Asymptotic notation

Definitions

T(N) = \mathcal{O}(f(N)) if there are positive constants c and n_0 such that T(N) \leq cf(N) when N \geq n_0. This is an upper bound.

T(N) = \Omega(g(N)) if there are positive constants c and n_0 such that T(N) \geq cg(N) when N \geq n_0. This is a lower bound.

T(N) = \Theta(h(N)) if and only if T(N) = \mathcal{O}(h(N)) and T(N) = \Omega(h(N)). This is a tight bound.

T(N) = \mathcal{o}(p(N)) if for all positive constants c there exists some n_0 such that T(N) < cp(N) when N > n_0. This is a strict upper bound.

Example

Let f(N) = N^2. Then, T(N) = \mathcal{O}(N^2) if there are positive constants c and n_0 such that T(N) \leq cN^2 when N \geq n_0.

Original author: Clayton Cafiero < [given name] DOT [surname] AT uvm DOT edu >

No generative AI was used in producing this material. This was written the old-fashioned way.

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