最大子串和问题

[LeetCode 53] Maximum Subarray

题目

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

测试案例

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

思路

最大子串和是一个动态规划问题。记，a[i - 1] 为 0 ~ i - 1 子串中最大子串和，sum[i - 1] 为以第 i - 1 个元素结尾的子串和的最大值。那么有如下递归式：
\[ \begin{align} sum[i] &= nums[i] + (sum[i - 1] > 0 \;?\; sum[i - 1] : 0) \\ a[i] &= max(a[i - 1],sum[i]) \end{align} \]

代码如下

class Solution {
    public int maxSubArray(int[] nums) {
        int n = nums.length, max = nums[0], sum = nums[0], temp;        
        for(int i = 1; i < n; i++){
            temp = nums[i] + (sum > 0 ? sum : 0);
            sum = temp;
            if(max < temp){
                max = temp;
            }
        }
        return max;
    }
}