@MISC{_bigoh,, author = {}, title = {Big Oh, Big Omega, and Big Theta}, year = {} }

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Abstract

We consider sequences f: N − → R or f: N − → N or sometimes functions f: R+ − → R. Here N denotes the positive integers, R the real numbers, and R+ the positive real numbers. For asymptotics, we are interested only in the behavior of f(n) for large values of n, so we need only that f be defined for n> k for some k. Asymptotic Relations Often, but not always, we use the following relations in comparing functions f(n) and g(n) that both approach infinity as n approaches infinity. Here are two equivalent notations that say that f(n) grows more slowly than g(n): f(n) ≺ g(n) if and only if f(n) = o(g(n)) if and only if lim n→∞ f(n) g(n) = 0. The second notation, o(g(n)), is Landau’s “little oh ” notation. For example, if 0 < p < q and b> 1, log n ≺ np ≺ nq ≺ bn. Here is how we denote that f(n) and g(n) have the same rate of growth: f(n) ≍ g(n) if and only if |f(n) | ≤ C|g(n) | and |g(n) | ≤ C|f(n)| for some C and all n> some k. A stronger relation says that “f(n) is asymptotic to g(n)”: f(n) ∼ g(n) if and only if lim n→∞ f(n) g(n) = 1. For example, n3 ≍ n3 and n3 ∼ n3