Asymptotic Notations and Basic Efficiency Classes

• We will discuss big O (O(g(n))), big Omega( Ω(g(n))) and big Theta, (Θ(g(n)))) notation.
  • These are actually sets of functions.
  • a function f(n) ∈ O(g(n)) if there exist positive constants c and n₀ such that f(n) ≤ cg(n) for all n > n₀
  • This says that past a certain input size (n₀) the function f(n) is no bigger then a constant factor of g(n).
  • 
  • We say that a function, and therefore the algorithm is O(g(n)) , but we really should say that it is in O(g(n)).
  • And I just labeled the axis with time, when it probably should be operation count, but we have concluded that these are the same.
  • Some examples.
    1. Is n ∈ O(n)? sure c=2 and n₀ = 1
    2. Is 3n+2 ∈ O(n)?, sure c=4, and n₀ = 2
    3. (we don't need optimal c and n₀, we just need them.)
    4. Is 1000n+40,000 ∈ O(n)?
    5. Is n² ∈ O(n)?
    6. Is 3n²+4 ∈ O(n²)?
    7. Is n ∈ O(n²)?
  • I think of n₀ as start up cost.
  • And c as the difference in the number of operations "inside the loop".
  • On page 57 he tells us that

                f(n)
    let k = lim ----
            n→∞  g(n)
    
    	

    • If k = 0, then f(n) ∈ O(g(n))
    • g(n) has a larger order of growth then f(n)
    • If k = c, then f(n) ∈ O(g(n)) and g(n) ∈ O(f(n))
    • f(n) and g(n) have the same order of growth
    • If k = &inifin; then g(n) ∈ O(f(n))
    • f(n) has a larger order of growth then g(n)
  • Big O bounds from above.
  • if f₁(n) ∈ O(g₁(n)) and f₂(n) ∈ O(g₂(n)) then f₁(n) + f₂(n) ∈ O(max(g₁(n),g₂(n))
• a function f(n) ∈ Ω(g(n)) if there exist positive constants c and n₀ such that cg(n) ≤ f(n) for all n > n₀
• f(n) is at least as bad as g(n)., or bound from below
• a function f(n) ∈ Θ(g(n)) if there exist positive constants c₀, c₁ and n₀ such that c₀g(n) ≤ f(n) ≤ c₁g(n) for all n > n₀
• f(n) and g(n) are "the same"