Algorithm analysis - find maximum value

Function MAX( n, x[] )
  best := x1
  for j := 2 to n do
     if xj > best then best := xj
  return best

How many times is the assignment   "best := x_j"   executed?

• The minimum number of times is 0.
• The maximum number of times is n-1.
• The average number of times is ... ?

Let A_n = the number of new maxima encountered.
Consider the case of n distinct numbers in which all n! possible relative orderings are equally likely.

• A₁ = 0

• A_j = A_j-1 + (either 1 or 0),     add 1 if x_j = max{ x₁... x_j },     add 0 otherwise

• Prob[ x_j is the max ] = 1/j

• therefore, avg of A_j = avg of A_j-1 + 1/j

• avg of A_n  =  ∑_{j=2 to n} 1/j  =  H_n - 1

H_n  ≤  1 + ∫₁ⁿ dx / x  =  1 + ln n

H_n  ≥  ∫₀ⁿ dx / (x+1)  =  ln (n+1)

H_n  =  ln n + γ + &Theta(1/n),     where γ = Euler's constant ≈ 0.577