O-notation€¦ · Theta (“as”) Notation : !!=!!!!. Intuition : f grows like g, i.e. between !...
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f(n) n 0 c 1 g(n) c 2 g(n) n O-notation † O, big Oh (upper limit) Notation : = . Intuition : f is smaller than g . When → ∞ : ≤ ∙ () . Definition : ∃ > , ∶ ∀ > ≤ ∙ () . Omega (lower limit) Notation : = . Intuition : f is larger than g . When → ∞ : ≥ ∙ () . Definition : ∃ > , ∶ ∀ > ≥ ∙ () . Theta (“as”) Notation : = . Intuition : f grows like g, i.e. between ∙ and ∙ . When → ∞ : ∙ () ≤ ≤ ∙ () . Definition : ∃ , > , ∶ ∀ ( > ) ∙ () ≤ ≤ ∙ () . o, little oh ‡ Notation : = . Intuition : f is a lot smaller than g . When → ∞ : ≪ () . Definition : →! () = . † More correctly, asymptotic notation. ‡ Onotation might not be asymptotically tight:2n 2 = O(n 2 ) is asymptotically tight, but 2n = O(n 2 ) is not. We use onotation (little oh) to indicate that our analysis is untight. n 0 f(n) cg(n) n n 0 f(n) cg(n) n
Transcript of O-notation€¦ · Theta (“as”) Notation : !!=!!!!. Intuition : f grows like g, i.e. between !...
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O-notation†
O, big Oh (upper limit) Notation : 𝒇 𝒏 = 𝑶 𝒈 𝒏 .
Intuition : f is smaller than g .
When 𝑛 → ∞ : 𝒇 𝒏 ≤ 𝒄 ∙ 𝒈(𝒏) .
Definition : ∃ 𝒄 > 𝟎 ,𝒏𝟎 ∶ ∀ 𝒏 > 𝒏𝟎 𝒇 𝒏 ≤ 𝒄 ∙ 𝒈(𝒏) .
Omega (lower limit)
Notation : 𝒇 𝒏 = 𝛀 𝒈 𝒏 .
Intuition : f is larger than g .
When 𝑛 → ∞ : 𝒇 𝒏 ≥ 𝒄 ∙ 𝒈(𝒏) .
Definition : ∃ 𝒄 > 𝟎 ,𝒏𝟎 ∶ ∀ 𝒏 > 𝒏𝟎 𝒇 𝒏 ≥ 𝒄 ∙ 𝒈(𝒏) .
Theta (“as”) Notation : 𝒇 𝒏 = 𝚯 𝒈 𝒏 .
Intuition : f grows like g, i.e. between 𝒄𝟏 ∙ 𝒈 and 𝒄𝟐 ∙ 𝒈 .
When 𝑛 → ∞ : 𝒄𝟏 ∙ 𝒈(𝒏) ≤ 𝒇 𝒏 ≤ 𝒄𝟐 ∙ 𝒈(𝒏) .
Definition : ∃ 𝒄𝟏, 𝒄𝟐 > 𝟎 ,𝒏𝟎 ∶ ∀ (𝒏 > 𝒏𝟎) 𝒄𝟏 ∙ 𝒈(𝒏) ≤ 𝒇 𝒏 ≤ 𝒄𝟐 ∙ 𝒈(𝒏) .
o, little oh ‡ Notation : 𝒇 𝒏 = 𝒐 𝒈 𝒏 .
Intuition : f is a lot smaller than g .
When 𝑛 → ∞ : 𝒇 𝒏 ≪ 𝒈(𝒏) .
Definition : 𝐥𝐢𝐦𝒏→!𝒇 𝒏𝒈(𝒏)
= 𝟎 .
† More correctly, asymptotic notation. ‡ O-‐notation might not be asymptotically tight: 2n2 = O(n2) is asymptotically tight, but 2n = O(n2) is not. We use o-‐notation (little oh) to indicate that our analysis is un-‐tight.
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