16.17. Contiguous Sequence

Description

You are given an array of integers (both positive and negative). Find the contiguous sequence with the largest sum. Return the sum.

Example:




Input:  [-2,1,-3,4,-1,2,1,-5,4]



Output:  6



Explanation:  [4,-1,2,1] has the largest sum 6.

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 $f[i]$ as the maximum sum of a continuous subarray that ends with $nums[i]$. The state transition equation is:

$$ f[i] = \max(f[i-1], 0) + nums[i] $$

where $f[0] = nums[0]$.

The answer is $\max\limits_{i=0}^{n-1}f[i]$.

The time complexity is $O(n)$, and the space complexity is $O(n)$. Here, $n$ is the length of the array.

We notice that $f[i]$ only depends on $f[i-1]$, so we can use a variable $f$ to represent $f[i-1]$, thus reducing the space complexity to $O(1)$.

Python3JavaC++GoTypeScriptJavaScriptSwift

class Solution:
    def maxSubArray(self, nums: List[int]) -> int:
        ans = f = -inf
        for x in nums:
            f = max(f, 0) + x
            ans = max(ans, f)
        return ans