Maximum Sum Subarray in JavaScript

Given an input array that may contain both positive and negative integers, find the sum of continuous subarray of numbers which has the largest sum.

Example:

Input: nums = [-2, -5, 6, -2, -3, 1, 5, -6]
Output: 7
Explanation: sum([6, -2, -3, 1, 5]) = 7

Note:

Your algorithm should run in O(n) time and use at most O(n) extra space.

Understanding the Problem

The core challenge of this problem is to find the maximum sum of a continuous subarray within a given array of integers, which may include both positive and negative numbers. This problem is significant in various applications such as financial analysis, where one might want to find the period with the maximum profit or minimum loss.

Potential pitfalls include misunderstanding the requirement for the subarray to be continuous and not considering negative numbers correctly.

Approach

To solve this problem, we can use Kadane's Algorithm, which is an efficient way to find the maximum sum subarray in O(n) time complexity. The algorithm works by iterating through the array and keeping track of the maximum sum of the subarray ending at the current position.

Naive Solution

A naive solution would involve checking all possible subarrays and calculating their sums, which would result in a time complexity of O(n^2). This is not optimal for large arrays.

Optimized Solution: Kadane's Algorithm

Kadane's Algorithm improves upon the naive solution by maintaining two variables: the maximum sum of the subarray found so far and the maximum sum of the subarray ending at the current position. This allows us to find the solution in a single pass through the array.

Algorithm

Here is a step-by-step breakdown of Kadane's Algorithm:

Initialize two variables: maxSoFar and maxEndingHere to the first element of the array.
Iterate through the array starting from the second element.
For each element, update maxEndingHere to be the maximum of the current element and maxEndingHere + current element.
Update maxSoFar to be the maximum of maxSoFar and maxEndingHere.
After iterating through the array, maxSoFar will contain the maximum sum of the subarray.

Code Implementation

/**
 * Function to find the maximum sum of a continuous subarray
 * @param {number[]} nums - The input array
 * @return {number} - The maximum sum of the subarray
 */
function maxSubArray(nums) {
  // Initialize maxSoFar and maxEndingHere to the first element
  let maxSoFar = nums[0];
  let maxEndingHere = nums[0];

  // Iterate through the array starting from the second element
  for (let i = 1; i < nums.length; i++) {
    // Update maxEndingHere to be the maximum of the current element and maxEndingHere + current element
    maxEndingHere = Math.max(nums[i], maxEndingHere + nums[i]);
    // Update maxSoFar to be the maximum of maxSoFar and maxEndingHere
    maxSoFar = Math.max(maxSoFar, maxEndingHere);
  }

  // Return the maximum sum found
  return maxSoFar;
}

// Example usage:
const nums = [-2, -5, 6, -2, -3, 1, 5, -6];
console.log(maxSubArray(nums)); // Output: 7

Complexity Analysis

The time complexity of Kadane's Algorithm is O(n) because it involves a single pass through the array. The space complexity is O(1) as it uses a constant amount of extra space.

Edge Cases

Potential edge cases include:

All negative numbers: The algorithm should return the maximum single element.
Single element array: The algorithm should return that element.
Mixed positive and negative numbers: The algorithm should correctly identify the subarray with the maximum sum.

Examples:

Input: nums = [-1, -2, -3]
Output: -1

Input: nums = [1]
Output: 1

Input: nums = [1, 2, 3, -2, 5]
Output: 9

Testing

To test the solution comprehensively, consider using a variety of test cases, including simple, complex, and edge cases. JavaScript testing frameworks like Jest can be used to automate the testing process.

Thinking and Problem-Solving Tips

When approaching such problems, it is essential to:

Understand the problem requirements and constraints thoroughly.
Start with a naive solution to get a basic understanding.
Look for patterns and optimizations to improve the solution.
Practice similar problems to enhance problem-solving skills.

Conclusion

In this blog post, we discussed how to solve the Maximum Sum Subarray problem using Kadane's Algorithm. We covered the problem definition, approach, algorithm, code implementation, complexity analysis, edge cases, and testing. Understanding and solving such problems is crucial for improving algorithmic thinking and coding skills.

Additional Resources

For further reading and practice, consider the following resources:

Maximum Sum Subarray III in O(n) Time Complexity using JavaScript

( Prefix Sums Approach )