What is the time complexity of Minimize the Maximum Difference of Pairs?

The optimal solution has a time complexity of O(n log n).

What data structures are used to solve Minimize the Maximum Difference of Pairs?

This problem uses Array, Binary Search, Dynamic Programming, Greedy, Sorting. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Minimize the Maximum Difference of Pairs?

Insight 1: Sorting the input array allows us to easily compute the differences between adjacent elements, which is crucial for finding pairs with minimal differences. Insight 2: Binary search can be used to efficiently search for the minimum maximum difference among all possible differences. We can use a 'can_form_p_pairs' helper function to determine if a given maximum difference is feasible.

Which companies ask LeetCode 2616. Minimize the Maximum Difference of Pairs in interviews?

This problem is frequently asked at Media.net, Navi.

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Minimize the Maximum Difference of Pairs - LeetCode 2616 Solution

Minimize the Maximum Difference of Pairs - Complete Solution Guide

Minimize the Maximum Difference of Pairs is LeetCode problem 2616, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

You are given a 0-indexed integer array nums and an integer p . Find p pairs of indices of nums such that the maximum difference amongst all the pairs is minimized . Also, ensure no index appears more than once amongst the p pairs. Note that for a pair of elements at the index i and j , the difference of this pair is |nums[i] - nums[j]| , where |x| represents the absolute value of x . Return the minimum maximum difference among all p pairs. We define the maximum of an empty set to be zero. Examp

Detailed Explanation

The problem requires finding 'p' pairs of indices from a given array 'nums' such that the maximum difference between the elements in these pairs is minimized. The absolute difference between the numbers at chosen indices forms a pair's 'difference'. The goal is to minimize the maximum of these individual pair differences, ensuring no index is used more than once. If 'p' is 0, the answer is 0.

Solution Approach

The solution uses a binary search approach. First, the input array 'nums' is sorted. Then, a binary search is performed on the range of possible maximum differences, which is from 0 to the difference between the largest and smallest elements in 'nums'. For each 'mid' value in the binary search, the 'can_form_p_pairs' function checks if it's possible to form 'p' pairs where the difference between each pair is at most 'mid'. If it is possible, we try a smaller maximum difference; otherwise, we try a larger maximum difference. The binary search continues until the minimum possible maximum difference is found.

Step-by-Step Algorithm

Step 1: Sort the input array 'nums' in ascending order.
Step 2: Initialize 'low' to 0 and 'high' to nums[n - 1] - nums[0], where n is the length of nums. These represent the minimum and maximum possible differences between pairs, respectively.
Step 3: Perform binary search between 'low' and 'high'.
Step 4: In each iteration of the binary search, calculate 'mid' as the average of 'low' and 'high'.
Step 5: Call the 'can_form_p_pairs' function with 'mid' as the maximum allowed difference.
Step 6: If 'can_form_p_pairs(mid)' returns true, it means we can form 'p' pairs with a maximum difference of 'mid' or less. Update 'high' to 'mid'.
Step 7: If 'can_form_p_pairs(mid)' returns false, it means we cannot form 'p' pairs with a maximum difference of 'mid'. Update 'low' to 'mid + 1'.
Step 8: Repeat steps 4-7 until 'low' is equal to 'high'. The final value of 'low' (or 'high') is the minimum maximum difference.

Key Insights

Insight 1: Sorting the input array allows us to easily compute the differences between adjacent elements, which is crucial for finding pairs with minimal differences.
Insight 2: Binary search can be used to efficiently search for the minimum maximum difference among all possible differences. We can use a 'can_form_p_pairs' helper function to determine if a given maximum difference is feasible.
Insight 3: The 'can_form_p_pairs' function determines if it is possible to form 'p' pairs such that the difference between each pair is less than or equal to the maximum difference given as input.

Complexity Analysis

Time Complexity: O(n log n)

Space Complexity: O(1)

Topics

This problem involves: Array, Binary Search, Dynamic Programming, Greedy, Sorting.

Companies

Asked at: Media.net, Navi.

Minimize the Maximum Difference of Pairs | LeetCode 2616

Medium

Array Binary Search Dynamic Programming Greedy Sorting

Time: O(n log n)Space: O(1)

Solution Code

Available in:

import java.util.Arrays;

class Solution {
    public int minimizeMax(int[] nums, int p) {
        if (p == 0) {
            return 0;
        }

        Arrays.sort(nums);
        int n = nums.length;

        
        
        
        
        
        
        

        
        
        
        

        
        
        
        
        

        int low = 0;
        int high = nums[n - 1] - nums[0];

        while (low < high) {
            int mid = low + (high - low) / 2;
            if (can_form_pairs(nums, mid, p)) {
                high = mid;
            } else {
                low = mid + 1;
            }
        }

        return low;
    }

    private boolean can_form_pairs(int[] nums, int max_diff, int p) {
        int count = 0;
        int i = 0;
        int n = nums.length;
        while (i < n - 1) {
            if (nums[i + 1] - nums[i] <= max_diff) {
                count += 1;
                i += 2;
            } else {
                i += 1;
            }
        }
        return count >= p;
    }
}

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Problem Info

Description

Problem 2616: Minimize the Maximum Difference of Pairs - Detailed explanation and solution approaches.

Topics

Array Binary Search Dynamic Programming Greedy Sorting

Companies

Media.netNavi

Stats

Difficulty:Medium

Languages:4

Companies:2

Problem #:2616

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

8 steps

1
Step 1: Sort the input array 'nums' in ascending order.
2
Step 2: Initialize 'low' to 0 and 'high' to nums[n - 1] - nums[0], where n is the length of nums. These represent the minimum and maximum possible differences between pairs, respectively.
3
Step 3: Perform binary search between 'low' and 'high'.
4
Step 4: In each iteration of the binary search, calculate 'mid' as the average of 'low' and 'high'.
5
Step 5: Call the 'can_form_p_pairs' function with 'mid' as the maximum allowed difference.
6
Step 6: If 'can_form_p_pairs(mid)' returns true, it means we can form 'p' pairs with a maximum difference of 'mid' or less. Update 'high' to 'mid'.
7
Step 7: If 'can_form_p_pairs(mid)' returns false, it means we cannot form 'p' pairs with a maximum difference of 'mid'. Update 'low' to 'mid + 1'.
8
Step 8: Repeat steps 4-7 until 'low' is equal to 'high'. The final value of 'low' (or 'high') is the minimum maximum difference.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n log n)

✨ Log-linear - Great for sorting algorithms!

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The time complexity is dominated by sorting the array, which is O(n log n). The binary search has O(log(max_diff)) iterations, where max_diff is the difference between the maximum and minimum elements in the array. Inside the binary search, we call the 'can_form_p_pairs' function, which iterates through the array once, taking O(n) time. Space complexity is constant as we're not using any additional data structures that scale with the input size. Although sorting might use extra space in some implementations, the dominant factor is constant extra space.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Dynamic Programming

While not the most efficient, a dynamic programming approach could theoretically be used to determine whether it is possible to select p pairs such that the maximum difference is minimized. Define dp[i][j] as whether it is possible to select j pairs from the first i elements of the array.

Time: O(n*p)Space: O(n*p)

Trade-offs: DP uses more space than binary search. It is unlikely to improve upon the speed of the binary search method. It can only be faster when number of pairs is very small compared to the number of elements in the array.

✨

Key Insights

1Insight 1: Sorting the input array allows us to easily compute the differences between adjacent elements, which is crucial for finding pairs with minimal differences.
2Insight 2: Binary search can be used to efficiently search for the minimum maximum difference among all possible differences. We can use a 'can_form_p_pairs' helper function to determine if a given maximum difference is feasible.
3Insight 3: The 'can_form_p_pairs' function determines if it is possible to form 'p' pairs such that the difference between each pair is less than or equal to the maximum difference given as input.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Binary Search

Halving the search space each step. Like finding a word in a dictionary!

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Greedy Algorithm

Making the best choice at each step without looking ahead.

Sorting

Arranging elements in a specific order (ascending/descending).

Edge Cases

Edge case 1: p = 0. If p is 0, the function should return 0 immediately because no pairs need to be formed.
Edge case 2: nums.length is small (e.g., 1 or 2). The binary search and pair formation logic should still work correctly.
Edge case 3: All elements in 'nums' are the same. The minimum maximum difference should be 0.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to sort the array before starting the binary search. Sorting is crucial for the 'can_form_p_pairs' function to work correctly.

✗Common mistake 2: Incorrect binary search implementation (e.g., infinite loop). Ensure that 'low' and 'high' are updated correctly in each iteration.

✗Common mistake 3: Integer overflow when calculating 'mid' in the binary search. Use `mid = low + (high - low) / 2` to avoid this.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Resource allocation. You have a set of tasks with varying resource requirements, and you want to pair them to minimize the maximum difference in resource usage within each pair.

🏦Real-world scenario 2: Data compression. Grouping similar data points together can reduce the amount of storage space required. This algorithm helps minimize the differences between elements in each group.

🎮Real-world scenario 3: Image processing. In image compression or noise reduction, you might group pixels together based on their color or intensity values. Minimizing the difference between pixel values within each group can lead to better image quality after compression or filtering.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: September 5, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 13

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 6

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

How do I avoid off-by-one errors in binary search?

Use the pattern: left = 0, right = n-1, while (left <= right), mid = left + (right - left) / 2. Decide if you need to find exact match, first occurrence, or last occurrence, and adjust your conditions accordingly. Test with arrays of size 0, 1, and 2.

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Minimize the Maximum Difference of Pairs"?

For this medium-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether binary search techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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