LeetCode 53 LintCode 41 Maximum Subarray

Table of contents

Description

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 <= 10^5
• -10^4 <= nums[i] <= 10^4

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.

Idea1

We could use dynamic programming to solve this question.

1. We could use two variables to track two states during the dynamic programming: the max sum ending at the current index and the result (max sum so far).
2. For each iteration, we add the current number to the maxHere, the max sum ending at the previous index. We compare that sum with the current number itself and take the maximum and set it to maxHere. For example, if maxHere was negative, the sum would definitely be smaller than the current number. Then we would just set the current number to maxHere.
3. Next, we take the maximum from the previous res and current maxHere to remember the max sum seen so far.
4. Finally, we return the result which records the maximum seen throughout.

Let’s use example 1 above and look at how maxHere and res would change.

array: [-2,1,-3,4,-1,2,1,-5,4]
look at index 0 number -2
maxHere needs to include at least the current number, because subarray is non-empty. so maxHere = -2
res = -2
look at index 1 number 1
maxHere = max(1, -2+1) = 1
res = max(-2,1) = 1

...

so maxHere and res values are below as we iterate through
maxHere: [-2,1,-2,4,3,5,6,1,4]
res: [-2,1,1,4,4,5,6,6,6]

Complexity: Time $O(n)$, Space $O(1)$.

Java

class Solution1 {
    int maxSubarrayDP(int[] nums) {
        int maxHere = 0, res = Integer.MIN_VALUE; // max sum ending at current index
        for (int n : nums) {
            maxHere = Math.max(n, maxHere + n); // note that the first item is the number, not 0
            res = Math.max(res, maxHere);
        }
        return res;
    }
}

Rust

use std::cmp::max;

/// leet 53

impl Solution {
    /// 0 ms, 3.34 mb.
    pub fn max_sub_array(nums: Vec<i32>) -> i32 {
        let (mut max_here, mut res) = (0, i32::MIN);
        for n in nums {
            max_here = max(max_here + n, n);
            res = max(res, max_here);
        }
        res
    }
}

Python

class Solution:
    """75 ms, 31.8 mb"""

    def maxSubArray(self, nums: list[int]) -> int:
        max_here, res = 0, -inf
        for n in nums:
            max_here = max(n, max_here + n)
            res = max(res, max_here)
        return res

Idea2

For the follow-up question, we could solve this question with divide and conquer.

We could use the prefix sum array data structure for a recursive solution. If we split a non-empty array into two halves, the maximum subarray can possibly be composed with three possible ways.

1. the entire subarray can be in the left half.
2. the entire subarray can be in the right half.
3. the subarray contains elements in both the left and right half.

We can calculate a forward maxHere array pre[], where pre[i] denotes max sum subarray ending at index i. And a reverse maxHere array suf[], where suf[i] denotes the maxi subarray starting at index i.

We use a recursive helper function to calculate the maximum subarray in the left and right halves. Then we take the maximum out of the three possibilities.

Complexity: Time $O(n)$, Space $O(n)$.

Alternatively, if we perform the calculation after the recursive call. We could implement a recursive solution with the complexity of Time $O(n\log n)$, Space $O(\log n)$ (recursive stack space).

Java

class Solution2 {
    int[] pre, suf;

    int maxSubArray(int[] nums) {
        int n = nums.length;
        pre = Arrays.copyOf(nums, n);
        suf = Arrays.copyOf(nums, n);
        for (int i = 1; i < n; i++) pre[i] += Math.max(0, pre[i - 1]);
        for (int i = n - 2; i >= 0; i--) suf[i] += Math.max(0, suf[i + 1]);
        return helper(nums, 0, n - 1);
    }

    /**
     * 72 ms, 58.05 mb.
     * Divide and conquer, O(N) time, O(N) space. T(N) = 2T(N/2) + O(1).
     * Alternative DnC solution available for O(NlgN) time and O(lgN) recursive stack space.
     */
    int helper(int[] nums, int left, int right) {
        if (left == right) return nums[left];
        int mid = (left + right) / 2;
        return Arrays.stream(new int[]{
                helper(nums, left, mid),
                helper(nums, mid + 1, right),
                pre[mid] + suf[mid + 1]}).max().orElseThrow();
    }
}