Progressive notation of the introduction to algorithms notes

Progressive notation

Θ (theta): Progressive tightening of the boundary. F (N) =θ (g (n)) indicates:? c1>0,c2>0,n0>0,s.t. N>N0,0≤C1G (n) ≤f (n) ≤c2g (n) was established. The limit is expressed as limn->∞f (n)/g (n) = c.

O (Greater Europe): progressive Upper bound. F (n) = O (g (n)) indicates: C>0,N0>0,S.T. Right? N>n0,0≤f (n) ≤CG (n) was established.

Omega (large Omega): progressive lower bound. F (N) =ω (g (n)) indicates:? c>0,n0>0,s.t. N>N0,0≤CG (n) ≤f (n) was established.

O (Small Europe): non-progressive tightening upper bound. F (n) = O (g (n)) means: right? c>0,?n0>0,s.t. N>n0,0≤f (n) <CG (n) was established. The main difference between f (n) = O (g (n)) and f (n) = O (g (n)) is that the former is set for any c>0,0≤f (n) ≤CG (n), and the latter is for a c>0,0≤f (n) <CG (n). In other words, f (n) = O (g (n)) can be expressed as when n->∞, LIMF (n)/g (n) = 0.

Omega (small Omega): non-progressive tight bound. F (N) =ω (g (n)) means: right? c>0,?n0>0,s.t. N>N0,0≤CG (n) <f (n) was established. The available limits are expressed as: when N->∞, LIMF (n)/g (n) =∞.

We remember f (n) = O (g (n)) to indicate that the function f (n) is a member of the Set O (g (n)), i.e. f (n) ∈o (g (n)). Note that f (n) =θ (g (n)) implies f (n) = O (g (n)), i.e. O (g (n)). Θ (g (n)). In the same way, Ω (g (n))? Θ (g (n)). Thus, the following important theorems can be derived:

For any two functions f (n) and g (n), we have f (n) =θ (g (n)), when and only if f (n) = O (g (n)) and f (n) =ω (g (n)).

The nature of progressive notation transitivity:

F (N) =θ (g (n)) and g (n) =θ (h (N)), then f (n) =θ (h (n))

F (n) = O (g (n)) and g (n) = O (h (n)), f (n) = O (h (n))

F (N) =ω (g (n)) and g (n) =ω (h (N)), then f (n) =ω (h (n))

F (n) = O (g (n)) and g (n) = O (h (n)), f (n) = O (h (n))

F (N) =ω (g (n)) and g (n) =ω (h (N)), then f (n) =ω (h (n))

Reflexive nature:

F (N) =θ (f (n))

F (n) = O (f (n))

F (N) =ω (f (n))

Symmetry:

F (N) =θ (g (n)) if and only if G (n) =θ (f (n))

Transpose Sex:

F (n) = O (g (n)) if and only if G (n) =ω (f (n))

F (n) = O (g (n)) if and only if G (n) =ω (f (n))

Since these properties are set for progressive notation, it is possible to compare the asymptotic comparison of two functions f and G with the comparison between Real and B:

F (N) =θ (g (n)) similar to a = b

F (n) = O (g (n)) similar to A≤b

F (N) =ω (g (n)) similar to A≥b

F (n) = O (g (n)) similar to a <b

F (N) =ω (g (n)) similar to a >b

However, the following properties of a real number cannot be carried to progressive notation:

Three-point: for any two real numbers a and B, the following three cases must be established: A<b,a>b or a=b.

Although two real numbers can be compared to each other, not all functions can be progressively compared. That is, for two functions f (n) and g (n) maybe f (n) = O (g (n)) and f (n) =ω (g (n)) are not true. For example, we cannot use progressive notation to compare N and N1+sin N

Progressive notation of the introduction to algorithms notes