Notes on Asymptotic Time Complexity

Thus, f is big O of g if f is bounded above by a constant multiple of g. Intuitively, f(n) = O(g(n)) means that f grows no faster than g as the index n grows. For example, the function f(n) = n2 is big O of the function g(n) = n3, while it is not true that g is big O of f in this case. Every function is big O of itself. The requirement that the inequality in the above definition of big O hold for all values of n is sometimes relaxed so that it only need apply for all n ≥ n0, for some number n0. This requirement is superfluous if g(n) > 0 for all n, because if f(n) ≤ Cg(n) for all n ≥ n0, one can ensure that f(n) ≤ C ′g(n) for the few remaining values n = 0...n0−1 by enlarging the constant C to C ′ if need be: just pick C ′ to be as large as the largest of the n0 values f(0)/g(0)...f(n0 − 1)/g(n0 − 1) (and no smaller than the original value of C).