What is the time complexity of Maximum Product of Three Numbers?

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

What data structures are used to solve Maximum Product of Three Numbers?

This problem uses Array, Math, Sorting. Understanding these concepts is essential for solving this Easy problem efficiently.

What are the key insights for solving Maximum Product of Three Numbers?

Insight 1: The maximum product can be formed either by the three largest positive numbers or by the two smallest (most negative) numbers multiplied by the largest positive number. The presence of negative numbers changes the nature of the problem. Insight 2: Sorting the array makes it easier to identify the smallest and largest numbers efficiently.

Which companies ask LeetCode 628. Maximum Product of Three Numbers in interviews?

This problem is frequently asked at FreshWorks, Infosys, Intuit, Nutanix, Roku and 2 more companies.

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Maximum Product of Three Numbers | LeetCode 628

Easy

Array Math Sorting

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

Solution Code

Available in:

class Solution {
    public int maximumProduct(int[] nums) {
        Arrays.sort(nums);
        int n = nums.length;
        return Math.max(nums[0] * nums[1] * nums[n - 1], nums[n - 1] * nums[n - 2] * nums[n - 3]);
    }
}

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Problem Info

Description

Problem 628: Maximum Product of Three Numbers - Detailed explanation and solution approaches.

Topics

Array Math Sorting

Companies

FreshWorksInfosysIntuitNutanixRokuSalesforceSiemens

Stats

Difficulty:Easy

Languages:4

Companies:7

Problem #:628

Problem Explanation

The problem asks us to find the maximum product of three numbers from a given integer array. The array contains at least three numbers and can have both positive and negative numbers. The goal is to return the largest possible product that can be achieved by multiplying any three elements in the array, respecting the constraints on the array's length and element values.

💡

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

The solution involves sorting the input array and then comparing two possible products: the product of the three largest numbers in the array and the product of the two smallest numbers and the largest number. The larger of these two products is the maximum product.

📋

Step-by-Step Solution

4 steps

1
Step 1: Sort the input array 'nums' in ascending order.
2
Step 2: Calculate the product of the three largest numbers in the array: nums[n-1] * nums[n-2] * nums[n-3], where 'n' is the length of the array.
3
Step 3: Calculate the product of the two smallest numbers and the largest number: nums[0] * nums[1] * nums[n-1].
4
Step 4: Return the maximum of the two calculated products.

💡

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 the sorting operation, which typically takes O(n log n) time using efficient sorting algorithms like merge sort or quicksort (Arrays.sort() in Java and sort() in C++ and Python are generally implemented using such algorithms). The rest of the operations (multiplication and comparison) take constant time. The space complexity is O(1) because the sorting is done in-place or uses a constant amount of 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

Linear Scan Approach

Instead of sorting, we can find the three largest and two smallest numbers in a single pass through the array. This involves maintaining five variables: min1, min2, max1, max2, and max3, and updating them as we iterate through the array.

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

Trade-offs: This approach is faster (O(n) vs O(n log n)) but more complex to implement correctly. It's beneficial when minimizing time is paramount, and the array size is large. The C code provides this implementation.

✨

Key Insights

1Insight 1: The maximum product can be formed either by the three largest positive numbers or by the two smallest (most negative) numbers multiplied by the largest positive number. The presence of negative numbers changes the nature of the problem.
2Insight 2: Sorting the array makes it easier to identify the smallest and largest numbers efficiently.
3Insight 3: We need to consider the edge case where the two smallest numbers are very large negative numbers, and the largest number is positive; their product can be greater than the product of the three largest numbers.

📚

Key Terms Explained

For Students

Array

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

Sorting

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

Edge Cases

Edge case 1: All numbers are negative. The algorithm still works correctly in this case because sorting will place the least negative (largest magnitude negative number) at the end, and multiplying the last three will result in the largest possible (most negative) product. The product of the first two and last elements will be the next largest of the two.
Edge case 2: Array contains zero. If array contains zero and two negative numbers then the product of two negative and one positive may not give maximum number. The logic will still work as array contains zero and the two negative numbers would be at the start of the sorted array.
Edge case 3: Only three elements in the array. The approach works correctly, as it simply sorts the three elements and multiplies them.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to consider the case where the two smallest numbers are negative and their product with the largest number yields the maximum product. Beginners might only consider the product of the three largest numbers.

✗Common mistake 2: Incorrectly implementing the linear scan approach (if chosen) leading to incorrect identification of the minimum and maximum numbers.

✅

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: Financial analysis where you want to find the three best-performing assets or the combination of assets that maximize returns, even if some assets have negative returns.

🏦Real-world scenario 2: Data analysis in scientific simulations to identify the combination of three parameters that leads to the most extreme or desired outcome, which could be maximizing or minimizing a certain metric.

💡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

Use this problem to demonstrate clean code practices. Variable naming and code organization matter.

⚠️ 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: August 28, 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

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 13

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 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 should I approach easy-level algorithm problems?

Easy problems build foundational skills. Focus on: 1) Understanding the problem completely before coding, 2) Writing clean, readable code, 3) Handling all edge cases (empty input, single element, etc.), 4) Analyzing time and space complexity. Don't skip complexity analysis even for simple solutions.

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 "Maximum Product of Three Numbers"?

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

🤖 Need help with this problem?

Maximum Product of Three Numbers - Complete Solution Guide

Problem Statement

Detailed Explanation

Solution Approach

Step-by-Step Algorithm

Key Insights

Complexity Analysis

Topics

Companies

Maximum Product of Three Numbers | LeetCode 628

Solution Code

Problem Info

Description

Topics

Companies

Stats

Problem Explanation

Think Before You Code

🤔 Ask Yourself

✏️ Try This First

Solution Approach

Step-by-Step Solution

Pro Tip for Students

Complexity Analysis

Time Complexity

Space Complexity

Why?

Big O Notation - Quick Guide

Alternative Approaches

Linear Scan Approach

Key Insights

Key Terms Explained

Array

Sorting

Edge Cases

Common Mistakes to Avoid

Learning Tip

Where is This Used in Real Life?

Author's Notes

About the Author

Community Tips & Insights

Frequently Asked Questions

What is the most efficient way to solve array problems?

How should I approach easy-level algorithm problems?

How should I practice algorithm problems effectively?

What is the optimal approach for "Maximum Product of Three Numbers"?

Maximum Product of Three Numbers | LeetCode 628

Solution Code

Problem Info

Description

Topics

Companies

Stats

Problem Explanation

Think Before You Code

🤔 Ask Yourself

✏️ Try This First

Solution Approach

Step-by-Step Solution

Pro Tip for Students

Complexity Analysis

Time Complexity

Space Complexity

Why?

Big O Notation - Quick Guide

Alternative Approaches

Linear Scan Approach

Key Insights

Key Terms Explained

Array

Sorting

Edge Cases

Common Mistakes to Avoid

Learning Tip

Where is This Used in Real Life?

Author's Notes

About the Author

Community Tips & Insights

Frequently Asked Questions

What is the most efficient way to solve array problems?