Other Measures

Notes

• O(g(n)) bounds above
• Ω(g(n)) bounds below
  • $\Omega(g(n)) = \{ f(n) : \text{ there exists constants } c > 0, n_0 > 0 \text{ such that } 0 \le c \times g(n) \le f(n) \text{ for all } n \ge n_0 \}$
  • This provides a lower bound on the performance of an algorithm.
  • We know, for example that insertion sort is Ω(n)
    • Because the outer loop runs from 2 to n.
  • This is used less frequently that O(n)
• Θ(g(n)); bounds from above and below
  • $\Theta(g(n)) = \{ f(n) : \text{ there exists constants } c_0, c_1 > 0, n_0 > 0 \text{ such that } 0 \le c_0 \times g(n) \le f(n) ≤ c_1 \times g(n) \text{ for all } n \ge n_0 \}$
  • This is a tight bound
• There are three other sets
  • $o(g(n)) = \{ f(n) : \text{ for any constant } c > 0, n_0 > 0 \text{ such that } 0 \le f(n) \le c \times g(n) \text{ for all } n \ge n_0 \}$
    • little-o
    • This says that g(n) dominates f(n)
    • n ∈ o(n²)
    • n² ∉ o(n²)
  • Definitions for ω and θ exist as well.
  • But I have not really ever seen them outside of the text book.