Codility 9.3 MaxDoubleSliceSum

Find the maximal sum of any double slice.

Task description
A non-empty array A consisting of N integers is given.

A triplet (X, Y, Z), such that 0 ≤ X < Y < Z < N, is called a double slice.

The sum of double slice (X, Y, Z) is the total of A[X + 1] + A[X + 2] + ... + A[Y − 1] + A[Y + 1] + A[Y + 2] + ... + A[Z − 1].

For example, array A such that:

    A[0] = 3
    A[1] = 2
    A[2] = 6
    A[3] = -1
    A[4] = 4
    A[5] = 5
    A[6] = -1
    A[7] = 2
contains the following example double slices:

double slice (0, 3, 6), sum is 2 + 6 + 4 + 5 = 17,
double slice (0, 3, 7), sum is 2 + 6 + 4 + 5 − 1 = 16,
double slice (3, 4, 5), sum is 0.
The goal is to find the maximal sum of any double slice.

Write a function:

def solution(A)

that, given a non-empty array A consisting of N integers, returns the maximal sum of any double slice.

For example, given:

    A[0] = 3
    A[1] = 2
    A[2] = 6
    A[3] = -1
    A[4] = 4
    A[5] = 5
    A[6] = -1
    A[7] = 2
the function should return 17, because no double slice of array A has a sum of greater than 17.

Write an efficient algorithm for the following assumptions:

N is an integer within the range [3..100,000];
each element of array A is an integer within the range [−10,000..10,000].

参考资料：
https://en.wikipedia.org/wiki/Maximum_subarray_problem

思路来源于https://www.cnblogs.com/wei-li/p/MaxSliceSum.html

1. 乍一看这个要比最大子段和高级，似乎要枚举最大字串的右边界i和中间点center，但其实只用枚举右边界i，右边界从i-1变成i之后，中间点只可能是center或者i-1中的一个，中间点为center的时候左边界不变最优，中间点为i-1的时候需要求最优左边界（这就是个内嵌的MaxSliceSum问题）；还有一种情况就是只保留A[i-1]一个元素。整体复杂度仍然是时间 O(N),空间 O(1)。
2. 还有一种思路就是只枚举中间点k，分别算以k-1结尾和k+1开头的最大字段和，然后再把两者拼起来，这种方法整体复杂度是时间 O(N),空间 O(N)。

def solution(A):
    # write your code in Python 3.6
    len_A = len(A)
    if len_A <=3:
        return 0
    max_left = 0 #中间点为i-1右边界为i时左段的最大值
    max_ending = 0 #右边界为i时两段最大值
    center = 1
    max_slice = 0
    
    for i in range(3,len_A):
        max_left = max(max_left+A[i-2],A[i-2]) #MaxSliceSum问题
        max_ending = max(max_ending +A[i-1],A[i-1],max_left) #MaxDoubleSliceSum问题
    #     if(max_ending == A[i-1]): #中间点不为i-1
    #         center = i-2
    #     elif max_ending ==max_left: #中间点为i-1
    #         center = i-1  
        max_slice = max(max_slice, max_ending)
    return max_slice