Other Asymptotic Algorithm Classes

Objectives

1. Discuss the classes Θ, Ω, o, θ, ω

Notes

• Last time we had: O(g(n)) = {f(n): there exist positive constants c and n₀ such that 0 ≤ f(n) ≤ c×g(n) for all n ≥ n₀}
  • After a point (n₀) f(n) is less than c×g(n).
  • We call this bounded above.
• I don't think we discussed this, but
  • is n+3 ∈ O(n²)?
  • Ie, is 3+n always less than n² beyond some point?
• So O(f(n)) is an asymptotically upper bound, and nothing more.
• Remember insertion sort:

  • SELECTION-SORT(A)
       Input: A an array of items with the comparison operator
       Output: the array A ordered by the comparison operator
    
     current ← 0
     while current < A.size
        small ← A[current]
        for next ← current +1 to A.size
           if A[next] < A[small]
               small ← next
        if small ≠ current
           swap(A[small], A[current])
        current ← current + 1
     

  • I hope we decided that this is O(n²) where n is the number of items in the array.
  • Is this also O(n⁴)?
• Sometimes we want to decide how good something can be.
• T(n) is Ω(g(n)) = {f(n): there exists positive constants c and n₀ such that 0 ≤ c× g(n) ≤ f(n) for all n > n₀}
  • This is bounded from below.
  • It says that the function (algorithm) can't get better than this.
  • Selection Sort is Ω(n²).
  • It is also Ω(n), but that is less interesting.
• For selection sort we can say something stronger
• T(n) is Θ(g(n)) = {f(n): there exists positive constants c₁, c₂ and n₀ such that 0 ≤ c₁× g(n) ≤ f(n) ≤ c₂× g(n) for all n > n₀}
  • Note this is the same as f(n) is O(g(n)) and f(n) is Ω(g(n)).
  • This is an asymptotic tight bound.
• We said that Gail-Shapley is O(n²) assuming all operations are constant
  • Do we have a lower bound (Ω) on G-S
  • Can we have a tight bound?
• O is the most commonly used notation, Ω and Θ less frequently.
• There are two other sets less commonly used
  • o(g(n)) = {f(n): for any positive constants c there exists a constant n₀ > 0 such that 0 ≤ f(n) < c×g(n) for all n ≥ n₀}
  • ω(g(n)) = {f(n): for any positive constants c > 0 there exists n₀ > 0 such that 0 ≤ c× g(n) < f(n) for all n > n₀}
  • Let f(n) = 4n²+3
  • f(n) is O(n²)
  • f(n) is o(n3), but not o(n²))
    • Let c = 1, 4n²+3 > c × n² no matter how large n is.
  • We say that n³ dominates n²
  • Your book doesn't mention o and ω
  • I had to go to CLRS and they dedicate TWO pages to the topic.
  • I am not sure that I have ever used these outside of this section.