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

You're missing the broader context in which those statements must have been made. Different kinds of problems have different demands, and often even have theoretical lower bounds on how much work is absolutely necessary to solve them, no matter the means.

For operations like sorting or scanning every element of a simple collection, you can make a hard lower bound of the number of elements in the collection for those operations, because the output depends on every element of the input. [1] Thus, O(n) or O(n*log(n)) are the best one can do.

For other kinds of operations, like accessing a single element of a hash table or linked list, or searching in a sorted set, the algorithm needn't examine all of the input. In those settings, an O(n) operation would be dreadfully slow.

[1] Others will note that sorting by comparisons also has an n*log(n) lower bound, from information-theoretic arguments. There are non-comparison based sorting algorithms that can beat this, for some types of input.