Asymptotic Bounds The Differences Between (Big-O, Omega and Theta) Properties

Scalability – When algorithm is applied to a large data set, will finish relatively quickly. Speed and memory usage – Measuring speed-we measure algorithm speed in terms of Operations relative to input size. Measuring Efficiency

Big O C*g(x) g(x) f(x) x y xoxo

Big O Definition: Let f(x) and g(x) be two functions; We say that f(x)  O(g(x)) if there exists a constant c, X o > 0 such that f(x)≤c*g(x) for all X ≥ X o. f (x) is asymptotically less than or equal to g(x) Big-O gives an asymptotic upper bound.

Big  C*g(x) g(x) f(x) x y xoxo

Big-Omega Definition: Let f (x)and g(x) be two functions; We say that f(x)   (g(x)) if there exists a constant c, X o ≥ 0 such that f(x) ≥ c*g(x) for all X ≥ X o f(x) is asymptotically greater than or equal to g(x) Big-Omega gives an asymptotic lower bound

Big Θ C 2* g(x) g(x) f(x) x y xoxo C 1 *g(x)

Big Θ Definition: Let f(x) and g(x) be two functions ; We say that f(x)   (g(x)) if there exists a constant c 1, c 2, X o > 0; such that for every integer x  x 0 we have c 1 g(x) ≥ f(x) ≥ c 2 g(x) F(x) is asymptotically equal to g(x) F(x) is bounded above and below by g(x) Big-Theta gives an asymptotic equivalence