In this blog post, we will explore the problem of minimizing the maximum difference between the heights of a given array of integers. This is a common problem in competitive programming and coding interviews and is featured in platforms like LeetCode. We’ll cover the problem statement, solution approach, and optimal algorithm with a step-by-step explanation to help you fully understand how to tackle this problem effectively.

The challenge is to minimize the maximum difference between the heights in an array after either increasing or decreasing the heights by a fixed value k. The aim is to adjust the heights in such a way that the difference between the maximum and minimum heights is minimized.

Problem Statement:

Given an array of integers heights[] of size n and an integer k, you are allowed to increase or decrease each element by at most k. The task is to minimize the maximum difference between the heights after modification.

Example:

Input:
heights[] = [1, 5, 8, 10], k = 2

Output:
5

Explanation:
The array can be modified to [3, 3, 6, 8]. The maximum difference is 8 - 3 = 5.

Understanding the Approach:

Step-by-Step Solution:

  1. Sort the Heights Array:
    Sorting helps us systematically look at the minimum and maximum heights in the array, making it easier to manipulate the array elements.
  2. Calculate the Initial Difference:
    Find the initial difference between the maximum and minimum heights (heights[n-1] - heights[0]).
  3. Iterate Through the Array:
    For each element in the array, find the possible new maximum and minimum heights after adding or subtracting k:
  • Maximum Height: It’s either the last element increased by k or the current element decreased by k.
  • Minimum Height: It’s either the first element increased by k or the current element increased by k.
  1. Update the Minimum Difference:
    Calculate the difference between the maximum and minimum heights after modifications and update the minimum difference found so far.
  2. Output the Minimum Difference:
    Return the minimum difference between the maximum and minimum heights.

Pseudocode:

def minimize_max_difference(heights, k):
    n = len(heights)
    if n == 1:
        return 0  # Only one element, no difference

    # Sort the array
    heights.sort()

    # Initialize the difference
    max_diff = heights[n-1] - heights[0]

    # Traverse through the array
    for i in range(1, n):
        # Calculate potential new max and min
        max_height = max(heights[i-1] + k, heights[n-1] - k)
        min_height = min(heights[0] + k, heights[i] - k)

        # Update the minimized max difference
        max_diff = min(max_diff, max_height - min_height)

    return max_diff

Detailed Explanation:

  • Sorting the Array:
    Sorting the array helps us manipulate the heights in a more manageable way by looking at the sequential changes in heights.
  • Initial Difference Calculation:
    By finding the initial difference between the highest and lowest values in the sorted array, we get an idea of the base difference that we need to minimize.
  • Traversing the Array for New Heights:
    For each height i, we calculate the new potential maximum and minimum heights. This is done by checking the effect of adding or subtracting k from the elements. We try to find combinations that minimize the difference between the new maximum and minimum.
  • Finding the Minimum Difference:
    By comparing all possible new differences between the maximum and minimum heights across the array, we arrive at the minimized difference.

Time Complexity:

The time complexity of this approach is O(n log n) due to the initial sorting of the array, and the subsequent traversal is O(n).

Space Complexity:

The space complexity is O(1) as we only use a few extra variables for comparison.

Code Implementation in Python:

def minimize_max_difference(heights, k):
    n = len(heights)

    if n == 1:
        return 0  # If there's only one element, there's no difference

    # Step 1: Sort the array
    heights.sort()

    # Step 2: Calculate the initial difference
    initial_diff = heights[n-1] - heights[0]

    # Step 3: Initialize the minimum difference
    min_diff = initial_diff

    # Traverse through the sorted array
    for i in range(1, n):
        # Calculate potential max and min heights
        max_height = max(heights[i-1] + k, heights[n-1] - k)
        min_height = min(heights[0] + k, heights[i] - k)

        # Update the minimized difference
        min_diff = min(min_diff, max_height - min_height)

    return min_diff

# Example Usage
heights = [1, 5, 8, 10]
k = 2
print("The minimized maximum difference is:", minimize_max_difference(heights, k))

Output:

The minimized maximum difference is: 5

This problem is an excellent example of how sorting and calculating potential maximum and minimum values play a critical role in minimizing differences in arrays. The approach is efficient and provides an optimal solution to minimize the maximum difference between the heights after adjusting them by k.

By following the approach mentioned above, you’ll be able to solve similar problems effectively, not only in coding interviews but also in real-world applications where minimizing differences or variations is required.

Happy coding! 🚀

Related LeetCode Problem:

This problem can be found on LeetCode as Problem 1505: Minimize the Maximum Difference Between Heights. Practicing this on LeetCode will help you understand various edge cases and enhance your algorithmic skills.

FAQs for “Minimize the Maximum Difference Between Heights

Q1: What does the problem “Minimize the Maximum Difference Between Heights” involve?
A1: This problem involves modifying the heights in an array by either increasing or decreasing each element by a fixed value k. The goal is to minimize the difference between the maximum and minimum heights in the array after the modification.

Q2: Why do we need to sort the array in the solution?
A2: Sorting helps to systematically compare the heights in ascending order, making it easier to calculate possible combinations of maximum and minimum heights after adding or subtracting k.

Q3: What is the significance of the value k in this problem?
A3: The value k represents the maximum amount by which you can either increase or decrease each height in the array. This gives you flexibility to adjust heights to achieve the minimum difference between the maximum and minimum values.

Q4: What is the time complexity of this solution?
A4: The time complexity is O(n log n) due to the sorting step. After sorting, the traversal through the array takes O(n) time.

Q5: Can I solve the problem without sorting the array?
A5: Sorting the array is essential to efficiently compare the minimum and maximum possible values across all combinations. Without sorting, it would be harder to optimize the solution and achieve a minimized difference.

Q6: Why is the initial difference calculated as heights[n-1] - heights[0]?
A6: This initial difference represents the difference between the maximum and minimum heights in the sorted array before making any modifications. It’s a baseline difference that will be compared against as you attempt to minimize it.

Q7: What is the strategy for finding the new maximum and minimum heights?
A7: For each height in the array, calculate two possible maximum and minimum heights after adding or subtracting k. By considering all combinations, the minimized difference is found by comparing all possible differences between these calculated heights.

Q8: How does adding or subtracting k to each element help minimize the difference?
A8: Adding or subtracting k to each element gives flexibility in adjusting the heights. This allows for reducing the gap between the maximum and minimum values in the array, effectively minimizing the overall difference.

Q9: What edge cases should be considered for this problem?
A9: Some edge cases include arrays with only one element (where the difference is 0), arrays with equal elements, and scenarios where k is 0 (no change possible). Ensure your solution handles these cases effectively.

Q10: How can I test my solution against different inputs?
A10: Test your solution with arrays of different sizes, including arrays with all elements equal, arrays with large values of k, and arrays with a single element. This will ensure that your solution is robust and handles all possible scenarios correctly.

Q11: Where can I practice this problem on LeetCode or similar platforms?
A11: This problem is available on platforms like LeetCode, under different titles such as “Minimize the Maximum Difference Between Heights.” Practicing on such platforms will give you access to various test cases and help you improve your algorithmic skills.

Q12: How do I know if my solution is optimal?
A12: An optimal solution will have a time complexity of O(n log n) or better and will effectively minimize the maximum difference between heights for all tested scenarios. Comparing your solution against others and checking for any inefficiencies can also help you determine its optimality.

These FAQs cover some common questions and clarifications related to the problem of minimizing the maximum difference between heights in an array. If you have more specific questions or edge cases, practicing on coding platforms and exploring community discussions can further enhance your understanding.

Read More ….

Array Sorting: Comprehensive Guide, Techniques, and LeetCode Solutions – https://kamleshsingad.com/wp-admin/post.php?post=5110&action=edit

Bulb Switcher III: A Comprehensive Guide and Solution – https://kamleshsingad.com/wp-admin/post.php?post=5096&action=edit

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