Is n or nlog(n) better than constant or logarithmic time?

Question

In the Princeton tutorial on Coursera the lecturer explains the common order-of-growth functions that are encountered. He says that linear and linearithmic running times are \"w

Answer 1

Algorithms with a sublinear behavior like O(1) or O(Log(N)) are special in that they do not require to look at all elements. In a way this is a fallacy because if there are really N elements, it will take O(N) just to read or compute them.

Sublinear algorithms are often possible after some preprocessing has been performed. Think of binary search in a sorted table, taking O(Log(N)). If the data is initially unsorted, it will cost O(N Log(N)) to sort it first. The cost of sorting can be balanced if you perform many searches, say K, on the same data set. Indeed, without the sort, the cost of the searches will be O(K N), and with pre-sorting O(N Log(N)+ K Log(N)). You win if K >> Log(N).

This said, when no preprocessing is allowed, O(N) behavior is ideal, and O(N Log(N)) is quite comfortable as well (for a million elements, Lg(N) is only 20). You start screaming with O(N²) and worse.