Goals

• Define the sets O, Ω and Θ for a given function.
• Explain the meaning and use of these sets.
• Explain the strengths and weaknesses of this notation.
• Given a function, state the set to which it belongs.

Asymptotic Notations

• Section 2.2, READ IT
• O(g(n)) = { f(n) : there exists constants c>0 and n₀>0 such that f(n) ≤ c g(n) for all n≥ n₀}
• So a this defines a set of functions, which are in some way equal.
  • At least in my mind,
  • n≥n₀ means we "ignore startup costs".
  • f(n) ≤ c g(n) means we ignore multiple instructions.
  • IE, we pick one critical instruction and count it.
• Consider these two algorithms:
  • Find Max/Min/Average

  • FIND_MAX_MIN_AVERAGE(A)
    
      min ← ∞
      max ← -∞
      sum ← 0
      count ← 0
      for i ← 0 to A.size()-1 do
         sum ← sum + A[i]
         if A[i] > max then
              max ← A[i]
         if A[i] < min then
              min ← A[i]
      average ← sum / n
     return max, min, average
    
    	

  • Standard Questions
  • What is n, what is the critical instruction?
  • Performance?
    • T(n) = n+3
    • T(n) = 3n+3
  • T(n) ∈ O(n)
  • OR T(n) = O(n)
  • How does this compare to sequential search?
• Big-O has a quirk:
  • T(n) = 3n+2
  • Is T(n) ∈ O(n)
  • Is T(n) ∈ O(lg(n))
  • Is T(n) ∈ O(n²)
  • Is T(n) ∈ O(2ⁿ)
  • Is T(n) ∈ O(n!)
• Look at some additional functions:
  • T(n) = c, for some constant c?
  • T(n) = 3n⁵ + 4n² + n - 15?
• We strive to get the "Tightest" upper bound possible.
• There are two other sets we talk about
• Ω(g(n)) = { f(n) : there exists constants c>0 and n₀>0 such that 0 ≤ c g(n) ≤ f(n) for all n≥ n₀}
• Θ(g(n)) = { f(n) : there exists constants c₀ > 0 and c₁ >0 and n₀>0 such that c₀ g(n) ≤ f(n) ≤ c₁g(n) for all n≥ n₀}
• We are generally most concerned about O(g(n)) but sometimes others.
• There are three others, little o, θ and ω but we will not concern ourselves with these.
• A property of O(g(n))
  • If t₁(n) ∈ O(g₁(n)) and t₂(n) ∈ O(g₂(n)), then t₁(n)+t₂(n) ∈ O(max(g₁(n),g₂(n))

  • t1(n) ∈ O(g1(n)) means
         there exists c1 and n1 such that
              0 < t1(n) ≤ c1 g1(n) for n > n1
    t2(n) ∈ O(g2(n)) means
         there exists c2 and n2 such that
              0 < t2(n) ≤ c2 g2(n) for n > n2
    let n3 = max(n1, n2)
    let c3 = max(c1, c2)
    So
      t1(n) + t2(n) ≤ c1 g1(n) +  c2 g2(n)
                    ≤ c3 g1(n) +  c3 g2(n)
                    ≤ c3( g1(n) + g2(n))
                    ≤ c3*2* max( g1(n) , g2(n))
    
    	  

• But what does this represent?

• COMPLEX_ALGORITHM
  
   execute some algorithm with time T1(n)
   execute some algorithm with time T2(n)
  
      

• Some other properties of O(g(n))
  • for any two functions t(n) and g(n)

            f(n)  | 0 then t(n) has a smaller order of growth than g(n)
    limn→ ∞ ---- = | c then t(n) and g(n) have the same order of growth
            g(n)  | &inifn; then t(n) has a larger order of growth than g(n)