n个数字的排序时间是否取决于数字的排列？

Question

Consider this problem:

A comparison-based sorting algorithm sorts an array with n items. For which fraction of n! permutations, the number of comparisons may be cn where c is a constant?

I know the best time complexity for sorting an array with arbitrary items is O(nlogn) and it doesn't depend on any order, right? So, there is no fraction that leads to cn comparisons. Please guide me if I am wrong.

Answer 1

This depends on the sorting algorithm you use.

Optimized Bubble Sort for example, compares all neighboring elements of an array and swaps them when the left element is larger then right one. This is repeated until no swaps where performed.

When you give Bubble Sort a sorted array it won't perform any swaps in the first iteration and thus sorts in O(n).

On the other hand, Heapsort will take O(n log n) independent of the order of the input.

Edit:

To answer your question for a given sorting algorithm, might be non-trivial. Only one out of n! permutations is sorted (assuming no duplicates for simplicity). However, for the example of bubblesort you could (starting for the sorted array) swap each pair of neighboring elements. This input will take Bubblesort two iterations which is also O(n).