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Unformatted text preview: Asymptotic Notations
TCSS343 — Autumn 2009
The asymptotic notations deﬁned below are used by computer scientists to denote the time eﬃciency of algorithms, in terms of the input size n. Especially big oh and big theta notation are frequently used in practice, but when you master them it is easy to understand big omega as well. This overview links the notation (a) with its intuitive meaning (b), its precise mathematical deﬁnition (c), as well as a limit test (d) that you can use to compare the order of growth of two functions in practice. Learn the overview by heart: look at it until you are convinced that you understand and have memorized it, then hide this handout and write down the overview on a blank sheet. Don’t forget to verify carefully whether you got everything right! Let t(n) and g (n) be nonnegative functions deﬁned on the set of natural numbers. 1. O-notation (big-oh) (a) t(n) is in O(g (n)) (b) t(n) grows no faster than g (n) (c) There exists a positive real number c and a nonnegative integer n0 such that t(n) ≤ c · g (n) (d) lim
t(n) n→∞ g (n) for all n ≥ n0 is 0 or a positive real number. 2. Ω-notation (big-omega) (a) t(n) is in Ω(g (n)) (b) t(n) grows at least as fast as g (n) (c) There exists a positive real number c and a nonnegative integer n0 such that c · g (n) ≤ t(n) (d) lim
t(n) n→∞ g (n) for all n ≥ n0 is a positive real number or ∞. 3. Θ-notation (big-theta) (a) t(n) is in Θ(g (n)) (b) t(n) grows at the same rate as g (n) (c) There exist positive real numbers c1 and c2 , and a nonnegative integer n0 such that c2 · g (n) ≤ t(n) ≤ c1 · g (n) (d) lim
t(n) n→∞ g (n) for all n ≥ n0 is a positive real number. 1 ...
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This note was uploaded on 11/09/2009 for the course TCSS TCSS-543 taught by Professor Decock during the Fall '09 term at University of Western States.
- Fall '09