Introduction to algorithms------asymptotic notation θ, 0, O, ω, ω detailed

Directory: 1. Asymptotic precision boundary notation: θθ (Big-theta) 2. Asymptotic upper bound mark: o O (Big-oh) 3. Asymptotic Nether notation: ωω (Big-omege) 4. Upper bound of non-asymptotic tightening: O (small-oh) 5. Non-asymptotic tight bound: Ω (small-omege) 6. Asymptotic notation Θ, 0, O, ω, ω Relationship 7. Reference 1. Asymptotic precision boundary notation: θθ (Big-theta)

Assume that the run-time expression of algorithm a T1 (n) t_1 (n) is: T1 (n) =30n4+20n3+40n2+46n+100 t_1 (n) =30n^4+20n^3+40n^2+46n+100
Suppose the run-time expression of algorithm B T2 (n) t_2 (n) is: T2 (n) =1000n3+50n2+78n+10 t_2 (n) =1000n^3+50n^2+78n+10
When the problem is large enough, such as n=100, the run time of the algorithm will depend mainly on the first item of the time expression, and the execution time of the other item is only a few one out of 10,000 of it, which is negligible. The constant coefficient of the first item, with the increase of N, also becomes unimportant to the execution time of the algorithm.
Thus, the run time of algorithm A can be recorded as: T1 (n) ≈n4 t_1 (n) ≈n^4, recorded as T1 (n) =θ (N4) t_1 (n) =θ (n^4); The run time of algorithm B can be recorded as: T2 (n) ≈n4 t_2 (n) ≈n^4, recorded as T2 (n) =θ (N4) t_2 (n) =θ (n^4).

The mathematical meaning of θθ
Mode one: Setting f (n) f (n) and g (n) g (n) is a function that defines the set of fields as natural numbers. If