归并排序归并阶段的Big O复杂度及特定合并场景的Big O问询

Hey there! Let's tackle your two questions about merge sort and merge operations clearly:

1. What's the time complexity (in Big O notation) of the merge phase in merge sort? #

The merge phase of merge sort runs in O(n) time, where n is the total number of elements in the two sorted subarrays being merged.

Here's why: during the merge, we have to iterate through every element exactly once to build the final sorted array. The number of comparisons we make can be as few as min(len(a), len(b)) (if one array is entirely smaller than the other) or up to n-1 (if elements alternate between the two arrays), but in either case, the total number of operations scales linearly with the total number of elements. Since Big O notation focuses on the asymptotic growth rate, we simplify this to O(n).

2. Given two lists a and b where len(a) + len(b) = 5, and the merge(a,b) operation only requires one comparison—what's the time complexity here in Big O notation? #

In this specific scenario, the time complexity is O(1) (constant time).

Wait, let's break this down: even though we do one comparison, the key point here is that the total number of elements is fixed at 5. No matter how the elements are arranged (as long as the merge only needs one comparison), all the operations we perform—like copying the remaining elements after the single comparison—are a fixed, constant number of steps. Big O notation describes how runtime grows as input size increases, but since our input size is fixed (never gets larger than 5), the runtime doesn't grow at all. So we categorize this as constant time, O(1).

内容的提问来源于stack exchange，提问作者JSK