Is it possible to calculate median of a list of numbers better than O(n log n)?

Question

I know that it is possible to calculate the mean of a list of numbers in O(n). But what about the median? Is there any better algorithm than sort (O(n log n)) and lookup middl

Answer 1

Just for fun (and who knows, it may be faster) there's another randomized median algorithm, explained technically in Mitzenmacher's and Upfall's book. Basically, you choose a polynomially-smaller subset of the list, and (with some fancy bookwork) such that it probably contains the real median, and then use it to find the real median. The book is on google books, and here's a link. Note: I was able to read the pages of the algorthm, so assuming that google books reveals the same pages to everyone, you can read them too.

It is a randomized algorithm s.t. if it finds the answer, it is 100% certain that it is the correct answer (this is called Las Vegas style). The randomness arises from the runtime --- occasionally (with probability 1/(sqrt(n)), I think) it FAILS to find the median, and must be re-run.

Asymptotically, it is exactly linear when you take into the chance of failure --- that is to say, it is a wee bit less than linear, exactly such that when you take into account the number of times you may need to re-run it, it becomes linear.

Note: I'm not saying this is better or worse --- I certainly haven't done a real-life runtime comparison between these algorithms! I'm simply presenting an additional algorithm that has linear runtime, but works in a significantly different way.