Big-Omega and Big-Theta: Understanding Asymptotic Notations, Slides of Computer Science

Download Big-Omega and Big-Theta: Understanding Asymptotic Notations and more Slides Computer Science in PDF only on Docsity!

Analysis of Algorithms

Big-Omega and Big-Theta

Big-Omega and Big-Theta Defined

• If f = Ω(g), then f is at least as big as g (or g is a lower bound for f) e.g. f(n) = n^3 and g(n) = n^2
• If f = Θ(g), f=O(g) and f = Ω (g) (or g is both an upper and lower bound. It is a “tight” fit) e.g. f(n) = n^3 + n 2 and g(n) = n^3

Big-Theta Asymptotic Tight Bound

• Theta means that f is bounded above and below by g; Big Theta implies the "best fit".
• f(n) does not have to be linear itself in order to be of linear growth; it just has to be between two linear functions.
• We will use Theta whenever we have enough information to show that the f(n) is both an upper and lower bound. Theta is a “stronger” statement than Big-Oh or Big-Omega.

Big-Theta Example

• Example: f(n) = n^2 - 5n + 13.
• The constant 13 doesn't change as n grows, so it is not crucial. The low order term, -5n, doesn't have much effect on f compared to the quadratic term, n^2.
• Q: What does it mean to say f(n) = Θ(g(n))?
• A: Intuitively, it means that function f is the same order of magnitude as g.