Discrrete mathematics for computer science asymptotic

Growth Rates of Functions 3/26/12 Asymptotic Equivalence • Def: ⎛  ⎞ f (n) : g(n) iff lim n ⎜  ⎟ ⎝   ⎠ For example, n                  Note that n2+1 is being used to name the function f such that f(n) = n2+1 for every n 3/26/12 An example: Stirling’s formula ⎛n⎞ n! : ⎜ ⎟ ⎝e⎠ 3/26/12      Little-Oh: f = o(g) • Def: f(n) = o(g(n)) if  lim  n   • For example, n2 = o(n3) since   lim      n     3/26/12 = o( ∙ ) is “all one symbol” • “f = o(g)” is really a strict partial order on functions • NEVER write “o(g) = f”, etc 3/26/12 Big-Oh: O(∙) • Asymptotic Order of Growth: ⎛  ⎞ f (n)      ⎜  ⎟  ⎝   ⎠ • “f grows no faster than g” • A Weak Partial Order 3/26/12 Growth Order 3n                       3/26/12 f = o(g) implies f = O(g)  because if  lim  n                  3/26/12 Big-Omega • f = Ω(g) means g = O(f) • “f grows at least as quickly as g” 3/26/12 Big-Theta: �(∙) “Same order of growth” f (n)                         So, for example, 3n      3/26/12  Rough Paraphrase • f∼g: f and g grow to be roughly equal • f=o(g): f grows more slowly than g • f=O(g): f grows at most as quickly as g • f=Ω(g): f grows at least as quickly as g • f=�(g): f and g grow at the same rate 3/26/12 Equivalent Defn of O(∙) “From some point on, the value of f is at most a constant multiple of the value of g” f (n)                3/26/12 Three Concrete Examples • Polynomials • Logarithmic functions • Exponential functions 3/26/12 Polynomials • A (univariate) polynomial is a function such as f(n) = 3n5+2n2-n+2 (for all natural numbers n) • This is a polynomial of degree (the largest exponent)  • Or in general f (n)    • Theorem:  – If a1 • Theorem: Any polynomial is o(any exponential) • If c