This is true for O(n) but not for o(n). f(n) is the class of O(g(n)) of g(n) if g(n) dominates f(n) for sufficiently large n. o(g(n)) strictly dominates f(n), t

This is true for O(n) but not for o(n). f(n) is the class of O(g(n)) of g(n) if g(n) dominates f(n) for sufficiently large n. o(g(n)) strictly dominates f(n), though, no matter how small n is. Big O notation, or Landau notation, is not just O(g(n)). It's also o(), Ω(), ω() and Θ(). It's understood, though, that n is considered to usually be rather large - another name for this notation is, after all, "asymptotic notation." In any case that you're really worried about speed, you should probably be calculating the speed of your algorithm directly rather than using mere asymptotic generalizations. Aside: I can't think of any algorithms with huge constants like you described; in theory, you're correct, but in practice, the asymptotic generalizations apply for n <= 1000 or even often 100.