leetcode -- Maximum Subarray

Find the contiguous subarray within an array (containing at least one number) which has the largest sum.

For example, given the array [−2,1,−3,4,−1,2,1,−5,4],
the contiguous subarray [4,−1,2,1] has the largest sum = 6.

More practice:

If you have figured out the O(n) solution, try coding another solution using the divide and conquer approach, which is more subtle.

[解题思路]

如果当前的和小于0，则在后面的数加上现在的和只会比原来的数要小，故舍弃

DP的思路(参考http://fisherlei.blogspot.com/2012/12/leetcode-maximum-subarray.html)：

O(n)就是一维DP.
假设A(0, i)区间存在k，使得[k, i]区间是以i结尾区间的最大值， 定义为Max[i], 在这里，当求取Max[i+1]时，
Max[i+1] = Max[i] + A[i+1],  if (Max[i] + A[i+1] >0)
                = 0, if(Max[i]+A[i+1] <0)，如果和小于零，A[i+1]必为负数，没必要保留，舍弃掉
然后从左往右扫描，求取Max数字的最大值即为所求。

public int maxSubArray(int[] A) {
        // Start typing your Java solution below
        // DO NOT write main() function
        int result = Integer.MIN_VALUE;
        int curSum = 0;
        for(int i = 0; i < A.length; i++){
            curSum += A[i];
            if(curSum > result){
                result = curSum;
            }
            if(curSum < 0){
                curSum = 0;
            }
        }
        
        return result;
    }

 (2) Divide and Conquer

二分法：将数组分成左右两部分，递归求两部分最大连续子数组，由于最大连续子数组可能横跨左右，故需对这种情况进行处理

假设数组A[left, right]存在最大值区间[i, j](i>=left & j<=right)，以mid = (left + right)/2 分界，无非以下三种情况：

subarray A[i,..j] is
(1) Entirely in A[low,mid-1]
(2) Entirely in A[mid+1,high]
(3) Across mid
对于(1) and (2)，直接递归求解即可，对于(3)，则需要以min为中心，向左及向右扫描求最大值，意味着在A[left, Mid]区间中找出A[i..mid], 而在A[mid+1, right]中找出A[mid+1..j]，两者加和即为(3)的解。

比较三种情况下得到的子数组和，取其中最大值

public int maxSubArray(int[] A) {
        // Start typing your Java solution below
        // DO NOT write main() function
 4         int max = Integer.MIN_VALUE;
        return maxArray(A, 0, A.length - 1, max);
    }
    
    int maxArray(int[] A, int left, int right, int max){
        if(left > right){
            return Integer.MIN_VALUE;
        }
        
        int mid = (left + right) / 2;
        int leftMax = maxArray(A, left, mid - 1, max);
        int rightMax = maxArray(A, mid + 1, right, max);
        
        max = Math.max(max, leftMax);
        max = Math.max(max, rightMax);
        
        int sum = 0, mlmax = 0;
        for(int i = mid - 1; i >= left; i--){
            sum += A[i];
            if(sum > mlmax){
                mlmax = sum;
            }
        }
        sum = 0; int mrmax = 0;
        for(int i = mid + 1; i <= right; i++){
            sum += A[i];
            if(sum > mrmax){
                mrmax = sum;
            }
        }
        max = Math.max(max, A[mid] + mlmax + mrmax);
        return max;
    }

时间复杂度： T(N) = 2*T(N/2) + O(N) 由主定理得 O(NlogN)