Problem Description
Given an integer array nums, find the subarray with the largest sum, and return its sum.
Key Insights
- The problem requires finding a contiguous subarray that has the maximum sum.
- A naive solution would involve checking all possible subarrays, leading to O(n^2) time complexity.
- An efficient approach involves using Kadane's algorithm, which operates in O(n) time.
- The divide and conquer approach can also be used, splitting the array into halves and combining results.
Space and Time Complexity
Time Complexity: O(n) for Kadane's algorithm, O(n log n) for the divide and conquer approach.
Space Complexity: O(1) for Kadane's algorithm, O(log n) for the divide and conquer approach due to recursion stack.
Solution
The problem can be efficiently solved using Kadane's algorithm, which maintains a running sum of the current subarray and updates the maximum sum found. When the running sum becomes negative, it is reset to zero, indicating the start of a new subarray. This approach uses constant space and runs in linear time.
For the divide and conquer method, the array is recursively split into two halves. The maximum subarray sum is computed for the left half, the right half, and the maximum sum that crosses the midpoint. The overall maximum is the maximum of these three values.