O gn ogn é um conjunto de fun çõ es ogn fn para

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o g(n) o(g(n))  é  um conjunto de fun çõ es o(g(n)) = {f(n) : para qualquer constante positiva  c > 0, existe uma                  constante  n > 0 tal que 0  f(n)  <  cg(n) para todo n   n 0 } f(n)   o(g(n)) se para qualquer c e um n 0 , f(n)  é  sempre menor  que cg(n) Defini çã o: f(n) = o(g(n)) se   Limite oferecido pela nota çã o O  pode ser restrito ou n ã o Limite oferecido pela nota çã o o  n ã é  restrito 0 ) ( ) ( lim = n g n f n
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Nota çã o o Exemplo: mostar que 2n = o(n 2 ) O que fazer? Encontrar  constante  n 0  tal que 2n <    cn 2  para todo c > 0 e n   n 0 ; ou Calcular limite... f(n) = o(n) ? f(n) = o(n 0.5 ) ? f(n) = o(n 1.5 ) ?
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Nota çã o ϖ Dada fun çã o g(n) ϖ (g(n))  é  um conjunto de fun çõ es ϖ (g(n)) = {f(n) : para qualquer constante positiva  c > 0, existe uma                  constante  n > 0 tal que 0  cg(n)  <  f(n) para todo n   n 0 } f(n)    ϖ (g(n)) se para qualquer c e um n 0 , f(n)  é  sempre maior  que cg(n) Defini çã o: f(n) =  ϖ (g(n)) se   Limite oferecido pela nota çã  pode ser restrito ou n ã o Limite oferecido pela nota çã ϖ  n ã é  restrito = ) ( ) ( lim n g n f n
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Nota çã ϖ Exemplo: mostar que n 2 /2 =  ϖ (n) O que fazer? Encontrar  constante  n 0  tal que n 2 /2 <    cn para todo c > 0 e n   n 0 ; ou Calcular limite... f(n) =  ϖ (n 2 ) ? f(n) =  ϖ (n 0.5 ) ? f(n) =  ϖ (n 1.5 ) ?
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Compara çã o de fun çõ es Propriedades de n ú meros reais se aplicam a compara çõ es  assint ó ticas Transitividade f(n) =  Θ (g(n)) e g(n) =  Θ (h(n))   f(n) =  Θ (h(n)) f(n) = O(g(n)) e g(n) = O(h(n))   f(n) = O(h(n)) f(n) =  (g(n)) e g(n) =  (h(n))   f(n) =  (h(n)) f(n) = o(g(n)) e g(n) = o(h(n))   f(n) = o(h(n)) f(n) =  ϖ (g(n)) e g(n) =  ϖ (h(n))   f(n) =  ϖ (h(n))
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Compara çã o de fun çõ es Reflexividade f(n) =  Θ (f(n)) f(n) = O(f(n))  f(n) =  (f(n))  ... e as nota çõ es o e  ϖ ?
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