minimization.
The candidate can correctly explain the impact of sorting and greedy choices.
The candidate handles edge cases effectively and ensures correctness through validation.

Common Pitfalls or Variants

Common pitfalls

Ignoring the importance of modifying the extremes (max and min) first.
Not validating the final array after changes, potentially leading to incorrect results.
Failing to account for edge cases such as very small arrays or large values.

Follow-up variants

Changing more than two elements and observing the impact on score.
Handling arrays with only one possible change, where the result is trivial.
Optimizing for specific constraints such as arrays with repeated elements.

FAQ

What is the key to solving the Minimum Score by Changing Two Elements problem?

The key is to minimize the difference between the maximum and minimum values by changing the largest and smallest elements, which can be efficiently done with a greedy approach.

How does sorting help in the Minimum Score by Changing Two Elements problem?

Sorting allows you to quickly identify the largest and smallest elements in the array, making it easy to apply the greedy strategy and minimize the score.

What are some edge cases to consider in this problem?

Edge cases include very small arrays, arrays with identical values, and arrays where one change results in the minimal score.

Can this problem be solved without sorting the array?

While sorting simplifies the process, you can also solve the problem by manually identifying the maximum and minimum values, though this may increase complexity.

What is the time complexity of the optimal solution for this problem?

The optimal solution has a time complexity of O(n log n) due to the sorting step, followed by constant time operations for validation.

terminal

Solution

Solution 1: Sorting + Greedy

From the problem description, we know that the minimum score is actually the minimum difference between two adjacent elements in the sorted array, and the maximum score is the difference between the first and last elements of the sorted array. The score of the array $nums$ is the sum of the minimum score and the maximum score.

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class Solution:
    def minimizeSum(self, nums: List[int]) -> int:
        nums.sort()
        return min(nums[-1] - nums[2], nums[-2] - nums[1], nums[-3] - nums[0])

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