插入排序的运行时复杂度的最佳描述是什么？

Question

I have understood it to be quadratic, but I am taking a multiple choice practice test for discrete math and the only four options are:

a) logarithmic

b) linear

c)linearithmic

d)polynomial

Answer 1

a) logarithmic = O(log n)

b) linear = O(n)

c) linearithmic = O(n log n)

d) polynomial = O(n ^k )

So O(n ² ) would be polynomial.

Answer 2

Inserting a new value into a list or tree of sorted values is O(log n).

Easiest to see with a straight sorted list. Lets say you want to insert x. Find the value at [n/2]. If this is greater than x, then the correct location lies on the left half. If [n/2] is less than x, then the correct location lies on the right. Repeat, checking the middle value each time. Each step halves the size of the interval to check until you find the correct value, giving logarithmic time for the insertion.

An insertion into a tree is similar; you have to traverse the tree to the correct node position, which requires traversing the average depth of the tree, which is proportional to log n