Algorithms for modern hardware?

Question

Once again, I find myself with a set of broken assumptions. The article itself is about a 10x performance gain by modifying a proven-optimal algorithm to account for virtual mem

Answer 1

I'll expand on GregS's answer: the difference is between effective complexity and asymptotic complexity. Asymptotic complexity ignores constant factors and is valid only for “large enough” inputs. Oftentimes “large enough” can actually mean “larger than any computer can deal with, now and for a few decades”; this is where the theory (justifiably) gets a bad reputation. Of course there are also cases where “large enough” means n=3 !

A more complex (and thus more accurate) way of looking at this is to first ask “what is the size range of problems you are interested in?” Then you need to measure the efficiency of various algorithms in that size range, to get a feeling for the ‘hidden constants’. Or you can use finer methods of algorithmic asymptotics which actually give you estimates on the constants.

The other thing to look at are ‘transition points’. Of course an algorithm which runs in 2n² time will be faster than one that runs in 10¹⁶nlog(n) times for all n < 1.99 * 10¹⁷. So the quadratic algorithm will be the one to choose (unless you are dealing with the sizes of data that CERN worries about). Even sub-exponential terms can bite – 3n³ is a lot better than n³ + 10¹⁶n² for n < 5*10¹⁵ (assuming that these are actual complexities).