Asymptotic Notation - Algorithm - Study Notes, Study notes of Algorithms and Programming

Download Asymptotic Notation - Algorithm - Study Notes and more Study notes Algorithms and Programming in PDF only on Docsity! Design and Analysis of Algorithms Chapter 2 Asymptotic Notation You may be asking that we continue to use the notation Θ() but have never defined it. Let’s remedy this now. Given any function g(n), we define Θ(g(n)) to be a set of functions that asymptotically equivalent to g(n). Formally: Θ(g(n)) = {f(n) | there exist positive constants c1, c2 and n0 such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n ≥ n0} This is written as “f(n) ∈ Θ(g(n))” That is, f(n) and g(n) are asymptotically equivalent. This means that they have essentially the same growth rates for large n. For example, functions like • 4n2, • (8n2 + 2n − 3), • (n2/5 + √n − 10 log n) • n(n − 3) are all asymptotically equivalent. As n becomes large, the dominant (fastest growing) term is some constant times n2. Consider the function f(n) = 8n2 + 2n − 3 Our informal rule of keeping the largest term and ignoring the constant suggests that f(n) ∈ Θ(n2). Let’s see why this bears out formally. We need to show two things for f(n) = 8n2 + 2n − 3 Lower bound f(n) = 8n2 + 2n − 3 grows asymptotically at least as fast as n2, 23