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 <= 105
-104 <= nums[i] <= 104
Follow up: If you have figured out the O(n) solution, try coding another solution using the divide and conquer approach, which is more subtle.
Solutions
Solution 1: Dynamic Programming
We define $f[i]$ to represent the maximum sum of a contiguous subarray ending at element $\textit{nums}[i]$. Initially, $f[0] = \textit{nums}[0]$. The final answer we seek is $\max_{0 \leq i < n} f[i]$.
Consider $f[i]$ for $i \geq 1$. Its state transition equation is:
Since $f[i]$ is only related to $f[i - 1]$, we can use a single variable $f$ to maintain the current value of $f[i]$ and perform the state transition. The answer is $\max_{0 \leq i < n} f$.
The time complexity is $O(n)$, where $n$ is the length of the array $\textit{nums}$. The space complexity is $O(1)$.