![Page 1: O-notation€¦ · Theta (“as”) Notation : !!=!!!!. Intuition : f grows like g, i.e. between ! ... (n2)is#asymptotically#tight,#but2n#=O(n2)is#not.#We#use#o@notation(little# oh)#toindicate#that#our#analysis#is#un@tight.#](https://reader034.vdocuments.us/reader034/viewer/2022042221/5ec7560e780b174dac548aae/html5/thumbnails/1.jpg)
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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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