合并k个排序的arrays的时间复杂度

Question

I have k sorted arrays, each length n and I want to sort them so there is one sorted array of length n*k. I know I can iteratively apply merge function from Merge Sort. However I don't understand why following code is O(nk^2) instead O(nk). I iterate over k array once and each merge subroutine has linear time with respect to its input. Where am I wrong? How to better understand running time of this algorithm?

def myMerge(left, right):
    l, r = 0, 0
    result = []
    for i in left+right:
        if left[l] <= right[r]:
            result.append(left[l])
            l += 1
            if l >= len(left):
                return result + right[r:]
        else:
            result.append(right[r])
            r += 1
            if r >= len(right):
                return result + left[l:]
    return result

#input
k = [[1,5,8],[2,7,9],[1,1,4]]

result = k[0]
for l in range(1,len(k)):
    result = myMerge(result,k[l])  

Answer 1

Both of your sentences are correct:

• you iterate over k array once
• each merge subroutine has linear time with respect to its input

However, the answer is O(nk^2). Why?

Because the length of the aforementioned input is not O(n). After the first loop it's already 2*n. After the second, it's already 3*n. And so on. So during the last iteration your array lengths are n*(k-1) and n which means each merge subroutine is O(nk) and therefore the entire algorithm is O(n*k^2)