What is the time complexity of Maximum Subarray?

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

What data structures are used to solve Maximum Subarray?

This problem uses Array, Divide and Conquer, Dynamic Programming. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Maximum Subarray?

Insight 1: A crucial observation is that if the current subarray sum becomes negative, it's better to start a new subarray from the next element, as adding a negative sum to any subsequent elements will only decrease the overall sum. Insight 2: Dynamic Programming can be used effectively. We keep track of the 'current maximum' sum ending at each element and the 'overall maximum' sum seen so far.

Which companies ask LeetCode 53. Maximum Subarray in interviews?

This problem is frequently asked at Accenture, Adobe, Amazon, Apple, Atlassian and 48 more companies.

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Maximum Subarray - LeetCode 53 Solution

Q: How to solve Maximum Subarray?

The provided code implements Kadane's Algorithm. It iterates through the array, maintaining two variables: `current_max` and `max_so_far`. `current_max` stores the maximum sum of a subarray ending at the current position. `max_so_far` stores the overall maximum sum encountered so far. At each step, we decide whether to extend the current subarray by including the current element or to start a new subarray from the current element. This decision is based on whether adding the current element to the current subarray sum increases the sum or makes it smaller than the current element itself. The `max_so_far` is updated in each iteration to keep track of the global maximum.

Maximum Subarray - Complete Solution Guide

Maximum Subarray is LeetCode problem 53, 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

Given an integer array nums , find the subarray with the largest sum, and return its sum . Example 1: Input: nums = [-2,1,-3,4,-1,2,1,-5,4] Output: 6 Explanation: The subarray [4,-1,2,1] has the largest sum 6. Example 2: Input: nums = [1] Output: 1 Explanation: The subarray [1] has the largest sum 1. Example 3: Input: nums = [5,4,-1,7,8] Output: 23 Explanation: The subarray [5,4,-1,7,8] has the largest sum 23. Constraints: 1 <= nums.length <= 10 5 -10 4 <= nums[i] <= 10 4 Follow up: If you have

Detailed Explanation

The problem asks us to find the contiguous subarray (a sequence of consecutive elements) within a given integer array that has the largest sum. We need to return only the sum of this maximum subarray, not the subarray itself. The input array can contain positive, negative, and zero values. The constraint specifies that the array will always have at least one element.

Solution Approach

The provided code implements Kadane's Algorithm. It iterates through the array, maintaining two variables: `current_max` and `max_so_far`. `current_max` stores the maximum sum of a subarray ending at the current position. `max_so_far` stores the overall maximum sum encountered so far. At each step, we decide whether to extend the current subarray by including the current element or to start a new subarray from the current element. This decision is based on whether adding the current element to the current subarray sum increases the sum or makes it smaller than the current element itself. The `max_so_far` is updated in each iteration to keep track of the global maximum.

Step-by-Step Algorithm

Step 1: Initialize `max_so_far` and `current_max` to the first element of the array.
Step 2: Iterate through the array from the second element.
Step 3: For each element, update `current_max` as the maximum between the current element itself and the sum of the current element and the previous `current_max`. This determines whether to start a new subarray or extend the existing one.
Step 4: Update `max_so_far` as the maximum between `max_so_far` and `current_max`. This ensures that `max_so_far` always holds the largest subarray sum seen so far.
Step 5: After the loop completes, return `max_so_far`.

Key Insights

Insight 1: A crucial observation is that if the current subarray sum becomes negative, it's better to start a new subarray from the next element, as adding a negative sum to any subsequent elements will only decrease the overall sum.
Insight 2: Dynamic Programming can be used effectively. We keep track of the 'current maximum' sum ending at each element and the 'overall maximum' sum seen so far.
Insight 3: The problem can be solved in O(n) time with O(1) space by using Kadane's Algorithm, which is a specific application of dynamic programming for this type of problem.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(1)

Topics

This problem involves: Array, Divide and Conquer, Dynamic Programming.

Companies

Asked at: Accenture, Adobe, Amazon, Apple, Atlassian, Autodesk, Barclays, Bloomberg, Cisco, Cognizant, DE Shaw, Goldman Sachs, Google, HashedIn, Huawei, IBM, Infosys, Intel, J.P. Morgan, LinkedIn, Meta, Microsoft, Morgan Stanley, Nike, Nvidia, Optum, Oracle, PayPal, PornHub, Qualcomm, Ripple, Rippling, SAP, Samsung, ServiceNow, Snap, Target, Tekion, Tesla, TikTok, Turing, Uber, Upstart, Vimeo, Visa, Walmart Labs, Wix, Yahoo, Zoho, Zomato, eBay, persistent systems, tcs.

Maximum Subarray | LeetCode 53

Medium

Array Divide and Conquer Dynamic Programming

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

Solution Code

Available in:

class Solution {
    public int maxSubArray(int[] nums) {
        int max_so_far = nums[0];
        int current_max = nums[0];

        for (int i = 1; i < nums.length; i++) {
            current_max = Math.max(nums[i], current_max + nums[i]);
            max_so_far = Math.max(max_so_far, current_max);
        }
        
        return max_so_far;
    }
}

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Problem Info

Description

Problem 53: Maximum Subarray - Detailed explanation and solution approaches.

Topics

Array Divide and Conquer Dynamic Programming

Companies

AccentureAdobeAmazonAppleAtlassianAutodeskBarclaysBloomberg

Stats

Difficulty:Medium

Languages:4

Companies:53

Problem #:53

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

5 steps

1
Step 1: Initialize `max_so_far` and `current_max` to the first element of the array.
2
Step 2: Iterate through the array from the second element.
3
Step 3: For each element, update `current_max` as the maximum between the current element itself and the sum of the current element and the previous `current_max`. This determines whether to start a new subarray or extend the existing one.
4
Step 4: Update `max_so_far` as the maximum between `max_so_far` and `current_max`. This ensures that `max_so_far` always holds the largest subarray sum seen so far.
5
Step 5: After the loop completes, return `max_so_far`.

💡

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)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The time complexity is linear because we visit each element in the input array only once. The space complexity is constant because we use only a fixed number of variables regardless of the input size.

📖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

Divide and Conquer

The problem can also be solved using a divide and conquer approach. The array is recursively divided into two halves. The maximum subarray can either be entirely within the left half, entirely within the right half, or cross the midpoint. The maximum subarray crossing the midpoint can be found in linear time. The algorithm then recursively solves the problem for the left and right halves and returns the maximum of the three possibilities.

Time: O(n log n)Space: O(log n) due to the recursive call stack.

Trade-offs: While the divide and conquer approach provides an alternative solution, it is generally less efficient than Kadane's algorithm (O(n)) due to the overhead of recursive calls. Divide and Conquer is useful when the problem naturally decomposes into subproblems that can be solved independently and combined efficiently. Kadane's algorithm is simpler to implement and more efficient for this specific problem.

✨

Key Insights

1Insight 1: A crucial observation is that if the current subarray sum becomes negative, it's better to start a new subarray from the next element, as adding a negative sum to any subsequent elements will only decrease the overall sum.
2Insight 2: Dynamic Programming can be used effectively. We keep track of the 'current maximum' sum ending at each element and the 'overall maximum' sum seen so far.
3Insight 3: The problem can be solved in O(n) time with O(1) space by using Kadane's Algorithm, which is a specific application of dynamic programming for this type of problem.

📚

Key Terms Explained

For Students

Array

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

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: Array with all negative numbers. The algorithm correctly identifies the largest single negative number as the maximum subarray sum. For example, if the input is [-1, -2, -3], the output will be -1.
Edge case 2: Array with only one element. The algorithm correctly returns that single element as the maximum subarray sum. For example, if the input is [5], the output will be 5.
Edge case 3: Array with all zeros. The algorithm correctly returns 0 as the maximum subarray sum (assuming handling of initial value).
Edge case 4: Large arrays with very large positive and negative numbers should still work correctly without integer overflow issues because the intermediate calculations (sums) could potentially become very large. Careful language choice and data type usage are important to avoid overflow.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to initialize `max_so_far` correctly. It's crucial to initialize it with the first element of the array, not zero, to handle cases where all elements are negative.

✗Common mistake 2: Incorrectly updating `current_max`. The update rule `current_max = max(nums[i], current_max + nums[i])` is critical. Using `current_max = current_max + nums[i]` alone would not account for the case where it's better to start a new subarray from the current element.

✗Common mistake 3: Assuming the array will always contain positive numbers. The algorithm must handle negative numbers correctly to find the maximum subarray.

✅

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: Stock price analysis: Finding the period with the maximum profit by identifying the subarray with the largest sum of price differences.

🏦Real-world scenario 2: Signal processing: Identifying segments of a signal with the highest energy or amplitude by finding the subarray with the largest sum of squared values.

🎮Real-world scenario 3: Bioinformatics: Identifying regions of a DNA sequence with the highest similarity to a target sequence by finding the subarray with the largest sum of similarity scores.

💡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

Consider multiple approaches before coding. Mentioning alternatives shows breadth of knowledge.

⚠️ 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: July 24, 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 14

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

@TechLeadJul 7

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jun 30

❓

Frequently Asked Questions

When should I use in-place modification vs creating a new array?

Use in-place modification when space complexity matters (O(1) space requirement) and the original array can be modified. Create a new array when you need to preserve the original data or when the problem involves significant restructuring that would complicate in-place logic.

How much time should I spend on a medium problem in interviews?

In a 45-minute interview, aim for: 5-10 minutes understanding and clarifying, 5-10 minutes discussing approach, 20-25 minutes coding, and 5-10 minutes testing. If stuck after 15 minutes of thinking, verbalize your thought process and ask for hints.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Maximum Subarray"?

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 divide and conquer techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?