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What is f(x) ≤ g(x) + O(1)? Inequalities With Asymptotics
Recently I came across asympotic inequalities of the form $f(x) \le g(x) + O(1)$ in Li and Vitanyi’s An Introduction to Kolmogorov Complexity and Its Applications, where this notion is used to discuss bounds on the complexity of strings. In this post I give a brief definition of what this notation means and how it relates to standard asymptotic notation. Particulary, I’m only going to talk about the case of $O(1)$, but the same ideas can be extended to $O(h(x))$ for any function $h(x)$.
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