Why are constants ignored in asymptotic analysis?

Question

Why are constants ignored in asymptotic analysis?

Answer 1

Constant factors are ignored because running time and memory consumption (the two properties most often measured using the O-notation) are much harder to reason about when considering constant factors.

If we define U( f(n) ) to be the set of all function g for which there exists an N such that for all N > n: g(n) <= f(n) (i.e. the same as O without the constant factor), it is much harder to show that an algorithm's running time is in U( f(n) ) than O( f(n) ).

For one thing, we'd need an exact unit for measuring running time. Using a CPU instruction as the basic unit would work, but that'd depend on the exact implementation of the algorithm as well as the processor architecture it runs on.

It's similar for memory consumption: different implementations of the same algorithm will differ in their memory consumption (by a constant factor). Further if an implementation uses a lot of pointers, the same implementation will use about twice as much memory on a 64-bit machine than on a 32-bit machine. But saying things like "this algorithm's memory consumption, when implemented using this C-code, is in O(23 * n) on a 32-bit Intel PC. It's in O(42 * n) on a 64-bit PC" is just not useful.

Ignoring constants allows us to reason about the properties of an algorithm in an implementation- and platform-independent manner.