Maximum Subarray

Problem

Given an integer array nums, find the contiguous subarray (containing at least one number) which has the largest sum and return its sum.

A subarray is a contiguous part of an array.

 

Example 1:

Input: nums = [-2,1,-3,4,-1,2,1,-5,4]
Output: 6
Explanation: [4,-1,2,1] has the largest sum = 6.

Example 2:

Input: nums = [1]
Output: 1

Example 3:

Input: nums = [5,4,-1,7,8]
Output: 23

 

Constraints:

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

Solution

/**
 * @param {number[]} nums
 * @return {number}
 */
var maxSubArray = function(nums) {
    let max = nums[0];
    const dp = [max];

    for (let i = 1; i < nums.length; i++) {
        if (dp[i - 1] > 0) {
            dp.push(nums[i] + dp[i - 1]);
        } else {
            dp.push(nums[i]);
        }
        max = Math.max(max, dp[i]);
    }
    return max;
};

We will implement a DP solution. Let dp[i] be the largest sum of the contiguous subarray ending and including nums[i].

• For the base case, if i = 0, then dp[0] = nums[0] since there's only a single element of nums available.
• For the recursive case:
  • If the previous largest sum dp[i - 1] is non-negative, the next max sum which includes nums[i] must be nums[i] + dp[i - 1] (since adding a non-negative number will never result in a smaller value).
  • If the previous largest sum dp[i - 1] is negative, the next max sum which includes nums[i] must be nums[i] itself (since adding a negative number will always result in a smaller value).