What is O value for naive random selection from finite set?

Question

This question on getting random values from a finite set got me thinking...

It\'s fairly common for people to want to retrieve X unique values from a set of Y values.

Answer 1

If you already have chosen i values then the probability that you pick a new one from a set of y values is

(y-i)/y.

Hence the expected number of trials to get (i+1)-th element is

y/(y-i).

Thus the expected number of trials to choose x unique element is the sum

 y/y + y/(y-1) + ... + y/(y-x+1)

This can be expressed using harmonic numbers as

y (H_y - H_y-x).

From the wikipedia page you get the approximation

H_x = ln(x) + gamma + O(1/x)

Hence the number of necessary trials to pick x unique elements from a set of y elements is

y (ln(y) - ln(y-x)) + O(y/(y-x)).

If you need then you can get a more precise approximation by using a more precise approximation for H_x. In particular, when x is small it is possible to improve the result a lot.