Notes on Theta notation

Here is a more precise definition of theta notation, along with another example.

(3) Theta (precise version)

• Definiton: A function f(n) is of exact order g(n), noted f(n) = Q(g(n)), if there exist positive constants c₁, c₂ and n₀ such that c₁*g(n) <= f(n) <= c₂*g(n) for all n >= n₀.

   

• Example: 

  

  COMMENTS:

  • The following chart shows both upper bound (2*nlgn) and lower bound (1*nlgn) functions for the formula (n + nlgn).

    

  • As for n₀, it must be chosen so that the resulting range of n is the intersection of the range for big-Oh and the range for omega.  The chart above shows that visually:
    • 2nlgn >= n + nlgn  -- for n >= 2
    • 1nlgn >= n + nlgn  -- for n >= 1

    So the intersection is n >= 2 = n₀.