Orders of Growth

• Or how bad are some functions?
• T(n) = some function in terms of n.
• Consider the following functions
  • log₂(n), or lg(n)
  • n
  • n lg(n)
  • n²
  • n³
  • n^k where k ∈ Z⁺ (CONSTANT INTEGER)
  • 2ⁿ
  • nⁿ
  • n!
• The book on page 46 gives us some examples of these values.
• note
  • log_a(n) = log_a(b) log_b(n)
  • or log_b(n) = log_a(n)/log_a(b)
  • Or log₂(n) = log₁₀(n)/log₁₀(2)
  • So the difference between log(n), ln(n), and lg(n) is just a constant.
• Read the paragraph on the middle of page 43.
• Circuit satisfiability problem
  • Given a circuit with n inputs, is there any input which will make the circuit produce a 1?
  • u = a ∧ ¬ a is not satisfiable
  • s = a ∨ ¬ a is satisfiable.
  • In general, the only way to determine if a circuit is satisfiable is to try all input values.
  • And there are 2ⁿ of those.
  • This is a genuinely hard problem as far as we know.
• The following table illustrates the difference in orders of growth.
  • Assume the input size doubles

    n	2n
    lg(n)	lg(2n) = lg(2)+lg(n) = 1+lg(n)
    n	2n (doubles)
    n lg(n)	2n lg(2n) = 2n(1+lg(n)) = 2n +2n lg(n)
    n2	(2n)2 = 4n2
    n3	(2n)3 = 8n3
    2n	22n = (2n)2
    nn	(2n)2n = 22n (nn) 2
    n!	(2n)!