The notes of Algorithms ---- Asymptotic Notation

O notation:

      O(g(n))={f(n): there exists consts c>0 no>0 such taht 0<=f(n)<=c*g(n) for all n>no}

      f(n)=O(g(n)) represents an anonymous function[f(n)] in that set[O(g(n))].

      Ex1. f(n)=n^3+O(n^2) means there is a function h(n) in O(n^2) such that f(n)=n^3+h(n);

      Ex2. n^2+O(n)=O(n^2) the '=' means is and it is not symmetric.

  Summary: the O notation represents the upper bounds.

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Ω notation:

      Ω(g(n))={f(n): there exists consts c>0 no>0 such taht 0<=c*g(n)<=f(n) for all n>no}

  Summary: the Ω notation represents the lower bounds.

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Θ notation:

      Θ(g(n))={f(n): there exists consts c1>0 c2>0 no>0 such taht 0<=c1*g(n)<=f(n)<=c2*g(n) for all n>no}

  Summary: Θ(g(n))=O(g(n))∩Ω(g(n)).

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o & ω:

      o(g(n))={f(n): for any positive consts c>0 there exists a constant no>0 such taht 0<=f(n)<c*g(n) for all n>no}

      ω(g(n))={f(n): for any positive consts c>0 there exists a constant no>0 such taht 0<=c*g(n)<f(n) for all n>no}

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O: <=

Ω: >=

Θ: ==

o:  <

ω: >