Maximum Subarray Leetcode Problem 53 [Python Solution]

Welcome to another LeetCode problem-solving guide! In this article, we’ll tackle the Maximum Subarray problem, which is Problem 53 on LeetCode.

We’ll provide you with a Python solution and a detailed explanation.

By the end of this guide, you’ll have a solid understanding of how to find the subarray with the largest sum within an array of integers.

Problem Overview

Given an integer array nums, our task is to find the contiguous subarray that has the largest sum and return that sum.

This problem is an excellent opportunity to explore an efficient approach for solving it.

Let’s start by looking at an example.

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 of 6.

Example 2:

Input: nums = [1]
Output: 1
Explanation: The subarray [1] has the largest sum, which is 1.

Example 3:

Input: nums = [5,4,-1,7,8]
Output: 23
Explanation: The subarray [5,4,-1,7,8] has the largest sum of 23.

Constraints:

1 <= nums.length <= 105
-104 <= nums[i] <= 104

Now, let’s dive into the details of solving this problem.

Problem Overview

The problem requires finding the maximum subarray sum within the given array.

The key challenge is to optimize this process and avoid inefficient brute-force approaches.

The brute-force method involves computing the sum of every possible subarray, which results in a time complexity of O(n^3), making it impractical for large arrays.

Understanding Constraints

Before we delve into the solution, it’s essential to understand the problem’s constraints.

The nums array can have a length ranging from 1 to 105, and each element can vary between -104 and 104. These constraints help us determine the efficiency and scalability of our solution.

Efficient Python Code Solution

Now, let’s present an efficient Python solution to this problem.

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def maxSubArray(self, nums: List[int]) -> int:
        max_sum = nums[0]  # Initialize max_sum with the first element of the array.

        current_sum = 0
        for num in nums:
            current_sum += num
            max_sum = max(max_sum, current_sum)

            # If the current sum becomes negative, reset it to zero.
            if current_sum &lt; 0:
                current_sum = 0

        return max_sum

In this solution, we use a linear-time algorithm to find the maximum subarray sum.

It employs a sliding window approach, where we iterate through the array while maintaining two variables: max_sum and current_sum.

This approach ensures that we achieve a time complexity of O(n).

1. Bruteforce Approach

Let’s briefly discuss the brute-force approach, even though we won’t be implementing it in our solution.

The brute-force method involves checking every possible subarray sum, which results in a time complexity of O(n^3).

This approach is highly inefficient for larger arrays and not suitable for solving this problem optimally.

2. Efficient Approach

Now, let’s explore the efficient approach we’ve implemented:

We initialize max_sum with the first element of the array, and current_sum is set to 0. These variables are used to keep track of the maximum subarray sum and the current subarray sum.
We iterate through the array, adding each element to current_sum.

At each step, we update max_sum with the maximum of its current value and current_sum.

This ensures that max_sum always contains the maximum subarray sum encountered so far.

If current_sum becomes negative at any point, we reset it to 0. This is a crucial step that allows us to discard negative prefixes and maintain a contiguous subarray.
Finally, we return the value stored in max_sum, which represents the maximum subarray sum.

Time and Space Complexity

Now, let’s analyze the time and space complexity of our efficient solution.

Time Complexity: The time complexity of our solution is O(n), where n is the length of the input array nums.

We iterate through the array once, performing constant-time operations at each step.

Space Complexity: Our solution has a space complexity of O(1) because we only use a constant amount of extra space to store variables like max_sum and current_sum.

It’s an in-place algorithm that doesn’t require additional data structures.

Reasoning Behind Our Approach

The efficient solution we’ve presented is based on a sliding window approach, which allows us to find the maximum subarray sum in a linear time complexity.

The key insights behind our approach are as follows:

Tracking Maximum Subarray Sum: We maintain a variable max_sum to keep track of the maximum subarray sum encountered so far.

This variable is continuously updated as we iterate through the array, ensuring that it contains the maximum sum.

Resetting Negative Prefixes: When current_sum becomes negative, we reset it to 0. This step is essential for dealing with negative prefixes that don’t contribute to the maximum subarray sum.

By resetting current_sum, we maintain a contiguous subarray.

Linear-Time Complexity: By iterating through the array only once, our solution achieves a time complexity of O(n), making it highly efficient, even for large arrays.

Conclusion

In this guide, we’ve tackled the Maximum Subarray problem (Problem 53 on LeetCode) using an efficient Python solution.

We’ve explored the key components of our solution, including the sliding window approach, tracking the maximum subarray sum, and resetting negative prefixes.

Efficiency is crucial when solving algorithmic problems, and our solution achieves a time complexity of O(n), making it suitable for large input arrays.

We hope this guide has been helpful in understanding how to find the maximum subarray sum efficiently.

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Happy coding!

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