The Asymptotic Notation Cheat Sheet, Cheat Sheet of Mathematics for Computing

Download The Asymptotic Notation Cheat Sheet and more Cheat Sheet Mathematics for Computing in PDF only on Docsity! 6.042/18.062J Mathematics for Computer Science October 19, 2004 Tom Leighton and Ronitt Rubinfeld The Asymptotic Cheat Sheet Asymptotic notation consists of six funny symbols used to describe the relative growth rates of functions. These six symbols are defined in the table below. f = !(g) f grows at the same rate as g There exists an n0 and constants c1, c2 > 0 such that for all n > n0, c1g(n) ! |f(n)| ! c2g(n). f = O(g) f grows no faster than g There exists an n0 and a constant c > 0 such that for all n > n0, |f(n)| ! cg(n). f = "(g) f grows at least as fast as g There exists an n0 and a constant c > 0 such that for all n > n0, cg(n) ! |f(n)|. f = o(g) f grows slower than g For all c > 0, there exists an n0 such that for all n > n0, |f(n)| ! cg(n). f = !(g) f grows faster than g For all c > 0, there exists an n0 such that for all n > n0, cg(n) ! |f(n)|. f " g f/g approaches 1 limn!" f(n)/g(n) = 1 The ! and ! notations are confusingly similar; qualitatively, functions related by ! must be even more nearly alike then functions related by !. The ! notation makes the table nice and symmetric, but is almost never used in practice. Some asymptotic relation- ships between functions imply other relationships. Some examples are listed below. f = O(g) and f = "(g) " f = !(g) f = o(g) # f = O(g) f = O(g) " g = "(f) f = !(g) # f = "(g) f = o(g) " g = !(f) f ! g # f = !(g)