最大连续子数组和(最大字段和)

最大连续子数组：

问题：

给定n个整数（可能为负数）组成的序列a[1],a[2],a[3],…,a[n],求该序列如a[i]+a[i+1]+…+a[j]的子段和的最大值。当所给的整数均为负数时定义子段和为0，依此定义，所求的最优值为： Max{0,a[i]+a[i+1]+…+a[j]},1<=i<=j<=n

例如，当（a[1],a[2],a[3],a[4],a[5],a[6]）=(-2,11,-4,13,-5,-2)时，最大子段和为20。

分治法：

def Max_cross_sum(list1,left,mid,right):
    left_sum = 0
    right_sum = 0
    temp = 0
    mid_1 = mid
    while mid_1>=left:
        temp += list1[mid_1]
        left_sum = max(left_sum,temp)
        mid_1 -= 1
    temp = 0
    mid_1 = mid + 1
    while mid_1<=right:
        temp += list1[mid_1]
        right_sum = max(right_sum,temp)
        mid_1 += 1
    return left_sum+right_sum
def Max_sum(list1,left,right):
    if left == right:
        return list1[left]
    mid = (left+right)//2
    left_sum = Max_sum(list1,left,mid)
    right_sum = Max_sum(list1,mid+1,right)
    cross_sum = Max_cross_sum(list1,left,mid,right)
    return max(left_sum,max(right_sum,cross_sum))    
list1 = [-2,1,-3,4,-10,2,10,-5,6]
Max_sum(list1,0,len(list1)-1)

动态规划法：

def Max_sum(lis1):
    temp = 0
    max_sum = 0
    for cnt in list1:
        temp = max(temp+cnt,cnt)
        max_sum = max(max_sum,temp)
    return max_sum
list1 = [-2,1,-3,4,-1,2,10,-5,1]
Max_sum(list1)