Minimum Score by Changing Two Elements

Problem Overview

LeetCode Problem involves finding the minimum possible score by changing exactly two elements in an array of integers. The problem essentially requires a judicious approach to altering elements to achieve an optimal set of conditions, making it a classic problem in the domain of Greedy Algorithm.

Understanding the Problem

Problem Statement

Given an integer array nums, you are tasked to find the smallest possible difference between the maximum and minimum elements of nums after changing exactly two elements. You can replace those two chosen elements by any integer value.

1. Input: An integer array nums.
2. Output: The smallest possible difference between the highest and lowest values in the array after changing any two elements.

Constraints:

• You cannot change more than two elements.
• You can only change the elements to any integer value.

Key DSA Concepts and Theory

Greedy Algorithm

The Greedy Algorithm is a problem-solving approach that builds up a solution piece by piece, choosing the next piece that offers the most apparent and immediate benefit (local optima), hoping that the results yield a global optimum. In this context, we aim to minimize the array's score by strategically altering specific elements to derive immediate benefits.

Core Idea

The problem can be dissected as a range minimization between the maximum and minimum values in the array after some modifications. To effectively reduce this range with limited changes, it is crucial to understand how these changes influence the difference and focus on bounding these values tightly.

Solution Approach

To solve the problem, follow these steps:

1. Initial Insights:

   • When the two extreme elements in an array are changed, this ideally impacts the range significantly. Therefore, the change should focus on these extremities.
   • In scenarios where n (the size of nums) is 4 or less, we can change all numbers to be the same, resulting in a range of 0.

2. Sorting and Selecting Candidates:

   • By sorting the array first, identify the most suitable candidates for replacement to minimize the difference between max and min.

3. Strategy:

   • After sorting, consider different pairs of changes that can lead to minimized differences. Specifically, adjust the smallest and largest values by considering intervals:
     • Discard top two maximums and bottom zero minimums.
     • Discard top one maximum and bottom one minimum.
     • And so on.
   • Use the sorted array to calculate possible minimum differences when only two changes are allowed.

C++ Code Example

#include <vector>
#include <algorithm>

int minDifference(std::vector<int>& nums) {
    int n = nums.size();
    if (n <= 4) return 0; // Changing all elements if n <= 4
    std::sort(nums.begin(), nums.end());
    // Considering four cases by changing two elements:
    return std::min({
        nums[n-1]-nums[2],  // Change first two minimums
        nums[n-2]-nums[1],  // Change one maximum and one minimum
        nums[n-3]-nums[0]   // Change first three maximums
    });
}

Java Code Example

import java.util.Arrays;

public int minDifference(int[] nums) {
    int n = nums.length;
    if (n <= 4) return 0; // All numbers can be made the same
    Arrays.sort(nums);
    return Math.min(
        Math.min(nums[n-1] - nums[2], nums[n-2] - nums[1]),
        nums[n-3] - nums[0]
    );
}

Time and Space Complexity Analysis

• Time Complexity: O(n log n), primarily due to the sorting operation. Afterwards, calculating the minimum difference takes constant time O(1).
• Space Complexity: O(1), as no additional space proportional to input size is used. Sorting might require additional space based on the algorithm used (like quicksort).

Common Mistakes to Avoid

• Not Sorting Correctly: If the sorting step is skipped or done incorrectly, it can lead to inaccurate calculations.
• Incorrect Pair Choices: Focusing on altering a wrong pair of numbers can easily lead to suboptimal solutions, particularly not considering edge case scenarios like very small arrays.

Similar Problems on LeetCode

• 1509. Minimum Difference Between Largest and Smallest Value in Three Moves - This problem similarly requires minimizing range differences with limited changes.

Additional Resources and References

• To explore more about Greedy Algorithms in a structured manner, consider studying classical problems like Interval Scheduling Maximizing and Activity Selection.
• For a deeper dive into array manipulation, consider exploring various sorting techniques, which are pivotal in the preprocessing phase of complex problems.