The Sets O(g), Θ(g), Ω(g)

• Let f and g be a functions from the non-negative integers into the positive real numbers
• For some real constant c > 0 and some non-negative integer constant n₀
  • O(g) is the set of functions f, such that:
    • f(n) ≤ c * g(n) for all n ≥ n₀
  • Ω(g) is the set of functions f, such that:
    • f(n) ≥ c * g(n) for all n ≥ n₀
  • Θ(g) = O(g) ∩ Ω(g)
    • Θ(g) is the asymptotic order of g or the order of g

Big-Oh Examples

f(n) ∈ O(g(n)) means that there are positive constants c and n₀ such that f(n) ≤ c*g(n) for all values n ≥ n₀

• Is n ∈ O(n²)?
  • Yes, c = 1, n₀ = 2 works fine
• Is 10n ∈ O(n)?
  • Yes, c = 11, n₀ = 2 works fine
• Is n² ∈ O(n)?
  • No; no matter what values for c and n₀ we pick, n² > c*n for big enough n

Given f ∈ O(h) and g ∉ O(h),
which of these are true?

1. For all positive integers m, f(m) < g(m).
2. For some positive integer m, f(m) < g(m).
3. For some positive integer m₀, and all positive integers m > m₀, f(m) < g(m).
4. 1 and 2
5. 2 and 3
6. 1 and 3

Another Way to Define Order Classes

• Comparing f(n) and g(n) as n approaches infinity...
• If limit as n approaches infinity of f(n)/g(n) is:
  • < ∞, including the case in which the limit is 0, then f ∈ O(g)
  • > 0, including the case in which the limit is ∞, then f ∈ Ω(g)
  • = c and 0 < c < ∞ then f ∈ Θ(g)
  • = 0 then f ∈ o(g)
    • read as "little-oh of g"
  • = ∞ then f ∈ ω(g)
    • read as "little-omega of g"